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User's guide to the magnetic unmixing software packet

1. H 11 Resample the coercivity distribution in the field range 23 0 53 6 The grey band around the curve in the following plot corresponds to the estimated error 2 sd C11 COERCIVITY DISTRIBUTION ON A LOG SCALE n 604 p Ho 4 E S j od o j H s 404 S 4 0 A 4 43 oO 4 N a all p O 204 Gq Bie 2d oO ge 4 eE eh r TIITII TT 25 30 35 40 45 50 magnetic field aa w11 ESTIMATED ERROR 104 8 10 4 E q J v J 1074 25e 30 35 40 45 50 magnetic field Q 11 Resample the coercivity distribution in the field range 26 0 54 0 The grey band around the curve in the following plot corresponds to the estimated error 2 sd COERCIVITY DISTRIBUTION ON A LOG SCALE n 604 p am 4 S 5 J 5 4 K H 404 g A od x4 yp oO 4 N od 4 yp 204 S D 4 fe 4 rar rar T T T TT 30 35 40 45 50 magnetic field ESTIMATED ERROR los o 104 4 J H J o 4 1014 T_T TTT T T_T a e E R S L E T L 30 35 40 45 50 magnetic field CODICA 5 0 reference manual 48 oy 12 a COERCIVITY DISTRIBUTION ON A LIN SCALE er J 5 5004 Pa al o 4004 pi 4 c 300 O a a 5 2004 N J s 4 1004 E 4 E J g J 20 40 60 80 100 magnetic field Range of fields spanned by the coercivity distribution 0 52 6 H 12 Resample the coercivity distribution in the field range 20 0 53 0 The grey band around the curve in the following plo
2. 0 5 0 0 5 1 1 5 2 0 0002 0 0004 0 0006 0 0008 0 001 0 0012 0 0014 0 0002 0 0004 0 0006 0 0008 0 001 0 0012 0 0014 1 Fig 3 Component analysis on a sample of urban atmospheric dust collected in Z rich Switzerland The component analysis is performed with logarithmic Gaussian functions Results of the component analysis are shown in a c and e The gray pair of line indicates the confidence limits of the measured coercivity distribution The blue line is the modelled coercivity distribution expressed as the sum of the logarithmic Gaussian distributions red green violet and light blue Confidence limits are plotted around each function Below each plot the difference between measured and modelled curve is drawn in blue the gray pair of curves indicates the amplitude of the measurement errors The mean quadratic residuals of each model are plotted in b d and f as a function of the amplitude of the logarithmic Gaussian function labeled with the same color The solutions plotted in a c and e represents the absolute minima of b d and f GECA 2 1 reference manual essa essa O O 7 1 4 1 2 1 0 0 8 0 6 0 4 0 2 1 4 1 2 1 0 0 8 0 6 0 4 0 2 4 0 0002 0 0004 0 0006 0 0008 0 001 0 0012 0 0014 d IN VY i Fig 4 Component analysis of the same sample as in Fig 3 The component analysis is performed with
3. The behavior of some coercivity distributions at H 0 might be discontinuous This is for example the case of an exponential distribution In such cases the coercivity distribution calcula ted by CODICA might show a large peak at H 0 This peak is an artifact produced by a field scale change You can remove this peak by choosing to calculate the coercivity distribution on a range of fields that does not include H 0 This problem is avoided if the magnetization curve is defined by enough points at H 0 At this points you have the same options as in 11 for choosing the range of fields and the number of points of the coercivity distribution and the coercivity distribution is plotted on the desired range CODICA 5 0 reference manual 40 H 13 Coercivity distribution on a power scale It is sometimes useful to plot the coercivity distribution on a field scale that is intermediate between a logarithmic and a linear scale The scale transformation defined by H H hereon called the power scale transformation is a suitable scale for this purpose The exponent p is a positive number Special cases are p 0 whose limit is the logarithmic scale and p 1 which is the linear scale A power scale will show different features of the coercivity distribution that may be hidden in the logarithmic and the linear scale representation see Fig 10 It is recommended to use a power scale to check if a coercivity distribution is unim
4. 10 4 N 104 N J v a 4 Cc Cc a J D 1 E ci E 574 5 4 0 prt m I r 0 50 100 150 200 250 300 1 10 100 AF field mT magnetic field mT 105 g h 082 ana Z J z J Ss O oen 5 06 5 4 N J No o T J 067 E J E a D 0 4 4 D J E J 044 02 J J 0 24 0 0 a a a 74 an ri T L T4117 444417 T 0 50 100 150 200 250 300 1 10 100 AF field mT magnetic field mT CODICA 5 0 reference manual 7 What is CODICA CODICA COercivity Distribution CAlculator is a program for the detailed analysis of magneti zation curves and the calculation of coercivity distributions CODICA takes advantage of the universal properties of magnetization curves and uses advanced mathematical tools to model magnetization curves and measurements errors without using restrictive assumptions The calcu lated coercivity distributions are free of distortions and typical artifacts of common filtering methods These are intrinsically inadequate for the analysis of asymptotic functions such as mag netization curves The confidence limits provided with the results are particularly useful for evaluating the significance of multicomponent mixing models based on coercivity distributions Fig 2 shows a comparison between the performance of CODICA and that of a commercial soft ware built in filtering method Fig 2 Performance comparison between CODICA and a commercial software built in filtering method a An artifi cial magnetization curve
5. Hj HEY 1 Jj 31 An interpolation dlx of the error free residuals at the points 2 2 is performed using a least squares collocation based on the covariance matrices C44 of d H C of d a and d H and C of e H defined as Caly CalHe HF kj Hl n Cocky C654 9 I len where 1 if 7 k and 6 0 else The interpolated residuals d d x d a are given by J 1 C C Ce 1d 33 with d d H d H The estimated errors of d are given by the diagonal elements of the error matrix E C3 C3 Caa C Ch 34 6 11 Filtering the residuals The autocorrelation model of the residuals calculated by CODICA is usually accurate However it is very difficult to find a model that applies to all possible situations and the human brain has a superior capacity in distinguishing a regular pattern from noise Therefore the user has the op portunity to correct the model proposed by CODICA to make the interpolated residuals dlx closer to the actual residuals or on the opposite to make d 2 smoother To do so the user is asked to enter his estimate 7 of the correlation length r If the user has the feeling that d a should be closer to the actual residuals he enters 7 lt 1 If the user has the feeling that d a still contain some noise he enters 7 gt r Let Ar r d be the difference between the actual residuals rand their interpolation d I
6. To run GECA 2 1 you need Mathematica 5 0 and later versions installed on a Windows OS At least 128 MB RAM and a 1 GHz CPU are recommended Install GECA 2 1 To install GECA 2 1 copy the source code file MAG_MIX_1 GECA Install Geca m and the file MAG_MIX_1 GECA Install components txt into the following directory C Wolfram Research Mathematica 5 0 AddOns StandardPackages Utilities whereby c depends on the installation of Mathematica on your computer WARNING The packages Codica m anf Geca m are incompatible Load only one of them in a Mathematica session If you need both programs load and use first CODICA When you are finished with CODICA close the Mathematica session Then open the same or another Mathematica notebook and load GECA This problem will be fixed with the next version of GECA GECA 2 1 reference manual 1 Introduction The program GECA GEneralized Coercivity Analyzer is part of the package MAG MIX GECA performs a component analysis based on special generalized functions which can fit the coercivity distributions of natural and artificial magnetic components particularly well It also performs a Pear son s y goodness of fit test to evaluate the number of functions required to model a coercivity di stribution Finally GECA performs an error estimation and calculate the cofidence limits for each model parameter Read carefully this manual to learn about GECA and take full advant
7. lp 2 Skewness s o3 o3 s 6sengq 1 q 1 1 856k 4 gt 0 left skewed s TE q lt 0 right skewed of f Kee m de Kurtosis s 03 03 3 kx2 p p gt 2 box shaped k e p lt 2 tip shaped of f aE m de Table 1 Relation between statistical distribution properties and distribution parameters for a SGG function Except for the median the relations are not analytical approximations are given in the case of small deviations from a Gaussian distribution GECA 2 1 reference manual Examples of SGG functions with different parameters are given in Fig 1 The parame ters of the coercivity distribution of some calculated and measured coercivity distribu tions are plotted in Fig 2 Fig 1 Examples of SGG distributions a Some particular cases with u 0 o 1 and q 1 are plotted The skewness of all curves is zero Furthermore p 1 for a Laplace distribution p 2 for a Gauss distribution and p defines a box di stribution b Some left skewed SGG distri butions with u 0 and o 0 5484 are plotted The SGG distribution with q 0 4951 is an excellent approximation of the logarithmic plot of a negative exponential di stribution 0 6 F 4 100 100 a osp PS a 141 110 62 n 7 0 1 5 5 o4 te n D 24 P 2 0 3 02 2 s lt 0 1 0 2 2 o Magnetite Halgedahl 1998 5 o Magnetite Bailey and Dunlop 1983 g e Detrital component lake sediments m Bi
8. mi 1 4 s1 0 45 ql 0 6 pl 2 al 0 7 ml 1 957 sl 0 235 ql 0 663 pl 2 Optimizing the distribution parameters Please wait Perform a component analysis 3 al 0 00075 m1 1 4 s1 0 45 q1 0 6 p1 2 a2 0 0007 Optimized distribution parameters al 0 9217 ml 1 509 sl 0 4882 ql 0 6235 pi 2 023 al 0 5348 mi 1 957 sl 0 235 ql 0 663 pi 2 Residuals modelled measured and measurement errors in of the maximum value of the coercivity distribution Model and data are significantly different Dashed lines delimitate the interval considered for the component analysis 2 8 0 63 1 6 ChiSquare points Confidence limits Refine your model GECA 2 1 reference manual 18 Enter initial parameters 4 Initial distribution parameters T al 0 9217 1 mil 1 509 s1 0 4882 ql 0 6235 pl 2 023 1 al 0 5348 i mi 1 957 si 0 235 ql 0 663 pi 2 0z 0 0 Optimizing the distribution parameters Please wait Perform a component analysis 4 al 0 0009217 m1 1 509 s51 0 4882 q1 0 6235 p1 2 023 a2 0 0005348 m2 1 957 s52 0 235 q2 0 663 p2 2 FindMinimum fmlim The minimum could not be bracketed in 50 iterations al 0 0006243 m1 1 311 s1 0 4483 q1 0 4878 p1 2 171 a2 0 0008625 m2 1 963 s82 0 2331 q2 0 7765 p2 2 107 Optimized distribution parameters al 0 7699 ml 1 412 s1 0 4699 gl 0 5537
9. tion that helps you in optimizing your measurements Does CODICA contain bugs Maybe The program code is almost 4000 lines long and its performance is not guaranteed under all circumstances However all possible user controlled options of CODICA have been tested on many measurements If a result is obtained it is guaranteed to be correct i e free of coarse errors such as units or scale errors CODICA 5 0 reference manual 11 2 Before using CODICA some suggestions to optimize your measurements Why remanence measurements The measurement of remanence curves is intrinsically more difficult than in field measurements such as hysteresis loops There are essentially two reasons that explain this fact The first reason resides essentially in the technical difficulty of switching on and off a magnetic field produced by an electromagnet in an exactly reproducible manner In fact many electromagnet control sys tems produce a slightly positive overshoot before reaching the programmed field intensity or a slightly negative overshoot when the field is turned off On the other hand pulse magnetizer be come slightly resonant at high fields so that small negative fields may be produced after the main discharge peak The second reason is that some samples are themselves a source of error especially when the remanent magnetization to be measured is much smaller than the saturation magnetization of the sample In such cases the smallest amount of ma
10. 41 with respectto H My H P H S d 2 Pa H da P A Pia H At 43 P H U H The estimated error 6M H of M H is 6 My H P H S d x P H H 6d a P H 44 The coercivity distribution 4g y H and its error 6 4 g H ona logy scale are given by 45 Mog g log H H My H n10 CODICA 5 0 reference manual The coercivity distribution H and its error 6 4 H ona H scale are given by Mey HP pH My Hf Ma H p H 8M H 60 CODICA 5 0 reference manual Appendix A 61 Tables of appropriate coercivity distribution units in the SI and the cgs system M is the mag netization unit magnetic moment per unit volume or unit mass H is the magnetic field unit Magnetization linear scale Magnetization generic unit M Magnetization SI unit A m Magnetization SI unit Am kg Magnetization cgs unit emu g Manetic field H generic unit H MH Manetic field H dimensionsless m kg avoid this SI unit A m eq 1 eq 1 combination Magnetic induction poH dimensionsless m kg avoid this SI unit T eq 1 X uo eq 1 X Ho combination SI unit mT eq 1 x 10 u9 eq 1 x 10 Lig Magnetic field H avoid this avoid this emu Oe g 7 gcs unit Oe combination combination ARM susceptibility susceptibility susceptibility susceptibility susceptibility linear scale generic unit M HY SI
11. The magnetic field is scaled according to the transformation H P H H 9 H q 3 where p gt 0 is the scaling exponent and H 5 if H 0 4 1 min H H H 5 if H gt 0 is a damping coefficient that avoids divergence at x 0 The inverse transformation is given by H P H H A 0 q 5 6 3 Model function for the magnetization curve The magnetization curve M H is modeled with the sum of two empirical functions S H and U H The function 17 S H a 5 tanh a arsinh H rs a a5 a 6 accounts for the sigmoidal shape of M H The symbol 1 means that CODICA uses the ma chine number gt 1 that is closest to 1 to avoid numerical instability problems The parameters a and a control the two horizontal asymptotes CODICA 5 0 reference manual 52 lim S H a 14sgna 2 4 a 7 H 00 The position of the median field is controlled by the parameter a and the width of S H by a The sign of a decides whether S H is a monotonically increasing or decreasing curve The parameter a gt 0 influences the tails of S H if a 0 the horizontal asymptotes are approached very slowly on the other hand if a gt oo the asymptotes are approached like a tanh function The parameter a gt 0 controls the asymmetry of S H S H S H is symmetric about H a if ag 1 A left skewed S H is obtained with 0 lt a lt 1 anda right
12. You can choose the entire ran ge of measurements or s to choose the range of fields spanned by the distribution fields span ned by the measurements To enter a different range type the range limits Example 1 300 Maximum range 1 00 100 vw type a In some cases the coercivity distribution is significantly different from zero on a smaller range You can choose this range by typing s You can also specify a different range by typing its limits If you enter the field range of the coercivity distribution manually keep in mind that you can only chose a range that is covered by measurements CODICA does not allow extrapolations outside this range A second prompt window a ask you TREE area Tenet to enter the number of points where i se Enter the number of sampling points in the chosen field CODICA should calculate the coercivity range a distribution These points will be equal Atleast 28 points are recommanded ly spaced within the range you chose previously CODICA suggests you to use a number of points that is at least as large as the number of measurement 300 However you are free to enter any number of points you desire After entering the field range and the number of points CODICA will recalculate and plot the coercivity distribution on the field range you chose The thickness of the curve small errors or the grey region around it large errors indicate
13. any part of the Mathematica notebook the notebook may hide the prompt window If you need to scroll through the notebook while the prompt window is open shift the latter to one side so that you can click on it when you are ready to enter your answer While running CODICA generates different types of outputs in the Mathematica notebook 1 progress messages black 2 information messages blue 3 warning messages red 4 criti cal warning messages bold red and 5 graphics CODICA will stop after critical warning mes sages because any further data process is impossible Critical warning messages indicate that your data cannot be analyzed probably because they do not represent a correctly measured magnetization curve At the end of a CODICA session useful results are saved into different files that can serve as input files for other programs of the MAG MIX package You can open these files with any other pro gram capable of reading numerical data such as Excel Gnuplot Kaleidagraph or Origin Exit CODICA You can exit CODICA before the program is finished by 1 typing abort in any prompt window produced by CODICA or 2 quitting the Mathematica Kernel These options are useful if CODICA is producing unexpected error messages or if you want to check the data file and restart again CODICA 5 0 reference manual 30 LH 1 Enter the path of the data file When you start CODICA a prompt win Wee LCT x dow asks you
14. sk1 0 8043 kul 1 149 1 a2 0 6556 MDF2 1 967 DP2 0 2165 Os sk2 0 2647 ku2 0 1122 QO 0 QO Calculating the confidence limits of each component Please wait Normalized components with confidence limits Distribution parameters 1 MDF1 1 428 DP1 0 4255 sk1 0 8043 kul 1 149 MDF2 1 967 DP2 0 2165 1 sk2 0 2647 ku2 0 1122 0 GECA 2 1 reference manual 22 Preparing data to an export format Please wait Save results to a log file Printing results to components dat Save end members Saving the coercivity distributions to WDKarm comp Column Column Column END 1 2 4 magnetic field component 1 Column 3 error of component 1 component 2 Column 5 error of component 2 GECA 2 1 reference manual 23 Loading CODICA and running GECA To run GECA open a new Mathematica notebook by clicking on the Mathematica program icon Type lt lt Utilities Geca on the input prompt In j and press the keys Shift Enter to load CODICA On the next input prompt type Geca and press the keys Shift Enter to start GECA From now on the program asks you to enter specific commands step by step In the following all GECA commands are explained in order of appearance Back to the program Enter the name of the data file The prompt window on the right x asks you to enter the name on the ile which contains che coercivity aa istribution data Type t
15. such measurements Each component is described by a 5 parameter model function and 15 parameters are needed for the three components The number of measurements corresponds barely to the number of parameters and the model is exactly determined Small measurement errors are sufficient to drive such a model into a completely wrong solution In fact a model with three components and a redundance factor of 5 five measurements for each model parameter requires 75 measurement points If you find this number excessive think about a Day plot the five parameters required two for the magnetization ratio two for the coercivity ratio and one for the paramagnetic correction are obtained from hysteresis loops and backfield curves consisting of a large number of points The reason for the difference between the common sense accuracy of a hysteresis loop and that of a remanent magnetization curve resides in the extra time required in the latter case to switch on and off the de magnetizing field Magnetization curves consisting of 80 measurements require few minutes to 5 hours running time depending on the de mag netization device and the degree of automation An interesting and extremely fast system for the acquisition of remanent magnetization curves has been developed by the paleomagnetic labora CODICA 5 0 reference manual 12 tory of the Kazan State University using a self made equipment informally named Pashameter after its constructor Ja
16. yn will ask you if you want to remove those points from further data processing gt The human brain has a unmatched visualizing and pattern recognition ability Therefore you will recognize outliers much easier and reliably than CODICA can If you think that the points marked by CODICA as outliers are normal measurements you can choose to keep these points for further data processing gt If you accept to remove the outliers CODICA will repeat all the previous data processing steps without these points It is strongly recommended to remove outliers for better final results Out lyers may bias many estimated parameters used by CODICA during the data processing gt After removing outliers it is possible that a second run through the data reveals other minor outliers You can repeat the process of removing outliers until you are satisfied with the result The case where the first and or the last point are considered as outliers needs special atten tion Often these points are not really outliers they rather reflect some particular processes oc curring at the beginning or at the end of the magnetization curve The initial part of a demag netization curve is possibly affected by viscosity effects On the other hand the last step of a de magnetization curve is probably performed after treating the sample with the same magnetic field as the field used to impart the magnetization Thermal activation effects m
17. 8 parameters and is flexible enough to account for a variety of magnetization curves including difficult cases given by mixtures of low and high coercivity minerals such as magnetite and hematite The best fit curve is plotted together with the measurements This fit is not the final model of the magnetization curve and you should not worry too much about discrepancies with the measured data A prompt window asks you whehter he ocaiKernel Input the scale used is acceptable or not The Pee scale suggested by CODICA is almost al l ways acceptable If you are not an expe rienced user you should accept this scale by typing the letter y for yes More experienced users can choose a different scale by typing n n CODICA 5 0 reference manual 33 gt The choice of a field scale is not critical for the quality final results so you should not worry too much about it CODICA automatically avoids critical scale transformations and accepting the suggested scale is seldomly a bad choice If you decide to enter the scaling exponent manually you should read section 6 3 and feel confident with the mathematics behind Avoid exponents gt 1 if possible A manual choice of the scaling exponent can be taken into consideration in the following cases 1 the scaling exponent suggested by CODICA is gt 1 2 the plotted best fit curve does not fit the measurements equally well at small and at large fields 3 yo
18. an automatic variation of the contribution of one function can have several local mi component nima for a given finite mixture mo Enter the number of this component or type n Help del One among them is a global minimum as well and is usually considered as the acceptable solu tion A solution which corresponds to a global minimum of the merit function is attained if the initial model is chosen to be close enough to the acceptable solution Since this solution is usually unknown in advance a sufficient number of ini tial models has to be tested in order x to ensure that at least one will con verge to a global minimum of the Accept the original component analysis red point p n merit function If you try 5 initial values for each parameter of a mo del with two SGG functions you should perform 5 10 optimi zations GECA performs a selected search for a global minimum of the merit function starting with the re sult of the last component analysis as initial model You can select one end member function whose am plitude a will be increased and de creased in steps of 1 100 of the total sample magnetization starting form the solution of the last component analysis After each step the new solution is taken as an initial model for the next component analysis As a result the merit function is plotted for all possible amplitudes of the selected SGG function In the program example the last solution of the componen
19. and France 1984 first realized that the coercivity distribution of synthetic assemblages of magnetite and hematite particles can be modeled with logarithmic Gaussian functions They implicitly introduced the logarithmic scale as a natural scale for coercivity distributions and the Gaussian function as the self similar PDF Detailed measurements of sediments and rocks demonstrated that the coercivity distributions of natural magnetic components are slightly skewed on a logarithmic scale Egli 2004 Their natural scale has been tentatively approximated by H H with p gt 0 5 for SD magnetite and p 0 2 for other natural magnetites Since the observed natural scales are pe culiar to each component and are probably not universal the logarithmic scale can be still con sidered as a fairly good scale for plotting coercivity distributions Some cases that confirm the fundamental hypotheses of component analysis are shown in Fig 9 The representation of a coercivity distribution on different field scale is a very effective tool for checking whether the distribution is strictly unimodal or it is the result of the overlapping of two or more magnetic components Subtle details that are completely hidden in one field scale be come evident if a different scale is chosen Egli 2003 discussed extensively the use of the power scale transformation defined by H H 12 with the scaling exponent p gt 0 For p 0 the power scale transformation c
20. coercivity distribution of a set of non interacting SD particles with a logarithmic Gaussian distribution of microcoercivities and grain sizes redrawn from R Egli Physics and Chemistry of the Earth 29 851 867 2004 The dashed solid curves indicate the calculated coercivity distribution without with thermal norm contribution activations a Effect of thermal activations on the switching field of the particles which is reduced by an amount called the fluctuation field arrows This effect also depends on how long the de magnetizing field is applied b Effect of thermal activations on the remanent T magnetization of the grains The magnetization of grains with a smaller microcoercivity decay faster with time arrows This effect depends also on the time lag between the acquisition of a magnetization and its measurement norm contribution 1 10 100 AF peak field mT 6 Shielding Ideally the sample and the magnetometer should be shielded against external varia tions of the magnetic field and other electromagnetic interferences 7 Mechanical unlocking Unconsolidated powder samples may contain magnetic grains that can rotate under the influence of a strong magnetic field Alternating fields are especially effective in shaking lose magnetic grains Powder samples should be firmly pressed in sample boxes Empty space in the box should be filled with a nonmagnetic material such as folded thin plastic foi
21. each plot the difference between model and measurements is plotted blue line to gether with the measurement errors pair of gray lines In a a model with one SGG function is evaluated The differences between model and measurements are too large and the model is rejected In c the a model with four SGG functions is rejected for the opposite reason the model fits the data unrealistically well for the given measurement errors A model with two SGG functions is represented in b In this case they statistics is compatible with the expected value within at a 95 confidence level and the model is accepted ChiSquare points 8 8 a tr Confidence limits 0 27 2 1 Pearson test not passed ChiSquare points 1 5 b Confidence limits 0 59 1 7 Pearson test passed ChiSquare points 0 37 c Confidence limits 0 52 1 6 Pearson test not passed GECA 2 1 reference manual 13 A model with a small number of parameters produces a residuals curve with few large oscillations The more parameters are included in the model the more oscillations characterize the residuals and the confidence limits of the x estimator become closer to the expected value Consequently mo dels with a too large number of parameters are rejected An example of Pearson s x test is shown in Figure 7 with the example of a sample of urban atmospheric particulate matter In Fig 7a the coercivty distribution is fitted
22. high fields i e the sample saturates at high fields 3 The slope of the curve at zero field is always finite i e the sample has a stable magnetization in a zero field CODICA proceeds essentially on three steps that are shown in Fig 3 with a simple example First a set of scale transformations is applied to the field and the magnetization Fig 3b After field and magnetization scales have been changed the magnetization curve becomes close to straight line and is said to be linearized Fig 3c A so called residual curve is obtained after removing the linear trend of the scaled curve by subtraction of a polynomial Fig 3d The residual curve has the characteristics of a stochastic signal because it oscillates more or less randomly around a mean value of zero The wiggles arise from small asymmetries of the original magnetization cur ve as well as from the measurement errors Measurement errors are easy to recognize in the residual curve since they are highly amplified This gives you the possibility to optimize and correct your experiments for optimal results The residual curve is then fitted with a method cal led least squares collocation which is a particularly effective model for stochastic non periodic signals Fig 3d The interpolated residual curve supposed to be free of measurement errors is transformed back into a magnetization curve Fig 3e and its first derivative called coercivity distribution Fig 3f The
23. least squares collocation method provides also a way to estimate the error associated to the operations described above and thus provides confidence limits for the results it produces Is CODICA difficult to use In the previous versions of CODICA the steps described above had to be controlled by the user which was asked to enter various rather cryptic parameters As a result the user was required to have some mathematical skills and a lot of endurance This new version of CODICA is fully auto mated You have to decide only the degree of smoothing used to interpolate the measurements and this operation is rather intuitive The complete analysis of a magnetization curve takes no more than a couple of minutes in addition to the computation time in case of curves with a large number of measurements To take full advantage of CODICA read carefully this manual which contains a practical guide through each step of the program with a real example The most interested users can read the technical reference which contains detailed information about the operations performed by CODICA and their theoretical background However this information is not necessary for a standard use of CODICA CODICA 5 0 reference manual 9 a ORIGINAL MEASUREMENTS b ASYMMETRIC BEST FIT red 154 15 4 c c 2 2 e g B 104 Z 10 4 c J c 4 J Da J oO oO 1 E J 54 54 a ey ea 0 50 100 150 200 250 300
24. options as in H 11 for choosing the range of fields and the number of points of the coercivity distribution to be plotted on a power scale gt You have the possibility to try different values of p Answer n to the prompt window asking you to choose a different range of fields Thereafter CODICA re asks you if you want to plot the coercivity distribution on a power scale Anwer y and enter a new power exponent in the next prompt window 14 Saving the results After plotting the coercivity distribution on al field scales and ranges you wanted CODICA is ready to save the results A prompt window will ask you to enter a name for the files where you want to save the data This name will be used to Site he a a tae Hes te yan write diterent miles whichare distingu Appropriate extensions will be added to each saved file shed by their extension If you do not want to save the data type exit to exit CODICA All files produced by CODICA are saved in the same directory where the original data file is located ot Local Kernel Input tape CODICA will save the filtered magnetization curve in a file with extension cum for cumulative distribution which is the integral of the coercivity distribution The coercivity distribution on a logarithmic scale is saved in a file with extension slog s for spectra log for logarithm If you chose to calculate the coercivity distributio
25. scale The thickness of the curve is proportional to the estimated error which is also plotted below Notice that the maximum error of the coercivity distribution is less than 4 The smoothness of the coercivity distribution suggests a single magnetic com ponent for this sample On the logarithmic scale however a small contribution from a high coercivity component is suggested by the inflection of the curve above 300 mT CODICA 5 0 reference manual 13 More effective than increasing the number of measurement points is the minimization of the experimental errors These errors arise from 1 magnetometer errors 2 rounding to an insuf ficient number of digits by the acquisition software 3 errors in the application of the de mag netizing field 4 imprecise sample positioning both in the field and in the magnetometer 5 time effects on viscous samples 6 insufficient shielding of external magnetic perturbations and 7 mechanical unblocking of magnetic grains in strong magnetic fields In the following we shall briefly discuss some of these experimental errors and how to reduce or avoid them 1 Magnetometer errors They are intrinsic to the magnetometer used 2 Digital rounding Most acquisition softwares give three digits results which are sufficient for all traditional purposes but are likely to produce nasty rounding effects in the saturation region of detailed magnetization curves Fig 5 SQUID magnetometers have a much higher
26. skewed S H is characterized by ag gt 1 The effect of the different parameters on S H is illustrated in Fig 11 a b aq l r r r r 0 1 5 0 5 2 0 2 4 Fig 11 Examples of the function S z a a a The parameters a az and a control the center the ampli tude and the left asymptote of S respectively b The slope of the central part depends on a c The way how the asymptotes are approached is controlled by a d The parameter a controls the symmetry of S The symmetric case is given by ag 1 ag gt 1 givesaright skewed S while a left skewed S is obtained with 0 lt a lt 1 The function U H a sgn a u a tn eosh a H a in2 as 8 ag accounts for an eventual highly unsaturated high coercivity component U H has the following two asymptotes CODICA 5 0 reference manual 53 lim U H 0 he 9 lim U H 2a sgn q H as 00 The parameter a is the field at which the two asymptotes of U H intersect a is proportional to the slope of U H gt gt a gt and ag controls the interval over which U H changes from a constant value of 0 to a line with slope 2a sgn a The effect of the different parameters on U is illustrated in Fig 12 a py Ab i 2 Fig 12 Examples of the function U x a a a7 dg a The parameter a controls the slope of the right asymptote b The interval within wh
27. the data stored in the file If satu ration is not reached within this range the calculated value is an underestimation of the total magnetization You can use the estimation of the total magnetization as a reference when you enter the initial distribution parameters Back to the program example GECA 2 1 reference manual 24 Set the fitting range GECA estimates a field range where the values of the coercivity xl distribution are significant As a Enter the range to consider for fitting significance limit a maximum min max TET i or type a to select the entire significan part of the relative error of 50 has been cho coercivity distribution Help sen for the values of the coercivity distribution You can enter a diffe rent range with the prompt window displayed on the right If the coer civity distribution was calculated from a demagnetization curve it is recommended to discard the data near the right end of the field range because they could be af fected by truncation effects Data outside the range you entered are displayed but are not considered for further calculations Back to the program example Enter initial distribution parameters You are asked to enter initial values for the parameters of the finite mixture model that will be used for the component analysis GECA uses a set of one to four SGG functions to fit the measured coer civity distribution Each SGG function is characterized by following five para
28. the measured coercivity distribution and more parameters should be included in the model On the other hand if py is too small the model fits the measured data unrealistically well and random effects produced by the measurement errors are included in some parameters which are not significant The model is accepted if y x belongs to the range of values given by the confidence limits The complex dependence of the merit function on the model parameters is illustrated in c for the case of a model with a fixed number of end member functions which approximately fit the coercivity distribution of all magnetic components These end member func tions produce a small misfit between model and data even is the measurement errors are not considered dashed curve Nevertheless there is only one stable solution of the component analysis green point which corresponds to an absolute minimum of If the measurement errors are taken into account the shape of becomes rather complex with numerous local minima 0 Some of these local minima represent possible solutions which fits the measu rements as good as the absolute minimum even if they do not model the coercivity distribution of the real components The absolute minimum red point represents a solution Oyy which is still close to the realitiy With larger measurement errors this could not be the case and a realistic solution may be given by a local minimu
29. the model is not able to account for the measurements other parameters should be added to reduce the misfit If the misfit is much smaller than the measurement error the model chosen for the component analysis is able to fit the measurements very well but it is not significant some of the model parameters do not have any phy sical meaning In this case you should decrease the number of model parameters by reducing the number of end members or by keeping some parameters fixed If the misfit has the same amplitude as the estimated measurement error the model may be adequate Nevertheless more than one solu tion witch satisfy this condition may exist An adequate parameter to test the significance of a component analysis is the y statistics GECA gives an estimation of 1 where l represents the degrees of freedom of the model For a correct model ne lt y lt Kiai where Xa Mica are the confidence limits at a given confidence level usually 95 GECA calculates the confidence limits with a 95 confidence level and displays them together with the estimation of x l If x gt Xia the model is significantly different from the measured data and GECA suggests you to refine it by adding more parameters If y lt Ta not all model parameters are significant and a warning message is displayed In this case you should reduce the number of parameters If ka lt x lt Ya GECA suggests to accept the model In the program example the component a
30. to other applications When you launch Mathematica a so called notebook is opened This notebook is initially empty A Mathematica notebook is simply a command shell to enter commands and see results You can save and edit the notebook and export graphics and other output results Every Mathematica notebook contains input lines numbered by In where is the input number and output lines numbered by Out where is the number of the corresponding input Everything you type in a notebook is interpreted as a command line and an input number is automatically assigned when you ask to run the command line You can do so by pressing the keys Shift Enter at the same time Example 1 Enter alone is used to enter a new line When you evaluate the first command of a notebook the so called Mathematica kernel is launched The kernel is the core of Mathematica that performs all the calculations requested through the input lines Example 1 A simple run with Mathematica Documentation Dateiordner 9 11 2003 4 25 PM 1 La u nch Ma thematica exe Configuration Dateiordner 9 11 2003 4 26 PM SystemFiles Dateiordner 9 11 2003 4 26 PM Addons Dateiordner 9 11 2003 4 26 PM Mathematica exe 112 KB Anwendung 6 12 2003 1 37 AM math exe 64 KB Anwendung 6 12 2003 2 29 AM MathKernel exe 108 KB Anwendung 6 12 2003 2 33 AM E binarytest 1KB Datei 4 14 2004 9 39 AM amp Mathematica 5 0 Untitled 1 STUDENT VERSION oog 2 An empty noteboo
31. very general and efficient method used for the analysis of signals and potentials in geodesy e The user can choose the range of fields in which the coercivity distribution is calculated This is a useful option if the results are used for component analysis References Egli R 2004a Characterization of individual rock magnetic components by analysis of remanence curves 1 Unmixing natural sediments Studia Geophysica et Geodaetica 48 391 446 back to text Egli R 2004b Characterization of individual rock magnetic components by analysis of remanence curves 2 fundamental properties of coercivity distributions Physics and Chemistry of the Earth 29 851 867 back to text Egli R 2004c Characterization of individual rock magnetic components by analysis of remanence curves 3 bacterial magnetite and natural processes in lakes Physics and Chemistry of the Earth 29 869 884 back to text Kruiver P P M J Dekkers and D Heslop 2001 Quantification of magnetic coercivity components by the analysis of acquisition curves of isothermal remanent magnetization Earth and Planetary Science Letters 189 269 276 back to text Moritz H Least Squares Collocation 1978 Reviews of Geophysics and Space Physics 16 421 430 back to text Robertson D J and D E France 1994 Discrimination of remanence carrying minerals in mixtures using isothermal remanent magnetisation acquisition curves Physic of the Earth and Pla
32. 0 1 2 3 4 5 6 magnetic field mT scaled magnetic field c 4 LINEARIZED DATA d 0 02 FILTERED RESIDUAL CURVE red 2 iS 4 3 3 mo a E i 0 04 5 0 T T T T T T T T 2 T T T T T T 0 2 4 6 0 1 2 3 4 5 6 scaled magnetic field scaled magnetic field e FITTED MEASUREMENTS red f COERCIVITY DISTRIBUTION ON A LOG SCALE 15 c il c 2 2 107 Fi Cc 7 Cc Dn 4 oD E 54 Ge 0 T TTT TTT T T1177 T rror T T 0 50 100 150 200 250 300 1 10 100 magnetic field mT magnetic field mT Fig 3 The working principle of CODICA on a simple example sample kindly provided by Christoph Geiss a Original AF demagnetization curve showing a characteristic MD shape b The field axis is rescaled in order to get a sigmoidal shaped curve To do so the scale is expanded at small fields c The magnetization is now rescaled in order to linearize the magnetization curve To obtain this result CODICA expands the magnetization scale near the beginning and the end of the magnetization curve The red line is the linear best fit to the data d The linear trend of the rescaled curve red line in c is subtracted to obtain the so called residuals As it can be clearly seen the measurement errors are quite evident in this plot The red curve is a best fit of the residuals that CODICA obtains from an autocorrelation model e A model for the error free magnetization curve red is obtained from the fitted residuals red curve in d by inv
33. 10 Example of the use of different field scales to check whether a coercivity distri bution is unimodal or is the result of two overlapped magnetic components The sample was a filter used to collect particu late matter from the atmosphere of a rural area in Switzerland Spassov et al 2004 Unlike other samples collected in polluted areas this sample was expected to contain only one magnetic component carried by the natural dust The coercivity distribution of this sample has been calculated with CODICA on a a linear scale b a logarith mic scale and c a field scale defined by H H Notice that on this last scale the coercivity distribution is not perfectly unimodal suggesting that the sample might contain a small contribution from far located anthropogenic pollution sources CODICA 5 0 reference manual 24 References Egli R 2003 Analysis of the field dependence of remanent magnetization curves Journal of Geophysical Research Solid Earth 108 B2 doi 2081 back to text Egli R 2004 Characterization of individual rock magnetic components by analysis of remanence curves 2 fundamental properties of coercivity distributions Physics and Chemistry of the Earth 29 851 867 back to text Robertson D J and D E France 1994 Discrimination of remanence carrying minerals in mixtures using isothermal remanent magnetization acquisition curves Physics of the Earth and Planetary Interiors 82 223 234 bac
34. A 5 0 can be installed exactly like the previous version Owners of a previous version may just replace the old source code file Codica m with the new one In the following a detailed list of the improvements and changes made in CODICA 5 0 is given e Acquisition curves are analyzed as such and not transformed into demagnetization curves e The error estimation does no longer require the evaluation of all measurement errors sources by the user A long experience on measuring magnetization curves demonstrated that the sources of measurement errors are extremely complex and difficult to predict The errors are now estimated empirically during the analysis of the magnetization curve e Various errors sources such as the lack of measuring points at critical regions of the magneti zation curve and digital truncation effects are recognized by CODICA A warning message is produced and the negative effects of the identified error sources are minimized as far as possible e All rescaling procedures applied by CODICA are now fully automated The user has the pos sibility of modifying the magnetic field scaling however a manual correction is almost never CODICA 5 0 reference manual 2 required CODICA also analyzes the distribution of the measurement points and suggests improvements for better results e The Butterworth low pass filter has been replaced with a least squares collocation method ba sed on Moritz 1978 Least squares collocation is a
35. SGG functions The same notation as in Fig 3 is used for the plots a Component analysis with one SGG function The modeled coercivity distributions is significantly different from the measured distribution 0 Mean quadratic residuals of a model with two SGG functions plotted as a function of the amplitude of one function Different local minima which correspond to stable solu tions of the component analysis are labeled with numbers The corresponding solutions are plotted in c d e and f The solution plotted in c corresponds to the global minimum of b and the resulting components are compatible with the coercivity distributions of natural dust red and combustion products of motor vehicles green The solutions corresponding to the other local mi nima of b are not realistic GECA 2 1 reference manual 8 The fundamental questions related to component analysis are e How many components are needed to fit a given coercivity distribution e Are multiple solutions possible If yes which solution is correct The answer to these questions is not simple In the example of Fig 4 the number of components and the identification of the correct solution among multipe solutions is evident However this is not al ways possible especially if good measurements are not available or if the coercivity distributions of individual components are too widely overlapped In this case some additional information is needed to put appropriate c
36. User s guide to the gt a MAG MIX 2005 by Ramon Egli magnetic unmixing software packet Release 1 April 2005 Includes the programs CODICA and GECA for the analysis of magnetization curves and coercivity distributions Runs on MS Windows with Mathematica 5 0 and later versions Preface The software package MAG MIX provides computer programs for the analysis of magnetization curves and coercivity distributions This first release includes the programs CODICA and GECA CODICA is a program that calculates the coercivity distribution of a magnetization curve and estimates the measurement errors GECA is a program for modeling a coercivity distribution as a linear combination of spe cial model functions that are supposed to represent the coercivity distribution of specific groups of magnetic particles called components The analysis of magnetization curves is a relatively recent and fast developing me thod used in environmental and rock magnetic studies to characterize materials that are a mixture of different magnetic minerals The success of unmixing such materials depends strongly on our knowledge about the fundamentals of mag netization processes and coercivity distributions This knowledge is evolving rapid ly and might provide more efficient and easy to use unmixing methods in the future The MAG MIX manuals contain detailed instructions for using the programs and provide basic knowledge about the theory of magneti
37. age from the different pos sibilities offered by the program to perform a component analysis and verify its significance This manual contains a theoretical part which gives you the background to understand the basic ideas of GECA and a practical part which guides you through each step of the program You can practice with the examples delivered together with this program GECA is designed to work optimally on coercivity distributions calculated with CODICA and stored in files with extention slog Click on the following topics to see the contents of this manual Theoretical background coercivity distributions e Finite mixture models e logarithmic Gaussian functions e Skewed Generalized Gaussian functions SGG e distribution parameters WwWNY ds NY an Some aspects of component analysis Performing and testing a component analysis e merit function e mean squared residuals e Chi square estimator e local and global minima of the merit function m CO CO CO e Pearson s Chi square goodness of fit test 1 A program example 14 Cautionary note 35 GECA 2 1 reference manual 2 Theoretical background coercivity distributions A group of magnetic grains with similar chemical and physical properties distributed around characteristic values is called a magnetic component Examples of magnetic components are pedo genic magnetite nanometric magnetite perticles with a wide grain size distribution and magneto s
38. ation maximization algorithm Geophys J Int 148 58 64 2002 Kruiver P P M J Dekkers and D Heslop Quantification of magnetic coercivity components by the analysis of acquisition curves of isothermal remanent magnetization Earth Planet Sci Lett 189 269 276 2001
39. by CODICA on the other gt The estimated error of the modeled magnetization curve is typically small and peaks at the median field where the slope of the magnetization curve is highest To understand this result keep in mind that measurement errors do not arise only from the magnetometer but also from applied field Errors in the applied field are more visible at places where the magnetization curve has a large slope 10 Calculate the coercivity distribution The coercivity distribution is calculated in a similar way as the magnetization curve starting from the first derivative of the modeled residual curve and inverting all th steps used to transform the magnetization curve into the residuals see section 6 13 for mathematical details You have several options for calculating the corcivity distribution on different scales and for choosing the appropriate range of fields covered by the coercivity distribution CODICA 5 0 reference manual 38 H 11 Coercivity distribution on a log scale Because of the particular importance of the logarithmic field scale CODICA will first display the coercivity distribution on this scale over the entire range of fields spanned by the measurements A prompt window will ask you to cho ose a range of fields over which CODI CA should recalculate the coercivity di ee ete en T stribution on a large number of points Type a to choose the entire range covered by the om Local Kernel Input
40. c moments spanning over more 0 4 normalized magnetization 0 2 than 6 orders of magnitude In the avera ged curve random errors are reduced by 50 T00 150 5300 250 a factor 25 with respect to a single mea AF peak field mT surement while systematic errors are un affacted The averaged curve shows dis 0 05 g continuities occurring at specific AF peak fields small inset shows the most evident discontinuity at 40 mT This problem in dicates some systematic errors in the applied field b Coercivity distribution calculated by finite differences of the cur o n 0 4 0 3 ve in a Discontinuities in a shows up as distinct peaks in the coercivity distribu tion See Appendix B for an empirical cor rection of these errors 0 24 normalized magnetization 0 14 50 100 150 200 250 AF peak field mT 4 Positioning errors The magnetic field produced by electromagnets is homogeneous only in a small region If the dimensions of that region are comparable to those of the sample a correct positioning becomes critical The same apply to the response function of the magnetometer Keep in mind that an error of 0 1 in both the applied field and the measurement is an upper limit for component analysis applications If the sample is placed by hand at each de magneti zation step errors arising from its misorientation are generally not negligible unless the sample fits firmly in a fi
41. cale a prompt window asks you for a new scaling exponent until you accept a solution gt It is strongly recommended to either accept the scaling suggested by CODICA or try several different scaling exponents You can compare the results by looking the mean standard deviation of the fitted curve reported in the title of the plot CODICA 5 0 reference manual 34 L 6 Calculate the residuals The functions used to fit the magnetization curve serves as a model to rescale the measurements in such a way that the magnetization curves is transformed into a nearly straight line This opera tion is called linearization of the magnetization curve The linear trend is then subtracted from the linearized magnetization curve to obtain so called residuals see section 6 6 for the mathematical details of this operation The residuals are just a kind of difference between the measurements and the model used by CODICA to fit them plotted on a suitable scale A typical residual curve contains several oscillations around a mean value of zero These oscillations are the sum of 1 very detailed features of the magnetization curve such as those arising from the overlapping of different magnetic components and 2 measurement errors Measurement errors are highly amplified in the residual curve and can be recognized clearly The scope of transforming the magnetization curve into the residual curve is that data processing on the latter is much more effi
42. caling exponent 1 00 Searching for an asymmetric best fit hyperbolic model Estimated contribution of a high coercivity component HYPERBOLIC MODEL red Mean sd 0 137 D 1e0 Ww N magnetization Pakiet ia e a eip e A H oO T T 0 20 40 60 80 100 scaled magnetic field Search a scale for the magnetization Please wait Iteration 1 Iteration 2 Iteration 3 Estimate th measurement Please wait rrors Calculate the residuals Subtract the residuals trend with a 2nd order polynomial Estimate the autocorrelation function of the residuals WARNING The residuals are almost random Estimated lower limit of the autocorrelation distance 17 6 Fit the residuals with an autocorrelation model WARNING CODICA detected som outlyers Please wait 2086 13 Please wait 2 586 13 Please wait CODICA 5 0 reference manual 44 M6 RESIDUAL CURVE estimated error in grey outlyers in red n w 5 d o 4 15 L 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 20 40 60 80 100 scaled magnetic field LL 7 Outlyers will be removed and the cleaned measurements will be analyzed again 7 The repeated steps are not shown here 8 RESIDUAL CURVE estimated error in grey n Ba oO 5 d n oO E 0 50 100 150 200 scaled magnetic field m8 Filter the residual curve using least squares collocation with correlation length 12 7 Estimate t
43. cient The residual curve is handled as a stochastic signal which is characterized by a so called autocorrelation distance The autocorrelation distance is the typical range of fields over which the error free residuals are oscillating around a mean value of zero CODICA uses a least squares col location method to model the error free residual curve and make a first estimate of the measu rement errors see sections 6 7 to 6 10 for the mathematical details The error free residual curve is characterized by the typical range of fields spanned by individual wiggles of the curve The results obtained as described above are plotted in a graphics with 1 the residuals dots 2 the least squares collocation model black line and 3 a first estimate of the measurement errors gray area around the points gt The appearance of the estimated measurement errors in the plot depends strongly on your data If you measured a sample whose magnetic properties are dominated by a single magnetic component the magnetization curve fits well with the model function used by CODICA In such case a large contribution to the residuals comes from the measurement errors Accordingly the estimated measurement errors are larger than the amplitude of the residual curve You should not be worried about this result it does not mean that the quality of the measurements was low On the other hand some mixtures of different magnetic components fit less well with th
44. d member set q 1 and aj fixed If you exactly know the para meters of one magnetic component you can model this component by an end member function with p 2 as initial values for the cor responding SGG function and keep fixed values of u o q and p Then only the magnetic contribution of this component given by a will be optimized these parameters fixed by entering It is recommended to start with a small number of end members and a small number of parameters the next promt window For exam Egamples q2 p2 or Help q2 p2 in the following prompt window You can choose every and to use independent information about the number of magnetic components and their properties You can then progressively increase the complexity of your model Keep in mind that the ple if you want to use a loga Last entered values will be taken for the fixed parameters combination of parameters to keep complexity of a model increases exponentially with the number of parameters to optimize If you GECA 2 1 reference manual 26 want to perform a component analysis with three SGG functions you have to deal with a solution space in 15 dimensions You will not have the possibility to perform a systematic solution search is such a space if you try 5 initial values for each parameter you should perform 5 3 x 101 optimizations You would probably find several stable solutions but only one among them is correct and has a physical mea
45. dd itivity This assumption does not hold in case of magnetic interactions between the magnetic grains of different components However magnetic interactions between different components are not likely to occur in natural samples since each component is expected to have a different origin and to hold different places within a nonmagnetic matrix On the other hand magnetic interactions within the same component are possible but they do not affect the linear additivity law Coercivity distributions of single magnetic components are described by probability density func tions PDF The shape of a PDF is controlled by a set of distribution centers with related dispersion parameters o with n N Special cases are given when n 1 p4 is the median o the mean deviation n 2 u is the mean the standard deviation and n oo H is the mid range and o the half range The dispersion parameter DP corresponds to o on a logarith mic field scale The symmetry of a PDF is described by the coefficient of skewness s where GECA 2 1 reference manual 3 s o Gee Symmetric distributions are characterized by s 0 and up 4 The curvature of a PDF is described by the coefficient of excess kurtosis k where k ot ih os 3 The Gaussian PDF is characterized by k 0 The description of non Gaussian PDF involves the use of functions with more than two independent parameters It is of great advantage if such functions maintain t
46. e magnetic particles In some cases such interpretation of natural samples can be misleading or inconclusive A less constrained approach to magnetic mineralogy models is based on the analysis of magnetization curves which are decomposed into a set of elementary contributions Each contribution is called a Magnetic component and characterizes a specific set of magnetic grains with a unimodal distri bution of physical and chemical properties Magnetic components are related to specific biogeo chemical signatures rather than representing traditional categories such as SD magnetite This unconventional approach can be regarded as a kind of principal component analysis PCA that gives a direct link to the interpretation of natural processes on a multidisciplinary level Since magnetic components rarely occur alone in natural samples unmixing techniques and rock magnetic models are interdependent Unmixing problems dealing with unknown components are strongly nonlinear and have usually multiple solutions Therefore an initial guess of the model parameters is required This guess relies on additional information about the geological and geochemical history of the sample Valuable information for rock magnetic and environmental studies can be obtained directly from the coercivity distribution of the sample which provides a richness of details hidden in the measurement curve Fig 1 Fig 1 next page Some applications of CODICA to rock magnetic and e
47. e model function used by CODICA The contribution of the measurement errors to the residual curve is small and the data are apparently cleaner gt The error estimation produced by CODICA at this point is not definitive and you might find it incorrect You should not worry about since the definitive error estimation is performed in a later stage of the data processing This error estimation is intended to give you an idea about the significance of the residuals As a general role of thumb the reliability of the error estimation is proportional to the number of measurements Since the modeled residual curve depends on the error estimate it might also be incorrect at this stage of the data processing 7 Remove outliers The magnetization curve might contain some odd measurements produced for example by inter ferences with other devices in the laboratory or by spikes in the electrical power supply Often these odd measurements are barely visible in a standard plot of the magnetization curve but become very evident when the residuals are plotted These odd measurements are recognized by CODICA 5 0 reference manual 35 CODICA as outliers when they fall apart the overall trend of the residuals by more than three times the estimated measurement error Outliers are marked as red points in the f X Local Kernel Input x plot of the residuals If some outliers x have been detected a prompt window Remove the detected outlyers
48. e normalized by their value at 10 mT GECA 2 1 reference manual 10 e O b Fig 6 Dependence of the merit function 8 on the parameters of the model chosen for fitting a coercivity distribution In a a noise free coercivity distribution with n 3 magnetic components is fitted with an adequate model with m end member functions The functions are assumed to re produce exactly the coercivity di stribution of each component If m lt n some components cannot Slightly inadequate model be considered into the model and With measurement noise e 0 gt 0 on the other hand Without measurement noise e 0 0 for a given combination Oun Of parameters when m gt n The number of compo nents can be easily guessed The situation becomes more complex in b where measurement errors are taken into account In this case there is always a misfit be O Bie 0 MIN min tween model and measurements and y decreases monotonically as the number of end members taken into account by the model is increased In this case the number of components is guessed with the help of a Pearson s x test According to this test e O is compared with the expected value of dashed line If Oy is compatible with the expected value within given confidence limits dotted lines the model is accepted If x is too large the modeled coercivity distribution is significantly dif ferent from
49. erting the mathematical functions used to transform the original measurement a into the residuals d f A coercivity distribution is calculated as the analytical derivative of e The thickness of the line corresponds to the estimated confidence limits of the coercivity distribution CODICA calculates the coercivity distribution on a logarithmic field scale as in f as well as on a linear scale CODICA 5 0 reference manual 10 Why is a compiled version of CODICA not available To run CODICA you need Mathematica 5 0 or a later version to be installed on a Windows system Of course a compiled version of CODICA would be more attractive However CODICA uses highly sophisticated mathematical routines that are embedded in Mathematica and are not easy to include in a compiled program Does CODICA make miracles Some users may be surprised by the poor performance of CODICA on some typical paleo magnetic quality magnetization curves which can be obtained with standard experiments However you should always consider that CODICA as any other software does not add a bit of information to your original measurements bad measurements will give poorly defined coerci vity distributions affected by large errors However CODICA finds the best fit to your data and helps to identify the reason of poor results You can repeat the measurements with a better sample and with optimized magnetization demagnetization steps This manual contains a sec
50. etween the grains If magnetostatic interactions are negligible the coercivity distribution of the grain assemblage is given by M H f Hyg pH H AH 4 For uniaxial SD grains as well as for MD grains Hg H u H Hg Stoner and Wohlfarth 1948 Xu and Dunlop 1995 It is reasonable to assume that this relationship has a general validity In that case 4 becomes a simple convolution between Hx and u Hx H ona logarith mic scale Mh f N F Ing Alh hg dhg 5 MH is a PDF function that depends on the distribution of all physical e g volume shape defects and chemical e g oxidation state parameters that control Hx It is typically a broad distribution The only relevant exception in nature is represented by magnetosomes whose size shape and composition is strictly controlled by bacteria 4 depends strongly on the domain state In SD par ticles i is a very narrow function with a peak at h log 0 5H MD particles are characte rized by a much broader u with a typical exponential shape that depends on the grain size and dislocation density Xu and Dunlop 1995 Both and Jyp where the index MD indicates MD particles depend on the statistical distri bution of physical and chemical processes that result form stochastic processes For seek of sim plicity we discuss the case of Consider an initial collection of identical grains all characteri zed by the same H The magnetization curve of s
51. f 1 lt ry CODICA uses a least squares collocation to obtain an interpolation Ar of Ar Ar Ca C3 Cae Ar 35 where the covariance matrices C and C4 are calculated using equations 30 32 with 7 instead of 7 The corrected interpolation d is then given by d d Azr 36 On the other hand if 7 gt 7 CODICA uses a least squares collocation to obtain a smooth version d of d 13 5 d C C C d Ar 37 CODICA 5 0 reference manual 59 with Ar as in equation 35 The covariance matrices C and C are calculated using equation 30 32 with 7 instead of r The first derivative d of d is needed to calculate the coercivity distribution and is obtained from equations 33 35 37 by replacing Cy with its derivative Chg given by The error estimate of d is given by equation 34 and that of d by the diagonal elements of the error matrix E Cha ChalCaa Cee Ci 39 with C3 le On Cale 2 k 7 1 N 40 oo 6 12 Filtered magnetization curve The filtered magnetization curve M H is calculated from d by inverting the steps described in sections 6 2 6 5 6 6 6 9 Hence M H S d H P H H U H Ht P n H PH The estimated error 6M H of M H is M H S d x P H H 6d z 42 where 6d x is the estimated error of d x 6 13 Coercivity distributions The coercivity distribution H on a linear field scale is the derivative of equation
52. gned to each PDF Strong differences exist between the results obtained with a linear combination of Gaussian distributions on the one hand and a linear combination of SGG distributions on the other Since finite mixture mo dels with non Gaussian coercivity distributions have not been reported in the literature it is not pos sible to decide from a priori information which kind of PDF should be used as a basis for a finite mixture model From the mathematical point of view all PDFs are equivalent since the goodness of fit which can be reached with a particular model depends only upon the total number of parameters assumed regardless of how they are assigned to individual components Generally the use of few PDFs with more distribution parameters instead of a large number of distributions with fewer distri bution parameters leads to results of the fitting model which are more stable against measurement errors The stable behavior of a fitting with SGG distributions can be explained by the fact that small deviations from an ideal coercivity distribution which arise from measurement errors are taken into account by variations in skewness and kurtosis rather than by variations in the contributions of the single components Obviously the values obtained for skewness and kurtosis may not be significant at all A similar stability can be obtained with Gaussian functions if some of them are grouped as if they were one component However it is
53. gnetization induced by random variations of the ambient field is sufficient to produce a noticeable effect in the measu rements So why use remanent magnetization to characterize a sample or to perform a compo nent analysis Component analysis can be performed on any kind of magnetization curves inclu ding hysteresis loops von Dobeneck 1996 Carter Stiglitz et al 2001 The problem with hysteresis loops is that they cannot be easily unmixed unless the properties of the individual components are known a priori Paradoxically the most suitable curves for component analysis are those obtained from weak field magnetizations such as ARM and TRM even if these magnetizations are among the most difficult to measure The advantage of using weak field magnetizations relies on their sensitivity to parameters such as grain size and oxidation state Egli 2004 Measuring remanent magnetization curves Errors in remanent magnetization measurements are mostly of technical nature and can be re duced by improving the measuring technique The standard way of measuring remanent magne tization curves has been established in the early story of paleo and rock magnetism and under went little changes despite the evolution of experimental equipments A classic de magnetiza tion curve consists of 10 15 measurement points more or less equally distributed over the coer civity range of the sample Imagine to perform a component analysis with three components on
54. he data processing L 4 Search an optimal field scaling exponent As discussed in section 3 the magnetic field can be rescaled to obtain a symmetric coercivity distribution This applies to a magnetization curve as well As a first step in the data processing CODICA looks for a scale transformation that gives the most symmetric magnetization curve The scale transformation used by CODICA is a power transformation H H q where q isa damping term and p is the scaling exponent see section 6 2 for more details Usually sym CODICA 5 0 reference manual 32 metric magnetization curves are obtained with 0 lt p lt 0 5 Accordingly CODICA explores systematically exponents p lt 1 This operation can take some time depending on the number of data If an optimal value is not obtained with p lt 1 CODICA will try with p gt 1 Since large values of p can produce numerical problems the search is limited to 0 01 lt p lt 2 7 This range is adequate for all types of magnetization curves However it can seldomly occur that the search for an optimal value of p fails In this case CODICA will assume p 1 The result of this optimization is an exponent P ym for best symmetry of the magnetization curve see section 6 4 The performance of CODICA is influenced by the distribution of the field values The ideal distri bution of field values produces equally spaced points when the magnetization curve is scaled for best symmetry Typicall
55. he general properties of a Gauss PDF the n th derivative should exist over and o lt for all values of n N Furthermore the Gaussian PDF should be a particular case of such functions A good candidate is the generalized Gaussian distribution GG known also as the general error distribution The Gaussian PDF is a spe cial case of GG distributions Other special cases are the Laplace distribution and the box distribution The GG distribution is symmetric s 0 In GECA a particular set of skewed genera lized Gaussian distributions called SGG is used to model single components A SGG function is given by 1 Jae q er QHP o TA 1 p e e 1 2 SGG z p 0 q p exp et 4 In 2 p 3 with z x u o x logH and 0 lt q lt 1 The GG distribution is a special case of 3 for q 1 and the Gauss distribution is a special case of 3 with q 1 and p 2 The relation between the distribution parameters u o q p and some statistical properties is given in Table 1 Distribution Definition Relation with the distribution Comments properties parameters Median Tip Tp ZH a is also called MDF or Tp f f z dz 2 j 3 MAF Mean O9 ly S H ae 0 856 generally not used in the by lH f f x ada literature bed forg lp 2 Standard Too 2 2 is also called DP o o 1 0 856k 1 s 3 2 deviation G f f x mY dx Is Oy 00 forg
56. he measurement errors CODICA 5 0 reference manual 45 LL 8 FILTERED RESIDUAL CURVE with cl 12 7 estimated error in grey 2 1 T 5 2 9 1 1 1 0 50 100 150 200 scaled magnetic field 8 Filter the residual curve using least squares collocation with correlation length 20 0 Estimate the measurement errors FILTERED RESIDUAL CURVE with cl 20 0 estimated error in grey residuals f 1 f 0 50 100 150 200 scaled magnetic field CODICA 5 0 reference manual B B JED D Xe T pam B 46 Calculate the fitted demagnetization curve MEASUREMENTS points WITH BEST FIT red 54 o d ob 4 H 44 S 4 5 4 o 4 o J cd 3 c J O A D 4 244 N 4 aa D hl oO 4 4 T 1 e 4 0 I T T T I T T T I T T T T T T T T T T T T T 0 20 40 60 80 100 magnetic field ESTIMATED BEST FIT ERROR 107 3 4 fe wh a 107 3 1034 Ty T T T T T T T T T T T T T T T T T T T T 0 20 40 60 80 100 magnetic field Calculate the coercivity distribution Please wait COERCIVITY DISTRIBUTION ON A LOG SCALE a 4 504 g J 5 J o 404 v J H TI a 305 o J ea w 207 N 4 a D J u 10 4 2 J D J o J amp 10t 10 magnetic field Range of fields spanned by the coercivity distribution 23 0 53 6 The coercivity distribution is significant over th ntire range of fields CODICA 5 0 reference manual 47
57. he path of the data file You can skip interme Example To read C users me data myfile dat you can Help diate directories if other files with the same name are not stored The data file should be an ASCII file with three columns of numbers se parated by spaces or tabulators The Jc jysers RAMON papers fiting SB32 arm dat file should not contain comment li nes or text in general The first co lumn is the scaled or unscaled field the second column is the value of the coercivity distribution for the corresponding field The third column is the relative error of the second column 0 1 means 10 er ror Output files of CODICA with extensions slin slog and spow are automatically accepted It is strongly recommended to run GECA only on CODICA output files with extension Slog Back to the program example enter C users myfile dat Plot the coercivity distribution The coercivity distribution is plotted together with the confidence limits given by the error estima tion stored in the file If the maximal measurement error is less than 5 of the peak value of the coercivity distribution only the confidence limit are plotted as a pair of gray lines With errors larger than 5 the coercivity distribution is plotted as a black line together with the confidence limits Within the plot an estimation of the total magnetization is given This estimation is obtained by integrating the coercivity distribution over the field range given by
58. hich is the identity function If the model function S H is close enough to M H the rescaled magnetization curve S M H is close to a straight line The residuals curve R H is thus defined as R H S M H H 14 with 1 dg S M sinh l artanh 1 2 Mans dy 15 ay as a3 CODICA calculates the smallest possible residuals by minimizing X wR Ay 16 k with respect to the parameters qa ag 6 7 A stochastic model of the residuals The residuals are the superposition of 1 a curve that represents the deviation of the model func tion S H from the measured curve and 2 a random signal e H that depends on the mea surement errors The properties of the residual curve are evaluated by calculating the autocova riance function Ca h m R H R H h dH 17 Since R H is a collection of discrete unevenly spaced points Cg h is calculate indirectly through the Fourier transform R of R R u RE AH Hh HY w RUA ei 18 o whereby Crh f IRF w Peos wh dw 19 CODICA 5 0 reference manual 55 A first simple estimate of e H is obtained by assuming that e H is an uncorrelated random signal In this case C h var e h where var e is the variance of e H and 6 h is the Dirac 6 function Under this assumption the variance of e H is given by var e C p 0 Cp Ah 20 whereby Ak is chosen to be the mean distance between the po
59. ich U changes its slope is controlled by ag 6 4 Searching the best scale for symmetry The symmetry of a magnetization curve M H is evaluated by fitting M H with the sum of the symmetric sigmoid S H and U H The scale H for which X wy M Hg So H U H 0 10 k with the parameters a ag chosen to obtain gt gt wy M Hg So Hg U HKP min 11 is defined as the most symmetric scale of M H CODICA evaluates 10 systematically for all power scales defined in 6 2 with 0 01 lt p lt 2 72 If CODICA does not find a solution of 10 within this range of p it just minimizes 11 with respect to a ag and to H 6 5 Model the magnetization curve After an appropriate scale H has been chosen CODICA models the magnetization curve with the two functions S H and U H by minimizing Sw M H Hi UEP 12 CODICA 5 0 reference manual 54 with respect to the parameters a ag The eventual contribution of a strongly unsaturated high coercivity component is subtracted from the magnetization curve to obtain the nearly saturated curve M H M H U H 13 6 6 Calculate the residuals The residuals are obtained by rescaling the magnetization axis In the ideal case the magnetiza tion curve M H is identical to the model function S H If the magnetization is rescaled using the inverse S of 5 the rescaled curve is given by S M H S S H H w
60. ight be responsi ble for a mismatch between the effectiveness of the magnetic field during acquisition and later demagnetization For example if a 200 mT field was used to impart an IRM your sample might be completely AF demagnetized already at 180 mT In this case the magnetization curve becomes suddenly flat near the last measurement point The first and the last point of a magnetization curve can therefore represent an anomalous trend or be real outliers 8 Filter the residuals At this point CODICA has calculated a model for the residuals and the measurement errors As already mentioned the human brain has a unmatched ability in visualizing and recognize pat terns The measurement errors are highly enhanced in the residual curve and your eyes are in stinctively able to see them By doing so you have the ability to tune the CODICA model of the residuals If CODICA underestimated the measurement errors you will find that the modeled residual curve follows too closely the measurement points and has an excessively irregular ap CODICA 5 0 reference manual 36 pearance On the other hand if CODICA overestimated the measurement errors the modeled residual curve is too smooth and important details of the residual curve are lost You can choose the degree of smooth amp g Local Kernel Input ness of the residuals model curve with a parameter called correlation length Enter the corre
61. in your choice CODICA propo ses a Starting value from its own estimate of the residual curve This is generally a good starting point gt There is a range for the possible values of 7 you can choose The lower limit is given by the maximum distance between two consecutive residuals the higher limit by the range of fields spanned by the measurements If you choose values close to the lower limit the filtered residuals CODICA 5 0 reference manual 37 will be very close to the measurements and will probably be affected by wiggles produced by the measurement errors On the other hand choosing values close to the upper limit gives you a straight line close to zero You might choose this latter option if you have the impression that the residuals are completely noisy gt It is recommended for new users to play with 7 by choosing different values in the entire per mitted range You will soon get a feeling of how least squares collocation filtering works It is also recommended to run CODICA to the end with different values of 7 to see the difference between the results obtained A dangerous temptation for many users is to choose too small values of 7 to keep all infor mation contained in the data You should remember that 1 magnetization curves are naturally smooth and do not change much over a few mT range except if you are measuring special mate rials such as thin films and 2 short range oscillations s
62. inimum 16 recommanded 64 Help civity distribution The standard de viation of the noise signal is chosen to be identical with the estimated measurement error for each value dole Bo ce Loe Be as f 64 civity distribution is fitted with the same set of end member functions used for the last component ana lysis whose solution is taken as the a initial model The result of the component analysis performed on the noisy coercivity distribution differs slightly from the result of the component analysis performed on the original coercivity distribution The process of adding and adequate noise component to the original data and fitting the resulting coercivity distribution is each distribution parameter These standard deviations are taken as an error estimation At the same time some statistical properties of the end member functions are calculated as well together with the related errors The error estimation performed by GECA is quite time consuming therefore you can choose the number of iterations to perform With 64 iterations an accuracy of 12 is expected for the error estimation The relative accuracy of the error estimation expressed in is given by 100 V7 where n is the number of iterations used The error estimation performed by GECA takes into account the effect of the measurement errors on a set of distribution parameters which is related to a particular local minimum of the merit function The effect of measurement error
63. intrinsic ac curacy which can be exploited by using a 5 digits reading Some acquisition softwares for the 2G cryogenic magnetometer may combine erroneously the flux count and the analog signal This error does not produce noticeable effects on standard measurements unless the magnetization of the sample corresponds to about one flux jump of the SQUID sensor Such software errors have to be removed if a component analysis is performed regardless of the sample s intensity 5 00000005 Fig 5 Example of digital rounding ef 9 es ee e fects on a magnetization curves mea 5 04 Y surement kindly provided by C Geiss 4 Digital rounding is particularly evident S 4 8 in the flat region of the curve small in 5 3 set Rounding to 3 digits can introduce 5 6 10 30 30 noticeable errors in the calculation of coercivity distributions CODICA can g 2 handle digital truncation to minimize errors however obtaining full digits 1 data is a preferable option T oO 0 r i r 2 9 0 0 0 9 _0 009 _0 _9 0 20 40 60 80 100 magnetic field mT 3 Errors in the applied field The precision of the field applied on the sample depends on the control unit of the magnet The field control of large magnets is critical Pulse magnetizers may become slightly underdamped at high fields and produce a small negative field after main discharge This results sometimes in IRM acquisition curves that decrease ant high fields The generation
64. ints in R H The first estimate of e H is then e H var e 6 8 Subtract a trend with polynomials The residual curve contains generally a trend that can be removed by subtracting the n 1 th order polynomial P H which minimizes So wR R H Pa 21 CODICA chooses n to be the number of significant local minima and maxima Hi H of the residual curve A local minimum or maximum H is considered significant if and only if RU RA gt max fec 22 k 1 lt j lt k 1 This means that the difference between two successive significant local extrema must be larger than the estimated error Fig 13 a b l I I l l T Ay H Hg Ay Hy Hg Fig 13 Example of a residual curve solid line with three local maxima minima The error is represented by the grey band around the residual curve In a the central local maximum is significant because the difference with the neighbor minima is larger than the error The opposite is true in b The trend free residuals r H RCA P H 23 CODICA 5 0 reference manual 56 are the superposition of 1 a more or less sinusoidal curve d H that represents the deviation of the model function S H from the measured curve and 2 a random signal e H that depends on the measurement errors r H d H e H 24 6 9 Rescale the residuals The noise free residual curve d H is supposedly more or
65. k appears File Edit Cell Format Input Kernel Find Window Help R 58 Untitled 1 STUDENT VERSION BAX 150 lt lt gt CODICA 5 0 reference manual 26 j l u n Mathematica 5 0 Untitled 1 STUDENT VERSION Joe 3 Type a command for example 1 2 File Edit Cell Format Input Kernel Find Window Help 5 Untitled 1 STUDENT VERSION og qJ A 1 2 E Mathematica 5 0 Untitled 1 STUDENT VERSION EEI 4 Press Shift Enter to evaluate the command Now 1 2 is Bea tel eS lect tl ecole Shr the first input In 1 and the result 3 the first output Out 1 E Untitled 1 STUDENT VERSION BAXI Inftj 1 2 lt Out 1 3 por Stare if Mathematica 5 0 te watnenatcaskere 5 After evaluating the first input the Mathematica Kernel is started automatically E Mathematica 5 0 test nb STUDENT VERSION BAX 6 You can save the notebook by clicking on the Menu File File Edit Cell Format Input Kernel Find Window Help and then Save By double clicking on the saved notebook A test nb STUDENT VERSION Joe you can launch Mathematica and load the notebook with all j stores inputs and outputs automatically 150 a lt Mathematica 5 0 test nb STUDENT VERSION Po File Edit Cell Format Input Kernel Find Window Help x 7 If you try to evaluate a wrong expressio
66. k to text Sato K B Bollobas W Fulton A Katok F Kirvan and P Sarnak eds 1999 L vy Processes and Infinitely Divisible Distributions Cambridge studies in advanced Mathematics 68 Cambridge University Press 479 pp back to text Stoner E C and E P Wohlfarth 1948 A mechanism of magnetic hysteresis in heterogeneous alloys Philosophical Transactions of the Royal Society of London Series A 240 599 602 back to text Xu S and D Dunlop 1995 Toward a better understanding of the Lowrie Fuller test Journal of Geophysical Research B Solid Earth 100 22533 22542 back to text CODICA 5 0 reference manual 25 4 Install and run CODICA 5 0 Requirements To run CODICA 5 0 you need Mathematica 5 0 and later versions installed on a Windows OS At least 128 MB RAM anda 1 GHz CPU are recommanded Install CODICA 5 0 To install CODICA 5 0 copy the source code file MAG MIX _1 CODICA Install Codica m into the following directory C Wolfram Research Mathematica 5 0 AddOns StandardPackages Utilities whereby c depends on the installation of Mathematica on your computer A short introduction to Mathematica all what you need to know about Mathematica is a software conceived to perform high demanding logic symbolic and numeric mathematic operations It integrates a numeric and symbolic computational engine graphics system programming language documentation system and advanced connectivity
67. l or calcium fluoride Badly sorted clay poor sediments such as loesses and glacial deposits are good candidates for mechanical unblocking effects In this case mix them with at least 40 of a non magnetic binding material such as calcium fluoride or wax CODICA 5 0 reference manual 16 References Carter Stiglitz B B Moskowitz and M Jackson 2001 Unmixing magnetic assemblages and the magnetic behavior of bimodal mixtures Journal of Geophysical Research 106 26397 26411 back to text Egli R 2004 Characterization of individual rock magnetic components by analysis of remanence curves 1 Unmixing natural sediments Studia Geophysica et Geodaetica 48 391 446 back to text Jasonov P G D K Nurgaliev et al 1998 A Modernized Coercivity Spectrometer Geologica Carpathica 49 224 225 back to text von Dobeneck T 1996 A systematic analysis of natural magnetic mineral assemblages based on modeling hysteresis loops with coercivity related hyperbolic basis functions Geophysical Journal International 124 675 694 back to text CODICA 5 0 reference manual 17 3 Basic theory of coercivity distributions all what you should know Coercivity distributions still represent an unusual way to analyze a magnetization curve Two major sources of confusion about coercivity distributions arise from the use of different field scales e g linear and logarithmic and from the related units The shape of a coercivity di
68. lation length of the residual curve Suggested values are around 12 7 that yeu ale asked to enter by a prompt Range of possible values 12 610 window EY Use the largest value that gives a fit within the error margins 610 gives a linear fit 20 The correlation length is used by the least squares collocation model as a parameter that indica tes the scale of the smallest details in the residual curve Filtering the residuals with a correlation length 7 means that features with an extension lt 7 are filtered out The larger is 7 the smoo ther will appear the filtered residual curve see section 6 11 for mathematical details After entering the correlation length CODICA will calculate the corresponding least squares col location model and a graphics with the new estimate of the residual curve and the measurement errors will be displayed A prompt window will ask if gt P j ge hae Bee ae om Local Kernel Input want to accept the displayed model Type my if you are satisfied with it or Accept the filtered residuals y n n if you want to enter a different value for the correlation length gt You can try as many models with different correlation lengths as you want until you are satisfied with the result gt If you should choose 7 so that the modeled residual curve follows the general trend shown by the residuals without being affected by the errors To help you
69. less sinusoidal Accordingly its Fourier spectrum d w peaks at a dominant frequency wg whereby 2r wy is the typical wavelength of the wiggles of d H CODICA applies an additional field scale transformation Hy P Hj Hi 9 Ht 9 q Hy 5 25 by choosing an exponent p that maximizes the peak of the Fourier transform r w of FH The new scale makes d H more similar to a sinusoidal function as possible Fig 14 sn i Ir 0 24 064 0 0 al 0 2 024 ag 20 40 60 80 d 0 5 4 0 0 0 20 40 60 80 Fig 14 Example of how CODICA rescales the residual curve to approach the sinusoidal curve a Original residual curve r H and b its power Fourier spectrum r H Notice how the wavelength of r H changes with H c A scale transformation H H makes the residual more similar to a sinusoidal curve d The corresponding Fourier spectrum shows a higher and more localized peak at the dominant frequency CODICA 5 0 reference manual 57 6 10 A least squares collocation model for the residuals Least squares collocation is used to interpolate a measured signal whose autocovariance function is known Hence the first step of least squares collocation consists in modeling the autocovariance function of r H d H e H Since d H is almost sinusoidal and e H is a stochastic signal CODICA models the corresponding autocovariance function
70. lusions about the occurrence of magnetic components in sediments Egli 2004a b c and gave raise to the need of a user friendlier and stable version of the program which is accessible also to non specialized users Several bugs have been already corrected in version 4 The use of CODICA for the analysis of different types of magnetic materials generated occasionally critical errors The main problems reported by users were e The incapability of reading a data file stored directly in a disk partition and not in a folder e g C myfile dat instead of C myfolder myfile dat e The instability of the fitting routines included in CODICA which led to fatal error in some cases for example with particular materials such as magnetic tapes e The need to enter manually special fitting parameters whose meaning is not intuitive and the need of special characters such as which are not available on all keyboards A profound revision of CODICA has been undertaken to make the program as most user friendly automated and stable as possible The result is a fully new version where the user is asked only to control the degree of smoothing of the final result Experienced users will appreciate the fully automated optimization of all the mathematical steps required to fit a magnetization curve and calculate the corresponding coercivity distribution New users will not need to train intensively with CODICA before getting useful results CODIC
71. ly the total magnetization of the sample Conver sely f H can be regarded as the coercivity distribution of a sample with unit magnetization Any PDF f x can be represented on different scales of x through a variate transformation y g x where y is the variate on the new scale In the case of coercivity distributions let H g H bea scaled magnetic field The coercivity distribution on the new scale is given by Og H OH Notice that cannot be plotted just by changing the scale of the horizontal axis because of the normalization property 2 Of special interest for the following discussion is the particular case given by h log H whereby 3 becomes M h 1n10 10 4 10 4 M H Mg H 3 In the following we designate with f h a PDF obtained from f H with h log H One could ask why a coercivity distribution should be calculated on different field scales The reason for a scale change has profound roots in the mechanism that generates a PDF in nature This mecha CODICA 5 0 reference manual 18 nism is so important for understanding coercivity distributions that the subject is discussed in the following The coercivity distribution of an assemblage of magnetic grains depends on 1 the statistical distribution H Hg of the microcoercivities Hg of the grains 2 the coercivity distribution u Hxg H of all grains with the same microcoercivity Hx and 3 the magnetostatic interactions b
72. m of lt 8 GECA 2 1 reference manual 11 If an adequate model is used to fit data affected by measurement errors Eyy gt 0 and Eyn 0 for m oo Fig 6b Two fundamental questions arise at this point 1 How many end member distributions should be considered for a component analysis 2 Is a particular solution close to the unknown real solution 0 These questions can be easily answered only if the model chosen for the component analysis is adequate and the measurement errors are sufficiently small The first condition can be approximati vely attained by using a set of SGG functions to model the coercivity distributions of the magnetic components SGG functions are able to reproduce all fundamental characteristics of the coercivity distribution of a single component median dispersion parameter skewness and kurtosis If the measurement errors are small enough the solution O y which corresponds to a global mini mum of 8 is close to the real solution Fig 6c In case of large measurement errors the real solution 6 may be close to one or more a local minima of 8 In this case additional independent information are needed to individuate the correct solution among all possible solutions in The problem of the number of end members to consider for a component analysis is evaluated with a Pearson s x goodness of fit test To perform this test the statistical distribution of the x estimator given in eq
73. ments on this lake f Coecivity distribution calculated from e and results of a component analysis colored areas Three components can be clearly distinguished the lo west coercivity component has been attributed to detrital magnetite the middle coercivity component to magneto somes that survived reductive dissolution and the high coercivity component to a high coercivity mineral such as hematite g AF demagnetization curves of ARM from samples of particulate matter collected from the atmosphere at three places in the city of Zurich Switzerland see S Spassov et al Geophysical Journal International 159 555 564 2004 h The coercivity distributions show the increasing contribution of a high coercivity component in more polluted areas GMA forest near Z rich WDK center of Z rich GUB highway tunnel CODICA 5 0 reference manual 6 a J b J 8 5 lt Tg lt 160 J ke 4 J T 804 4 Ye 4 c amp 5 E 4 2 5 604 DN J g 30 40 50 2 4 S J T J 5 7 414 N J D J 3 J q c 20 4 J D J J 7 E J 0 m r 0 4 20 40 60 80 100 120 0 20 40 60 80 100 AF field mT magnetic field mT 1 04 J c d J 254 J 20 S g Bo o Soag x asd gt 154 go 5 5 J y 2 104 E 4 m 5 IRM J ARM 0 0 4 qo d 4 SS 0 20 40 60 80 100 120 20 40 60 80 100 AF field mT magnetic field mT e f 3 154 154 J Z J G
74. meters amplitude a the area under the SGG function which is equivalent to the magnetization of a component whose coercivity distribution is represented by this function median this parameter corresponds to the median value of the function also called median destructive field MDF or median acquisition field MAF parameter for the standard deviation o this is the principal parameter which controls the standard deviation of the SGG function also called the dispersion parameter DP parameter for the skewness q this is the principal parameter which controls the skewness of a SGG function with 1 lt q lt 1 Positive values of q generate left skewed functions negative values of q generate right skewed functions Symmetrical functions are characterized by q 1 Generally real coercivity distributions are characterized by 0 5 lt q lt 1 If you do not have independent informations about the starting parameters set a value of q near 1 parameter for the kurtosis p this is the principal parameter which controls the kurtosis of a SGG function A logarithmic Gaussian distribution is characterized by p 2 More squared functions are generated with p gt 2 less squared distributions by p lt 2 Common values for real coerci vity distributions are given by 1 6 lt p lt 2 4 If you do not have independent information about the starting parameters set p 2 GECA 2 1 reference manual 25 Enter the parame
75. mponent analysis since it justifies the use of model functions for the coercivity distributions of each magnetic component The above discussion shows that there is a well defined relationship between the natural scale of a PDF and the stochastic process that generated it We recall the description of a stochastic pro cess given by 6 The general form of 6 deals with a variate X with PDF f whose change with time is given by X t dt X t s X Zdt 9 where Z is a variate described by a PDF y Equation 9 is called a stochastic process If s is independent of X the PDF of X t dt is given by the convolution of f and The reiteration CODICA 5 0 reference manual 21 of 9 generates an infinitely divisible PDF as t oo In the general case of s depending on X we shall consider the variate transformation Y g Z whereby Y t dt g X t s X Zdt Y t g X s X Zdt 10 A self similar PDF is obtained from 10 if g X s X is independent of X whence g x f dx s z a1 defines the natural scale of the stochastic process 9 Conversely a PDF that is self similar on a scale defined by g x might result from a stochastic process with s x 1 9 z The hypothesis that coercivity distributions can be modeled with self similar PDF on an appro priate field scale is difficult to confirm for natural magnetic components since rocks and sedi ments are almost always mixtures of at least two components Robertson
76. n an error message appears X test nb STUDENT VERSION Jog In 1 Inverse 2 Inverse matsq Argument 2 at position 1 is not a nonempty square matrix More Out 1 Inverse 2 1 150 a lt When you try to evaluate a wrong expression an error message appears Usually the kernel is still running properly after encountering an error however in case of very complicated calculations the kernel may stuck In such cases quit the kernel from the task list or by invoking the task manager Load CODICA in a Mathematica session To work with CODICA you need to load the program from a Mathematica session Open a new Mathematica notebook by clicking on the Mathematica icon An empty window will appear Type exactly the following string lt lt Utilities Codica whereby the symbol is the grave accent If your keyboard does not provide this accent open an example notebook provided with this manual You find this string at the beginning of the notebook Press the keys Shift and Enter CODICA 5 0 reference manual 27 at the same time to upload CODICA After CODICA is uploaded a welcome message appears on the screen To run CODICA type Codica followed by Shift Enter After this step CODICA interacts with the user via promt windows CODICA 5 0 reference manual 29 5 CODICA 5 0 tutorial Introduction This tutorial is intended to provide you with a basic knowledge of CODICA The f
77. n on a linear scale or a power scale files with extensions slin s for spectra lin for linear and or spow s for spectra pow for power will be created as well n gt CODICA saves always the last coercivity distribution calculated on a particular scale If you want to save the results of a CODICA session be sure that the last coercivity distribution you asked CODICA to calculate on a given scale is the distribution you want to save gt You can save the coercivity distributions plots produced by CODICA by clicking on the corresponding graphics in the Mathematica notebook The selected plot will be surrounded by a selection rectangle On the top menu bar of Mathematica select Edit gt Save Selection As gt Format whereby Format is a graphics format Full quality images are obtained using the EPS format CODICA 5 0 reference manual 42 A program example Notice Numbers to the left refer to sections in the text with the corresponding explanation Click on the link text on the left to return to the corresponding section of the manual text D x lt a B NED N B N B w B B w B rx i rx lt lt Utilities Codica CODICA v 5 0 for Mathematica 5 0 and later versions Distributed with the package MAG MIX release 1 04 04 2005 Copyright 2005 by Ramon Egli All rights reserved Codica C MAG MIX Codica Examples tape da
78. nalysis with one SGG function is inadequate to model the measurements within the given error margins Back to the program example GECA 2 1 reference manual 28 Enter initial distribution parameters 2 logarithmic Gaussian functions It is possible to run a new compo pa nent analysis with different initial x parameters After obtaining your Enter the initial distribution parameters first result the input prompt on the i a1 m1 s1 g1 p1 right asks you for one of the follo a2 m2s2q2p2 Help wing actions 1 type new initial parameters 2 re enter the initial Type parameters of the previous compo Type nent analysis 3 enter the result of previos component analysis as a set of initial parameters Since the component analysis with one component was not adequate a more complex model with two lo garithmic Gaussian functions is u sed The initial values for skewness and kurtosis are setted to zero by entering q 1 and p 2 for both components These parame ters are kept fixed by entering x f ql1 pl q2 p2 in the second pro f ql pl q2 p2 P Enter the list parameters you want to keep fixed during to re enter previous initial parameters to enter the parameters of the last model ETA 0 0007 1 5 0 4 1 2 0 0007 1 9 0 25 1 2 mpt window Initial parameters for optimization a and o are guessed until a Examples q2 p2 or th Help Last entered values will be taken for the fixed
79. netary Interiors 82 223 234 1994 back to text CODICA 5 0 reference manual 3 Index HLL lg iadete UT ci d 21g iepeereeeteeteesse trent eesrn sce a PRON O S SNe A UIT SU ORIEN NUNES a EDN SN RIST NESTON 5 2 Before using CODICA some suggestions to optimize your measurements sscsssecseecseecseecsseeesees 11 3 Basic theory of coercivity distributions all what you should KNOW ssssssssssssssssssssseseesesssssssseeseesssseseseee 17 VAS CAIN and run CODICA5 Oirrinn aaah caso atiis 25 2 CODICA S O tUtONAN tate ca she cecteceet eda ceexc ek cec sedate ese tecc sacs tee ese cests ance cer adidir a adie ini ei ai aiaei drda adada 29 6 Technical FETOEE VICE i ccestscesscucrcsssvaseaciacciteriguainssesdasesstaamessassdetaareeadnssiaineisesiataadecteageda tanned detde padise Utena 51 6 1 W ighted fitsarana arrisera enire ri E Nnr rE ANERE Eer PERE eE EA RESSE ATER EE arria ai 51 6 2 Scaling the magnetic TIE G sasssieessunnuanwmannrnicosiouniecndanatuneniauammaamanuniin 51 6 3 Model function for the magnetization curve secssssssseccssesssecssesssecssecssecsseccssecssccsucessesessesseeeseeeseecseees 51 6 4 Search the best field scale for symmetry sssesssssssrssssssssessesssssssssssssnsssessessssseressseereesereeersereseeeneeneesnsensessss 53 6 5 Model the magnetization Curve ccsescsessscsscssecsssssecssessscssscsssesssescssscenccsscenccsucenscsuccsscsuccsscanecascessensceseetss 53 6 6 Calc lat th residuals ssreiissssinsn
80. ning The parameters have a hierarchical structure a controls the amplitude of an end member u the position along the field axis and the width q the symmetry and p the curvature The amplitude is the most important parameter the curvature is the less important You can start with fixed values of p or fixed values of q and p Use logarithmic Gaussian functions to model magnetic components which are not saturated in the field range of the measured coercivity distribution In the program example the measured coercivity distribution is similar to an asymmetric unimodal probability density function There is no direct evidence for more than one magnetic component Therefore initial parameters for one SGG function have been entered in the program example Since only 5 parameters have to be optimized the component analysis is relatively simple and only one stable solution is expected Therefore it is not necessary to start with a modeled coercivity distri bution which is very close to the measured data Back to the program example Perform a component analysis GECA performs a component ana FA Local Kernel Input xj lysis by optimizing the initial para meters in a way that minimizes the ar npa dal ae 100 iterations squared residuals between model and measurements by using a Le Help venberg Marquardt algorithm The parameters to optimize are displa yed together with the correspon ding initial values If the initial value
81. not always evident which distributions group together and multiple solutions are often possible The aspects discussed above are illustrated with the examples of Fig 3 and Fig 4 Both figures show the results of a component analysis performed with GECA on the coercivity distribution of a sample of urban particulate matter In Fig 3 the component analysis is performed with logarithmic Gaussian functions Four logarithmic Gaussian functions are needed to fit the measured data so that the misfit between model and data is compatible the measurement errors However it is impossible to identify these four distributions with an equivalent number of magnetic components In Fig 4 the component analysis is performed with SGG functions The mea surements are already well fitted with one SGG function however the measured and the modeled coercivity distributions differens significantly This model could be adequate to describe low precision measurements of the same sample Two SGG functions fit the data within the margins given by the measurement errors However multiple solutions are possible but only one solution minimize the difference between model and measurements The other solutions imply rather uncom mon shapes for the coercivity distribution of the individual components which are not likely occur in natural samples GECA 2 1 reference manual 6 a b 0 5 0 0 5 1 1 5 2 0 0002 0 0004 0 0006 0 0008 0 001 0 0012 0 0014 c d
82. nvironmental studies The left plots show the original measurements the right plots are coercivity distributions calculated with CODICA The thickness of the curves corresponds to the error estimate of CODICA a AF demagnetization curves of a Tiva Canyon Tuff that contains acicular magnetite in the SP SSD grain size range Measurements have been started 8 5 and 160 hours after an ARM was imparted see the IRM Quaterly 14 3 2004 for more details about the Tiva Canyon Tuff b Coercivity distributions calculated from a Notice the bimodal character of the 8 5 h curve showing the magnetization of viscous and more stable particles The difference of the to curves inset shows the coercivity distribution of the viscous particles with relaxation times between 8 5 and 160 h c Modified Lowrie Fuller test performed on a sample of intact MV1 magnetotactic bacteria and d the corresponding coercivity distributions of ARM and IRM Notice that the two magnetizations are identical except for a low coercivity IRM contribution which may be related to collapsed or not well formed chains of magnetosomes sample kindly provided by D Bazylinski The properties of this low coercivity contribution are very similar to those of clustered SD particles e AF demagnetization of an ARM imparted to an anoxic sediment from lake Baldeggersee Switzerland see R Egli Physics and Chemistry of the Earth 29 869 884 2004 for a detailed discussion about magentic measure
83. odal A prompt window will ask you neu Se CW want to calculate the coercivity distri bution ona power s cale Typ e y if you Tee rics aid distribution on a power field scale are interested in this option Another prompt window will ask you to Te OU enter the exponent p of the power Enter the exponent p of the field scale transformation scale transformation You can enter any Hee oe or eas positive number Numbers close to zero Exponent for the most symmetric curve 2 28 will give results that are similar to those Exponent chosen by CODICA 1 00 on a logarithmic scale The exponent should be a positive number 2 28 The behavior of some coercivity distributions at H 0 might be discontinuous for power scales with p gt 1 This is for example the case of an exponential distribution In such cases the coercivity distribution calculated by CODICA might show a large peak at H 0 especially if p gt 1 This peak is an artifact of changing the field scale used by CODICA into a power scale You can remove this peak by choosing to calculate the coercivity distribution on a range of fields that does not include H 0 The use of p gt 1 5 is not recommended After you entered a value for p CODICA will first display the coercivity distribution on this scale over the entire range of fields spanned by the measurements CODICA 5 0 reference manual 41 At this points you have the same
84. of alternating fields for AF demagnetization is very sensitive to electromagnetic interferences Short pulsed interferences are cancelled through thermal activations if the decay rate of the alternating field is small enough As a role of thumb avoid decay rates higher than 4 mT s To calculate the decay rate of your equipment take the peak AF field and divide it by the time required by the AF field to decay from its maximum value to zero do not include the ramping up time Some AF demagnetization devices are more precise than other We noticed some systematic small errors produced by the AF demagnetization system of 2G Fig 6 These errors are not noticeable in standard applications but are significant when detailed demagneti CODICA 5 0 reference manual 14 zation curves must be obtained We do not know the origin of these errors but since they are systematic a correction formula is provided see Appendix B Pass trough devices are even more critical since they are faster and sensitive to the speed of the sample through the magnet Fig 6 a Average of 606 AF demag netization curves that have been normali gy as oO zed by their initial magnetization Both AF 0 8 f demagnetization and measurements ha ve been performed using a 2G cryogenic magnetometer with a 300 mT AF demag netizing system AF settings ramp rate 0 6 5 dwell time 3 These curves include ARMs and IRMs of different samples with magneti
85. ogenic magnetite lake sediments 04 Natural dust PM 10 i a Urban pollution PM 10 4 7 10 20 40 70 median destructive field jz mT Fig 2 Coercivity distribution parameters u o q and p for the AF demagnetization of IRM in various synthetic and natural samples Numbers beside the points indicate the grain size in um a Scatter plot of p and o The dashed line indicates the value of o for a negative exponential distribution b Scatter plot of q and p The cross point of the dashed lines corresponds to the values of q and p for a logarithmic Gaussian distribution All samples show significant deviation form a logarithmic Gaussian distribution All parameters of sized magnetite are intermediate be parameter for the kurtosis p 2 4 Y ho 9 o L o Magnetite Halgedahl 1998 L o Magnetite Bailey and Dunlop 1983 t Detrital component lake sediments m Biogenic magnetite lake sediments Natural dust PM 10 5 exponential a Urban pollution PM 10 distribution 0 4 0 6 0 8 parameter for the skewness q Z H 8 tween those of a logarithmic Gaussian distribution and those of an exponential distribution GECA 2 1 reference manual 5 Some aspects of component analysis The result of a component analysis depends upon the PDF chosen to model the end member coer civity distributions and particularly on the number of parameters assi
86. ome independent information about the coercivity distribution expected for the individual magnetic components In the program example point number 46 is entered which corresponds exactly to the global minimum of the merit function Back to the program example Perform a component analysis representing a global minimum The set of distribution parame Local Kernel Input x ters which corresponds to a glo bal minimum of the merit func tion is taken as initial model for the last component analysis In the program example this is the final solution which represents a finite mixture model which is compatible with the measure ments and with independent in formations about the properties of the individual magnetic com ponents In other cases you may not accept this solution and enter a set of initial parameters which Accept this component analysis y n Help corresponds to other values of the merit function until a satisfactory result is obtained Back to the program example GECA 2 1 reference manual 33 Perform an error estimation You can choose to perform an error ma i Local K l Input x estimation of the last component a S Sess itildhihenbdieslead ide nalysis GECA will perform the er Perform an error estimation of the distribution parameters ror estimation by adding a random 2 a ae Type n or enter the number of error simulations to perform noise signal to the measured coer m
87. omes prismatic magnetite with a very narrow grain size distribution between 40 nm and 80 nm Magnetic components have a simple shaped unimodal distribution of coercivities Commonly the coercivity distribution of a single magnetic component is modeled with a logarithmic Gaussian func tion log 11 20 G x 1 0 exp 1 N2T0 In the literature x is identified with the magnetic field H u is the median destructive or acqui sition field Hj However not all coercivity distributions can be modeled appropriately with 1 Experimental and called also MDF and MAD respectively and o the dispersion parameter DP theoretical coercivity distributions of single components are better described by distribution func tions with four parameters The two additional parameters control the skewness and the squareness of the distribution The coercivity distribution f H of a mixture of different magnetic components may be considered as a linear combination of the coercivity distributions of the single components f A Si M f A 0 2 i l where c and M are the concentration and the saturated magnetization of the i th component respectively and f H 0 is the corresponding coercivity distribution with the parameters 0 0 0 Equation 2 is called a finite mixture model and f H 0 are the so called end members Equation 2 assumes that the magnetization of all components adds linearly linear a
88. on for another time unit dt whereby equation 6 is now applied to the two micro coercivites that resulted from the previous step A coercivity distribution with four peaks is ob tained in this way Fig 8c If the simulation of this stochastic process is allowed to proceed for a long time a clear trend characterizes the resulting coercivity distribution Fig 8d l a n 1 1 2 b n 2 D Fo 8 Q Q 0 1 1 6 1 6 3 8 D 5 8 O 1 8 normalized microcoercivity normalized microcoercivity Fig 8 Stochastic model of a coercivity distribution redrawn from R Egli Physics and Chemistry of the Earth 29 851 867 2004 a An initial set of identical particles has the same microcoercivity Hg 1 The probability of having Hx 1 is obviously 1 b A random process changes the microcoercivity of the particles by a small amount 6 H over a given time Now Hy 1 6 or Hy 1 6 with equal probability c The same random process affects the microcoercivity over the next time interval whereby each microcoercivity value changes again by 6 Four micro coercivities values are obtained in this way d The probability distribution of the microcoercivities after six time intervals with 6 H 0 1H This distribution approaches fairly well the limit case of an infinite number of time intervals which is given by a logarithmic Gaussian PDF As the alteration process is going on the coercivity distribution converges to a cha
89. on of each parameter is stored as well if you decided to run an error estimation with GECA An example of the content of the log file is displayed below A components txt Editor oj x Datei Bearbeiten Format file name al wDKarm slog 0 0007989 MDFL s1 7 ql pl DPL 0 00002064 1 428 0 01136 0 4739 0 002868 0 5562 0 004 Back to the program example Save the end members to a file The end member distributions to FAPTA TT E xj gether with their confidence limits can be stored in a separated file as a list of columns with the numeri cal values of each function The file Hep will have the same name as the file where the original data for the coercivity distribution were stored with extention cum This file will be stored in the same directory as the data file An example is given below Back to the program example Save the coercivity distributions to a file yn EL wokarmcomp wordPad TT Datei Bearbeiten Ansicht Einf gen Format pela Sle al lalele BI field comp 1 abs error comp abs error 0 62572 1 67553e 6 2 18e 8 46818e 24 5 81e 23 0 623455 19e 8 66975e 24 6 0 621189 21e 8 88759e 24 6 0 618924 22e 8 123e 24 ts 0 616658 1 23e 8 37737e 24 7 2 2 2 2 Dr cken Sie F1 um die Hilfe aufzurufen GECA 2 1 reference manual 35 Cautionary note GECA 2 1 has been tested more than 500 times with coercivity distributions of various ar
90. ons w d and d is used by GECA as an improved merit function with respect to 6 since the randomizing effect of the measurement errors on the weighting factors r is removed Generally the merit function has several ae to stable solutions 0 local minima which correspond wah min Of the component analy sis Among these minima there is an absolute mi nimum Ey ElOumn Depending on the star ting values 0 of 0 one of these solutions is at tained by GECA If the model used for component analysis is ade quate and if there are no measurement errors Emn 0 Let n be the number of magnetic components and m the number of end member functions used in the model Then gt 0 for m lt n and un 0 for m gt n so that the number of components can be easily guessed Fig 6a In case of an inadequate model the end mem ber functions cannot reproduce exactly the coerci vity distribution of all magnetic components and Em gt 9 even without measurement errors 9 4 fH a 3 2 1 10 10 10 H 10 10 10 H Fig 5 Mean measurement error of the coercivity distribution of six samples of loess soil lake se diments marine sediments and atmospheric particulate matter The absolute error 5f H and the relative error 5f H f H are plotted in a and c respectively In b the absolute error is normalized by the square root of f H The field unit is mT All curves ar
91. onstraints to the number of end members and to their distribution para meters Performing and testing a component analysis When component analysis is performed a modeled coercivity distribution f x 6 with parameters 0 0 9 is compared with the measured coercivity distribution given by a set of numerical values x f 5f with measurement errors 5f A solution of the component analysis is repre sented by a set of values of 8 which minimizes a so called merit function The merit function is an estimation of the difference between the modeled and the measured curve 0 if the model is identical with the measurements Examples of are the mean squared residual N f z 8 4 i 1 used for a least squares fitting and the x estimator fa 18 7 2 a i x 8 5 D aed used for a minimum y fitting GECA uses following weighted version of the x estimator w aea 2 2 a l where r 6f f are the relative errors In this case measurement points affected by a large relative error are less considered for the component analysis Equation 6 can be rewritten as N oe i 1 4 y c fE O GECA 2 1 reference manual If f x originates from the sum of a finite number of elementary contributions f is a Poisson distri buted variable and 8f f An experimental confirmation of this assumption is shown in Fig 5 After these considerati
92. onverges to the logarithmic transformation H ln H Thus 12 provides a set of scales that contain the logarithmic and the linear scales as a special case Fig 10 shows an example where the the power scale transformation has been used to check the unimodality of a coercivity distribution CODICA 5 0 reference manual magnetization scaled field unit magnetization field unit magnetization scaled field unit MV1 magnetosomes a p 1 1 596 0 20 40 60 80 100 magnetic field mT pedogenic magnetite b 4 p 0 514 a 1 832 0 2 4 6 8 10 0 514 scaled magnetic field mT clay in glacial till p 0 205 a 1 813 1 1 5 2 2 5 3 3 5 scaled magnetic field mT 7 22 Fig 9 Validation of the stochastic model for the coercivity distribution of some na tural magnetic components First a field scale transformation H H has been applied to the measured coercivity distribu tions thick black lines The scaling expo nent p has been chosen so that the trans formed coercivity distribution is symmetric e g its skewness is zero The estimated measurements errors are given by the thick ness of the black line The scaled coercivity distribution has been fitted with a L vy self similar PDF lt z o amp with width parame ter o and shape parameter a red line Special cases of x are the Gaussian PDF a 2 and the Cauchy PDF a 1 The parameter a cont
93. or vehi cles and from waste incineration fp 90075 1 4 0 45 0 6 2 0 0007 1 957 0 235 0 663 2 The coercivity distribution of the combustion products can be model led by a SGG function with u 1 96 o 0 235 q 0 66 p 2 These parameters are kept fixed during the component analy sis Only the magnetization of the Enter the list parameters you want to keep fixed during K combustion products given by a Stan is unknown and is optimized to i Examples q2 p2 or Help gether with the unknown parame ples 192 p 2 Last entered values will be taken for the fixed parameters ters of natural dust Back to the program example Perform a component analysis one component is known This component analysis is characterized by a much better agreement with the measurements if compared to the previous results Six distribution parameters have been optimized The same number of parameters has been used to perform a component analysis with two logarithmic Gaussian functions nevertheless 7 1 was almost one order of magnitude larger This model is still signi ficantly different from the measurements A reason for that could arise from small variations in the properties of the same magnetic component collected from different places Back to the program example GECA 2 1 reference manual 30 Enter initial distribution parameters 2 SGG functions To take into account small varia i tions in the magnetic prope
94. parameters relatively good agreement with the measured data is obtained Back to the program example Perform a component analysis 2 logarithmic Gaussian functions The component analysis with two logarithmic Gaussian functions is inadequate to model the measu rements within the given error margins The misfit between model and measurements is larger than that obtained with one SGG function even if there is one more parameter to optimize This example shows that logarithmic Gaussian functions are generally not suitable for modelling the coercivity distribution of single magnetic components A good agreement between measurements and model is achieved only with 4 logarithmic Gaussian functions Figure 3 Unfortunately these functions can not be related to the coercivity distributions or real magnetic components Back to the program example GECA 2 1 reference manual 29 Enter initial distribution parameters one component is known The different sources of magnetic Local Kernel Input x minerals for the sample taken as example are known from indepen Enter the initial distribution parameters dent investigations on urban atmo al ml s1 ql pl 7 spheric dust samples collected in a pee Help the same region The two main sources are given by natural dust Type i to re enter previous initial parameters and by the products of combustion Type t to enter the parameters of the last model processes mainly from mot
95. pi 2 206 6847 962 sl 0 2329 7221 pl 2 053 al mi ql orc OMH Residuals modelled measured and measurement errors in of the maximum value of the coercivity distribution Dashed lines delimitate the interval considered for the component analysis ChiSquare points 1 5 Confidence limits 0 59 1 7 Model and data are compatible You may accept this component analysis GECA 2 1 reference manual Systematic solution search Perform an automatic variation of the contribution of component 2 This process takes several minutes Please wait Decreasing contribution of component 2 Increasing contribution of component 2 Residuals as a function of the contribution of component 2 Every 10th point in gray first point is 21 red point is the starting solution 19 GECA 2 1 reference manual ul al 1 m1 ql Ts al m1 0 ql 0 0 0 Optimizing the distribution parameters Please wait 20 Choose initial parameters Ds 428 5562 oF g oO 7989 26556 29167 21265 Initial distribution parameters si 0 4739 pl 2 108 sl 0 2297 pl 2 052 Perform a component analysis 5 al 0 0007989 m1 1 428 s1 0 4739 q1 0 5562 p1 2 108 a2 0 0006556 m2 1 967 s2 0 2297 q2 0 7265 p2 2 052 Ts al 1 ql Ta al m1 0 gil 0 0 Ox Model and data are compatible You may accept this component analysi
96. points is more difficult to estimate It is identical with the number of measurements if the measurement errors are equivalent to an ergodic noise signal that is when the autocorrelation of the noise signal is equivalent to a Dirac 6 function GECA 2 1 reference manual This is often not the case with real measure ments where entire groups of measured points are affected by the same error Furthermore the coercivity distributions calculated by CODICA are low pass filtered and an autocorrelation of the remaining measurement errors is unavoida ble GECA estimates the degrees of freedom of the fitting model by evaluating the residuals curve which results from the difference be tween the model and the measurements The residuals curve contains a certain number of random oscillations around a mean value of ze ro To reproduce these oscillations a minimum number of points is necessary whose spacing defines the Nyquist frequency of the signal GECA sets 1 equal to the number of zero crosses of the residuals Obviously the shape of the residuals curve depends on the model chosen for component analysis Fig 7 Examples of Pearson s x test on the component analysis of a sample of urban at mospheric particulate matter The gray and the blue curves are the measured and the modelled coercivity distributions respectively Curves labeled with other colors represent the coerci vity distributions of individual end members Below
97. racteri zed by the same field CODICA detects multiple measurements and calculates an average value A warning message is displayed You should ignore this message if the data file has been produced automatically by a measurement system However if the measurement steps were performed by hand you should consider the possibility that some steps were remeasured and only the last value of a multiple measurement might by correct In this case you should exit CODICA and check the data file manually gt More than 300 measurements Error free magnetization curves are smooth The amount of data produced by some automatic system does not increase the resolution of a magnetization curve the highly redundant data are helpful only in reducing the effects of measurement errors The least squares collocation method used by CODICA to filter the data is very efficient in removing measurement errors however the amount of mathematical operations required is proportional to the square of the number of measurements Accordingly the computation time increases enormously for large dataset If the dataset contains more than 300 measurements CODICA uses a moving average filter to reduce the number of data This procedure does not discard informa tion from the original file and you should not be worried about the quality of the results A war ning message remembers you that CODICA is dealing with a reduced amount of points 3 Evaluate the properties of the magneti
98. racteristic shape that depends only on s PDFs obtained from a stochastic process as described above have an interesting property when represented on an appropriate scale they are self similar This means that if f z o is a self similar PDF with width parameter the convolution of two such PDFs gives again the same self similar PDF f 2 0 f 0 f x 0 05 In other words the sum of random variates described by a self similar PDF is a variate with the same self similar PDF Such PDFs have very peculiar mathematical properties that make them so important in CODICA 5 0 reference manual 20 statistics and in the description of stochastic processes The most famous self similar PDF is the Gaussian or Normal function x u 20 7 exp y 1 ARS with mean u and variance a This PDF can be generated by using 6 with a constant s to describe a so called additive stochastic process The process defined by s x Hy Fig 8 is a multi plicative stochastic process that generates the logarithmic Gaussian or Normal PDF _In o A 20 g 1 N x u 0 Re with median jz and dispersion parameter The logarithmic Gaussian PDF itself is not self similar but it is transformed into the self similar Gaussian PDF through the scale transformation x ln z Thus z lnz can be regarded as the natural scale of the logarithmic Gaussian distribution This example can be generalized to the following statement a stochas
99. rameters Please wait Perform a component analysis 1 a1 0 0015 ml 1 7 s1 0 6 ql 0 5 pl 2 2 Optimized distribution parameters al 1 454 m1 4 744 s1 0 6875 ql 0 4256 pl 3 076 Residuals modelled measured and measurement errors in of the maximum value of the coercivity distribution Dashed lines delimitate the interval considered for the component analysis ChiSquare points 8 8 Confidence limits 0 52 1 9 Model and data are significantly different Refine your model GECA 2 1 reference manual m1 ql al m1 qa Optimizing the distribution parameters Please wait al 0 0007 m1 1 5 s1 0 4 a2 0 0007 m2 1 9 s2 0 25 al 16 Enter initial parameters 2 Ce sS ero RR NI Initial distribution parameters sl 0 4 pl 2 sl 0 25 pl 2 Perform a component analysis 2 Model and data are significantly different Refine your model Optimized distribution parameters al 0 9101 m1 12517 s1 0 4363 ql 1 pl 2 al 0 5428 ml 1 948 sl 0 2124 ql 1 pl 2 Residuals modelled measured and measurement errors in of the maximum value of the coercivity distribution Dashed lines delimitate the interval considered for the component analysis ChiSquare points 27 Confidence limits 0 52 1 9 GECA 2 1 reference manual 17 Enter initial parameters 3 Initial distribution parameters al 0 75
100. rols the squareness of x a tip shaped PDFs with heavy tails are obtained with lt 2 a Coerci vity distribution obtained from the AF de magnetization of ARM for cells of the mag netotactic bacterium MV1 mixed with pure kaolin Notice that a scale transformation was not required The coercivity distribution is heavily tailed as indicated by the low a b Coercivity distribution obtained from the AF demagnetization of ARM for pedo genic magnetite in a soil developed from glacial till in Minnesota The coercivity di stribution of the pedogenic component has been obtained by subtracting the contribu tion of the glacial till from the demagneti zation curve c Coercivity distribution ob tained from the AF demagnetization of ARM for the clay fraction of a glacial till in Minnesota Notice that the coercivity distri butions in b and c are plotted on dif ferent field scales but are characterized by almost the same L vy self similar PDF with awvl1 8 CODICA 5 0 reference manual _ iy magnetization um kg g magnetization uAm kg magnetization pA m gt kg A Oo D S S A 0 50 100 150 magnetic field mT aN N 2 oO poe Pp O S S S E D D E S E S E E 300 100 200 250 fo 1 10 magnetic field mT N w DA UI oO Oo oO oO potirirrtirrirrtirriitiriity e S A L E S S E O S S S S E S 200 1 10 100 magnetic field mT 23 Fig
101. rties of x combustion products the results of Enter the initial distribution parameters o the previous component analysis a1 m1 s1 q1 p1 are taken as initial values for a new a2 m2 s2 q2 p2 Help component analysis where all 10 distribution parameters are optimi Type i to re enter previous initial parameters 6699 zed This is done by typing r to Type t to enter the parameters of the last model recall the result of the last component analysis Back to the program example Perform a component analysis 2 SGG functions The model is now much more FAPTA x complex and the search for a stable m e a than 50 itera el ea iterations essage appears and you are asked to stop or con __ Hep tinue for other 50 steps Finally a stable solution is reached The di stribution parameters of the combu stion product did not change more than 10 with respect to the initial values and the model is now com patible with the measurements wi thin the measurement errors The solution of this component analysis is accepted since it is compatible with the magnetic properties assu xj med initially for the combustion products Back to the program example Help Accept this component analysis y n GECA 2 1 reference manual 31 Perform a systematic solution search Stable solutions of the component p i i s analysis correspond to local mini x ma of the merit function The merit Perform
102. s ml OPO GOH measure 21989 428 5562 6556 967 265 ment Optimized distribution parameters sl 0 4739 pl 2 108 sl 0 2297 pl 2 052 Residuals modelled measured and errors in of the maximum value of the coercivity distribution Dashed lines delimitate the interval considered for the component analysis ChiSquare points 1 5 Confidence limits 0 59 1 7 GECA 2 1 reference manual 21 Calculating statistical parameters of the distributions Perform an error estimation Perform an error estimation of the distribution parameters with 64 error simulations Accuracy of the error estimation 12 This process takes several minutes time Please wait Error estimation of the statistical parameters Please wait Calculating the confidence limits of the components Please wait Parameters of component 1 Parameters of component 2 a 0 7989 0 021 a 0 6556 0 021 p 1 428 0 011 u 1 967 0 0038 o 0 4739 0 0029 o 0 4739 0 0029 q 0 5562 0 0047 q 0 7265 0 0065 p 2 108 0 019 p 2 052 0 0082 MDF 1 428 0 011 MDF 1 967 0 0038 mean 1 374 0 012 mean 1 957 0 0044 DP 0 4255 0 0027 DP 0 2165 0 0024 skewness 0 8043 0 013 skewness 0 2647 0 015 kurtosis 1 149 0 026 kurtosis 0 1122 0 024 Result of the component analysis Total magnetization 1 455 0 029 Li al 0 7989 1 MDF1 1 428 DP1 0 4255
103. s on the merit function is not considered Therefore the error estimation performed by GECA has to be considered as a lower limit for the real error of each parameter Finally all distribution parameters are displayed together with the estimated errors Additionally statistical parameters like the dispersion parameter DP the mean the skewness and the kurtosis are displayed with the related errors The result of the component analysis is plotted again together with the confidence limits of each end member function Finally the normalized end member distributions are plotted together with their confidence limits The area under the curve of each end meber distribution is equal to one in this last plot of the field The new noisy coer repeated several times GECA calculates the standard deviation of the component analysis results for convergence of the component analysis to parameters which correspond to other local minima of the Back to the program example GECA 2 1 reference manual 34 Save the component analysis results in a log file You can save the component ana gz eee ae xi nents txt You will find this Print results to the log file components dat y n file in the same folder where the program package CODICA is in stalled The log file contains the re sults of all component analysis you decided to save in form of a list of distribution parameters and statis tics for each end member distribu tion The error estimati
104. s the estimated 95 confidence interval of the coercivity distribution The error estimate is also plotted below the coercivity distribution gt You can choose to recalculate the coercivity distribution on a different range of fields CODICA 5 0 reference manual 39 A prompt window wil ask you if you TACLI want to choose a different range of TPP R a i i T fields Type n if you are satisfieed with Sii sae ia the result If you type y CODICA will ask you to enter a new range of fields Then it will recalculate and the coercivity distribution on the new ran ge This process is repeated until you type n in the prompt window to the right 12 Coercivity distribution on a linear scale It is sometimes useful to plot the coercivity distribution on a linear scale A linear scale will show different features of the coercivity distribution that may be hidden in the logarithmic scale repre sentation see Fig 10 A prompt window will ask if gt Prony ae r v kaa aX Local Kernel Input want to calculate the coercivity distri bution on a linear scale Type my if you Calculate the coercivity distribution on a linear scale y n are interested in this option If you accepted to calculate the coercivity distribution on a linear scale CODICA will first display the coercivity distribution on this scale over the entire range of fields spanned by the measu rements
105. s were carefully chosen the search for a solution is performed in a reasonable time with no more than 100 iterations Otherwise the search will take more than 100 iterations or it will converge to an absurd solution If a global or a lo x cal minimum of the squared resi duals is not reached within 100 ite rations a warning message appears and you will be asked to continue the search or stop it and plot the solution given by the last iteration It is recommended to perform at least 200 iterations The numerical values of the parameters are shown every 100 iterations and you can check how they change and if they converge to a meaningful result Accept this component analysis y n GECA 2 1 reference manual 27 If a convergence to a stable solution cannot be obtained interrupt the search for a solution by typing n in the prompt window and choose other initial parameters The result of the component analysis is displayed exactly like the initial model The same colors are used to label the end members Additionally the difference between the model and the measu rements blue line is plotted below the result of the component analysis together with the measu rement errors pair of gray lines The difference between model and measurements called misfit in the following should be of the same order of magnitude as the estimated measurement error If the misfit is much larger than the estimated measurement error
106. s with Cy h do exp h r1 cos w h C h eg exp h r where d var d and ef varfe are the variances of d and e respectively r and r are the so called correlation distances of d and e respectively and 277 w is the dominant wave length of d Assuming that d and e are uncorrelated C h Ca h Celh 27 26 On the other hand according to 19 C h f7 r w Peos wh dw 28 To find a best fit correlation model of C h CODICA minimizes the squared difference between 27 and 28 tax Himin 2 J C h eC C h dh 29 with respect to the parameters d o Tj T and w The correlation distance 7 of the residuals is assumed to be the value of h for which C h is reduced to the half C 7 0 5C 0 If either ey gt 0 8d or r lt 7r CODICA consider the residuals to be dominated by a noise signal produced by the measurement errors In this case the estimate of C h using 26 is not relia ble and is replaced by C h do exp h m 29 whereby C 7 0 5C 0 If gt gt r is known a more accurate estimate of e H by evaluating the variance of the difference r between r H and the moving average Sor exp E HEP 8 r H Epa Hye 30 If the same moving average filter is applied to r r ry we obtain following estimate of e H CODICA 5 0 reference manual 58 SO r H7 FAP P exp H7 HEY r a 2 7 X 4x 4 X exp
107. siseiianieininsannainanaiaiiiaiais 54 6 7 A stochastic model for the residuals ssssssssessersssesreesssessesssssssssssssessesssesssssrsrersrereesereeereerseeeseeeseenesensessss 54 6 8 Subtract a trend with polynomials sssssssssessssssssssssssesresessssseseeesrsnssssseneeeerennsssseneererensssesstreerersnessssstnet 55 6 9 Rescale the residuals sssssssssssssssssssssssseesseeseerseereercereseoeneeeseessenssssssssssse sse sssesretsreeeeroereerereereeeteeeeereseeeneesses 56 6 10 A least squares collocation model for the residuals eessssssssssssssseesseesssssssnsssssssseesseeesseesssnsssnssssesse 57 6 11sFiltering th resid als ssssrsosoneisunennemtannanten unai Eaa GRE 58 6 12 Filtered magnetization curve ssssesssesseeserseeeeereseeereeseeessessessssssessssssesssesserseeterrrereereeeeetserseeereeseeeseeseeesesssee 59 6 13 Co rcivity distribtUiOrsS ssis aaora saN Kaanaa aAa e iaa eaa aiaa ainaani s 59 LE ELS EAs eaan a a a a a aD re Ser 61 APPENA E ninni es a testers as A RRN ee es 62 CODICA 5 0 reference manual 5 1 Introduction Why magnetization curves Rocks and sediments inevitably contain mixtures of magnetic minerals grain sizes and weathe ring states Most rock magnetic interpretation techniques rely on a set of value parameters such as susceptibility and isothermal anhysteretic remanent magnetization ARM or IRM These para meters are usually interpreted in terms of mineralogy and domain state of th
108. sonov et al 1998 With this equipment complete acquisition curves with thousands of measurements up to 500 mT can be obtained in a few minutes Fig 4 a MEASURED IRM ACQUISITION CURVE b RESIDUALS dots AND INTERPOLATION red 200 1 1504 x E J BS 100 8 oS E 504 0 TTT tt T T T T T T me 0 100 200 300 400 500 0 2 4 6 8 10 magnetic field mT magnetic field mT c COERCIVITY DISTRIBUTION ON A LINEAR SCALE d COERCIVITY DISTRIBUTION ON A LOG SCALE ke J J rl La 3005 5 5 D uv J 8 08 gt 200 4 T O 4 E 4 A x E 04 100 4 O 4 4 Coe a re SSS oL Se EEr 0 100 200 300 400 10 100 10 4 5 5 oy 3 o 4 107 4 a n m r 0 100 200 300 400 10 100 magnetic field mT magnetic field mT Fig 4 Example of a IRM acquisition curve measured with the coercivity spectrometer described in Jasonov et al 1998 courtesy of R Enkin and subsequent coercivity analysis with CODICA a Raw measurements A typical mea surement up to 500 mT consists of 1500 steps acquired in few minutes b The residual curve calculated by CODICA shows clearly the measurement errors The increasing amplitude of these errors at high fields indicates that the ap plied magnetic field account for a large portion of the errors The red line is the fit calculated by CODICA c and d are the corresponding coercivity distributions calculated on a linear respectively logarithmic
109. stribu tion as well as its unit depend on the scale used for the magnetic field Appendix A reports tables for the appropriate units to use for coercivity distributions In the following the mathematical and physical meaning of a coercivity distribution is discussed Understanding this meaning is very important for a correct interpretation of the results obtained with CODICA Let define the coercivity distribution H of a magnetization curve M H as the absolute value of the first derivative of M H OM H whereby the factor y depends on the type of magnetization curve and accounts for the Stoner Wohlfarth relationships in non interacting SD particles Stoner and Wohlfarth 1948 Thus y 1 for all acquisition and AF demagnetization curves and y 1 2 for DC demagnetization curves also called backfield curves Coercivity distributions can be regarded as the statistical distribu tion of so called switching fields In some cases the switching field can be identified with the field required to switch the magnetization of a single magnetic particle This is for example true for an assemblage of non interacting SD grains However this simplification does not apply to MD parti cles or assemblages of strongly interacting grains The statistical character of a coercivity distribu tion is formalized by assuming H to be proportional to a probability density function PDF m H Mof H 2 where the proportionality constant M is simp
110. t Check the measurements ORIGINAL MEASUREMENTS Szi 1e0 A magnetization N Oo ao maaa a 0 20 40 60 80 100 magnetic field Evaluate the properties of the measured curve WARNING CODICA detected possible digital truncation effects List of points where jumps arising from digital truncation have been identified point field detected jump 6 2 5 G01 12 Geo CELON 20 9 5 0 01 23 12 5 0 01 Some points have been discarded resampled in order to minimize digital truncation effects Estimated initial magnetization 5 126 0 010 Estimated final magnetization 0 004773 0 0033 extrapolated Estimated median field 40 03 Searching for an optimized field scale Scan scaling exponents lt 1 Please wait Scan scaling exponents gt 1 Please wait Field scale Transformation Mean sd of fit original H H 1 01 for equally spaced points H H 0 849 1 04 for the most symmetric curve Be 4HFl oops 28 0 5475 suggested scale H Hils 7 lasa 0 899 43 CODICA 5 0 reference manual B 4 B r3 DED pam Loa B oO B oN B oN Searching for an asymmetric best fit hyperbolic model Estimated contribution of a high coercivity component HYPERBOLIC MODEL red Mean sd 0 120 magnetization 1le0 N Ww gt uo ja O nH m 200 300 400 scaled magnetic field E 0 100 Manually chosen field s
111. t analysis red point is close to the global minimum of the merit function The sharp steps of the merit function are an effect of the sudden convergence of some distribution parameters to a different local minimum You are asked to accept the solution of the last component analysis indicated by a red point if it corresponds to a global minimum of the merit function Back to the program example GECA 2 1 reference manual 32 Choose initial distribution parameters from the systematic solution search You can check the distribution pa rameters which corresponds to va rious values of the merit function previously plotted The merit func tion was calculated for 100 points black dots in the last plot every 10th point is gray You can enter the number of the point which cor responds to a particular value of the merit function you are interested in In this way you can explore the solutions which correspond to va rious local minima and to the glo bal minimum of the merit function This option is particularly useful in 4 Local Kernel Input a x Choose the distribution parameters corresponding to a particular point in the optimization plot by entering the number of the desired point Hep the case that several local minima exist which are close to the global minimum Due to the measure ment errors a meaningful solution could be given by one of these local minima You may evaluate different solutions with s
112. t corresponds to the estimated error 2 sd COERCIVITY DISTRIBUTION ON A LINEAR SCALE n p 4 600 G 5 m J v i 400 e 6 A al 4 a N 3 200 G D fo g 4 a CD a T 25 30 35 40 45 50 magnetic field ESTIMATED ERROR lots o 1073 3 4 J o J 107 3 a a AEE E GBE SE a ay zar amna ma SE S S P a SE a A AE ONE o a A a A SEEE N E S i 25 30 35 40 45 50 magnetic field CODICA 5 0 reference manual 49 H 13 COERCIVITY DISTRIBUTION ON A H HA2 28 SCALE ar G T v F E A jj 6 N H p v G D 6 20 40 60 80 scaled magnetic field Range of fields spanned by the the coercivity distribution 1 47 51 6 H 13 Resample the coercivity distribution in the field range 18 0 50 0 The grey band around the curve in the following plot corresponds to the estimated error 2 sd COERCIVITY DISTRIBUTION ON A H 2 28 SCALE mw 23 4 p 4 s 4 amp f 5 4 m 24 1 2 o 4 a 4 c 1 54 6 4 ea 4 p 4 oO 4 N oa So LA yp 4 rat Z D 4 0 54 a a a a A S S A S D S E O D E E E E i ql 1 20 25 30 35 40 45 50 magnetic field ESTIMATED ERROR 104 4 E o J H J 4 4 oO 10 43 Sto yeas eyes Ss e rn e E T T 20 25 30 35 40 45 50 magnetic field CODICA 5 0 reference manual 50 i A Save the fitted measurements in file tape cum 1 column magnetic field 2 column magne
113. ters as an ordered PARATOI i x list a u o q and p of the first component a u 0 q and p of the second component and so on a1 m1 s1 g1 p1 Enter the initial distribution parameters Help as in the example given in the prompt window shown to the right The end member distributions defi ned by the initial parameters you entered are plotted with different colors red green violet and light 0 0015 1 7 0 6 0 5 2 2 blue The modeled coercivity di stribution is given by the sum of all end members and is plotted in blue together with the measured coerci vity distribution black gray The initial parameters should be chosen so that the modeled coercivity di re stribution is as close to the measu Cone eae aera red coercivity distribution as possi ble You can enter the initial para meters either with some knowledge about the magnetic components which are contributing to the mea sured distribution or by try and er ror In this last case you can reenter new initial parameters until you get a satisfactorily result After ente ring the initial parameters you are asked to keep some parameters fixed during the optimization If ra you want to optimize all parame MESA SLGEGII xi ters type Otherwise enter the Enter the list parameters you want to keep fixed during symbols for the fixed parameters in optimization r iii rithmic Gaussian function for the second en
114. tic process produces a PDF that is self similar on an appropriate scale hereon called the natural scale of that process A wide and very general class of self similar PDF is represented by the so called L vy stable PDFs Sato et al 1999 These functions are symmetric about their median ju If we limit our considerations to this class of self similar PDF we obtain a more useful statement a stochastic process produces a PDF that is symmetric if represented on its natural scale The representation of a coercivity distribution on its natural scale offers the advantage of a great simplification since ad ditional parameters for the asymmetry of the PDF are not necessary The stochastic nature of coercivity distributions brings us to the concept of magnetic component If we define a magnetic component as an assemblage of magnetic grains with a common origin and a common biogeochemical history the coercivity distribution of such a component is the re sult of the stochastic nature of all processes that led to the formation of the magnetic grains e g processes of nucleation and growth and their subsequent chemical physical alteration and se lection According to this definition of a magnetic component and the above discussion about stochastic processes the coercivity distribution of a magnetic component is represented by a sym metric self similar PDF on an appropriate field scale We shall consider this statement as the funda mental hypotheses of co
115. tificial and natural samples and the most different combinations of initial parameters Nevertheless there is a remote possibility that particular uncommon data or parameter sets will produce evaluation pro blems In this case blue written warning messages appear on the Mathematica front end If more than one of these messages is displayed you may force quit the Kernel of Mathematica as follows in the top menu bar choose Kernel gt Quit Kernel gt Local You can also exit from GECA at any time just by typing abort in any input prompt window GECA 2 1 does not work on previous versions of Mathematica 5 0 because of a substantial rede finition of a minimum search routine embedded in Mathematica 5 0 Updates of Mathematica will generally not affect the functionality of packages such as GECA wich is expected to run on later versions Use GECA 1 1 on previous versions of Mathematica 5 0 In case of problems write to the author Ramon Egli at the address given at the beginning of the source code file Geca m Please save and send a copy of the Mathematica session you were using when a problem arises together with the data file you analyzed with GECA References Egli R Analysis of the field dependence of remanent magnetization curves J Geophys Res 102 doi 10 1029 2002JB002023 2003 Heslop D M J Dekkers P P Kruiver and I M H Oorschot Analysis of isothermal remanent magnetization acquisition curves using the expect
116. tion combination CODICA 5 0 reference manual Appendix B 62 Suggested empirical field correction table for AF demagnetization curves obtained with a 2G cryogenic magnetometer with build in AF unit Range of AF fields mT Suggested correction factor A Hedi H A 0 99 0 10 29 9 0 254 30 39 9 0 465 40 49 9 0 104 50 69 9 0 104 70 79 9 0 25 80 89 9 0 10 90 109 9 0 65 110 139 9 0 95 140 149 9 0 35 150 179 9 0 55 180 300 0 55 GECA GEneralized Coercivity Analyzer Version 2 1 User s manual Improtant notice This version 2 1 of GECA GEneralized Coercivity Analyzer is almost identical with the previous version 2 0 that is described in this manual Since GECA is going to be deeply revised in the near future this manual has not been updated Please consider the following differences of GECA 2 1 with respect to the manual e Since some users reported difficulties in using the characters and in the prompt windows their need has been eliminated in GECA 2 1 Please ignore these characters in all examples provided in the manual For example if the manual tells you to enter 0 2 type 0 2 instead Accordingly is replaced by no character just click the OK button of the prompt window without typing anything e You can now model a coercivity distribution with a maximum number of 5 components instead of 4 Install GECA Requirements
117. tization 3 column absolute fit error Save the log scaled coercivity distribution in file tape slog 1 column log magnetic field 2 column coercivity distribution unit magnetization 3 column relative error Save the linear scaled coercivity distribution in file tape slin 1 column magnetic field 2 column coercivity distribution unit magnetization field 3 column relative error Save the linear scaled coercivity distribution in file tape spow 1 column magnetic field 2 28 2 column coercivity distribution unit magnetization field 2 28 3 column relative error Thank you for using CODICA Are you satisfied Please report eventual problems or suggestions to the author CODICA 5 0 reference manual 51 6 Technical reference 6 1 Weighted fit Since the measurement points are not necessarily equally spaced an unweighted fit would be biased by the parts of the magnetization curve where the points are closely spaced To avoid this bias all data fits performed by CODICA are weighted to avoid this effect In a least squares fit the function e S w u f a a 1 k is minimized with respect to the parameters a whereby Y are the points to be fitted k 1 N f x a is the parameterized fitting function and ie ea aa 2 lt k lt N 1 Ty 1 2 To T Ty Ty_ w 2 ba iE N N 1 Ty Ly Ty Ly are the fitting weights 6 2 Scaling the magnetic field
118. to enter the full path of the file that contains your data Type reaped ia cae a exactly the complete path with correct upper and lower cases and including the extension of the file e g txt or dat To avoid typing long paths it is recommended to store your data in a easily accessible folder e g C users myself datafile dat Example C users data myfile dat C MAG MIX_1 Codica Examples D ata tape dat CODICA can read any file that contains at least two columns of numbers Header and footer lines will be ignored The first column is supposed to contain the field values the second column the corresponding measurements Eventual further columns to the right will be ignored The colums must be separated by at least one of the following characters space tabulator or the punctua tion signs CODICA starts to read the file at the first row that has the appropriate for mat and stops when it encounters a row that has not the appropriate format e g a footer line or at the end of the file Therefore header lines must not begin with numbers in a column format gt If CODICA experiences problems in reading the file remove the header If CODICA does not find the file or if the file format is wrong a critical error message is displayed and the program is stopped L 2 Check the measurements CODICA checks the validity of the measurements Valid measurements represent any kind of magneti
119. u want to see how the magnetization curve is affected by different values of the scaling exponent gt For illustrative purposes only the scale suggested by CODICA in this example was rejected 5 Entering a scaling exponent manually This step is necessary only if you wand to discard the field scale suggested by CODICA A prompt window asks you to enter a bes Local Kernel Input Eg positive scaling exponent in the range between 0 0001 and 5 You should con Enter a positive value for the field scaling exponent sider very carefully the reasons for ente Possible range of values 0 0001 5 ring exponents gt 1 5 The exponent you enter in this window is not definitive you will have the possibility to reject it and make a different choice The performance of CODICA is not guaranteed for scaling exponents gt 2 7 gt A criterion for choosing the scaling exponent is given by the difference between the measu rements and the best fit curve If this difference is highest at small fields you should choose a smaller scaling exponent On the other hand if the difference is highest at large fields you should choose a larger scaling exponent CODICA will rescale the measurements according to the exponent you entered and recalculate a best fit with an asymmetric hyperbolic function After the results are plotted a prompt window will ask you to accept or reject the field scale If you do not accept the field s
120. uation 5 is considered The x estimator is a statistical variable which is distributed ac cording to a y distribution with M k 1 degrees of freedom being N the number of indepen dent points to fit with a given model and k the number of model parameters The expected value of the x estimator is N k 1 The confidence limits at a confidence level a generally a 0 95 p 2 2 ih are given by XN k 1a and on a ae with T trad cep 8 2 XN k l p If x2 gt ae the model differs significantly from the measurements The model should be refined by adding new parameters eventually by considering an additional end member function If x lt EER the differences between model and measurements are unrealistically small An excessive number of parameters allow the model to include random effects of the measurement errors Consequently some of these parameters are not significant The model should be revised to include a smaller number of parameters eventually by reducing the number of end members or by keeping some parameters fixed If Xa ria S aes Xy 11 a the model is acceptable To calculate the y estimator with equation 5 some knowledge about the measurement errors amp and the number of independent data points is necessary The measurement errors are automatically estimated with CODICA when a coercivity distribution is calculated from an acquisition demagneti zation curve The number of independent data
121. uch as wiggles are produced by mea surement errors that do not have a completely random appearance so called pink noise gt Ideal white noise is completely uncorrelated and is easily recognized and removed How ever measurement errors can be correlated for example if they arise from fluctuations of the ambient field during measurements In this case the measurement errors have their own correla tion length Least squares collocation can remove measurement errors efficiently if the correla tion length of the noise is much smaller than the correlation length of the residuals 9 Calculate the filtered measurement curve Once you accepted a model for the residuals CODICA is ready to calculate the filtered sup posedly error free magnetization curve To do so all previous steps are inverted starting from the filtered residuals in order to obtain again a magnetization curve see section 6 12 for mathema tical details The magnetization curve is plotted together with the original measurements The estimated error of the modeled magnetization curve is plotted as well gt If the quality of the measurement was good you probably will not notice any difference be tween the raw measurements and the magnetization curve calculated by CODICA However the differences become evident if you compare a numerical derivative of the magnetization curve obtained using the raw data on one hand and the magnetization curve calculated
122. uch grains is a step function and the corre sponding PDF is given by 6 H Hx where 6 is the so called Dircac 8 function Fig 8a Ima gine now that the magnetic grains are subjected to an alteration process such as oxidation or corrosion that changes H over time This change might consist of a systematic term that reflects the mean effect of the alteration process to Hg and a random or stochastic term The latter depends on the heterogeneity of the alteration process which might be more effective at some portions of space that are for example more exposed to a certain substance The mean effect of the alteration process will be ignored in the following without loss of generality A short time unit d after alteration began the microcoercivity of the particles is given by H t dt H t 5 H dt 6 where s is a function that describes how the stochastic alteration process depends on Hx For example we can assume that the relative change of H after a unit time is let say 10 In that case s 0 1H Another choice could be that of a constant s which is unrealistic because it produces negative values of H t dt Since 6 describes a random process we expect half of CODICA 5 0 reference manual 19 the grains to increase their microcoercivity while the microcoercivity of the other half decreases The coercivity distribution is now given by two peaks Fig 8b The same alteration process is allo wed to go
123. unctions of the program are illustrated step by step with an example based on real data Each step is marked by a book symbol followed by a number e g 1 Click on the book symbol or other interactive topics to jump to the example at the end of this section You also find this example in the file MAG MIX_1 Codica Examples Example nb provided with this manual You can open this file and run the example by yourself to familiarize with CODICA Change the input parameters to see how the results are affected To run the example you need to copy the folder MAG MIX_1 Codica Examples data provided with this manual onto your computer This folder contains the data used for the example The leading example of this tutorial was chosen to activate all the options of CODICA In a typical run with your data you will encounter only some of these options This tutorial provides you with all the important informations you need to run CODICA and un derstand the basic principles of the data processing Details about how CODICA performs the data processing are provided in section 6 You do not need to read that section unless you are interested in the software development or in the mathematics behind the calculation of coer civity distributions Structure of CODICA CODICA interacts with you through prompt windows When a prompt window appears you are asked to enter your answer in the prompt window If a prompt window is open and you click on
124. unit none SI unit m kg cgs unit emu Oe g Manetic field H M Hy generic unit H Manetic field H Atm m ke AT avoid this SI unit A m eq 1 eq 1 combination Magnetic induction oH Av m m kg A avoid this SI unit T eq 1 X Ho eq 1 X fy combination Sl unit mT eq 1 x 10 u9 eq 1 x 10 pug Magnetic field H avoid this avoid this emu Oe g gcs unit Oe combination combination Magnetization Magnetization Magnetization Magnetization Magnetization logarithmic scale generic unit M Slunit A m Sl unit Am kg cgs unit emu g Manetic field H M generic unit H Manetic field H A m Am kg avoid this SI unit A m eq 4 eq 4 combination Magnetic induction poH A m Am kg avoid this SI unit T eq 4 eq 4 combination SI unit mT eq 4 eq 4 Magnetic field H avoid this avoid this emu ot gcs unit Oe combination combination ARM susceptibility susceptibility susceptibility susceptibility susceptibility logarithmic scale generic unit M H Slunit none SI unit m kg cgs unit emu Oe g Manetic field H MH generic unit H Manetic field H dimensionsless m kg avoid this SI unit A m eq 4 eq 4 combination Magnetic induction oH dimensionsless m ke avoid this SI unit T eq 4 eq 4 combination SI unit mT eq 4 eq 4 Magnetic field H avoid this avoid this emu Oe g gcs unit Oe combina
125. was created by simulating the magnetization curve solid line of a logarithmic Gaussian coercivity distribution Measurements dots has been cal culated using a random number generator and assuming Gaussian errors with a standard deviation of 0 002 for the magnetization measurements and a standard deviation of 2 for the magnetic field The red curves in b and c show the coercivity distribution calculated from the simu lated measurements in a using the software Origin and magnetic field mT CODICA respectively The black curve in both plot is the error free logarithmic Gaussian distribution used to simulate the measurements The coercivity distribution calculated using Origin was obtained by first order nume rical differentiation and subsequent 5 points FFT low pass filtering FFT low pass filters based on a different number of points gave worse results Notice the distortions at low fields in b a 1 0 magnetization b 1 24 origin 5pt FFT magnetization 1 10 100 magnetic field mT A magnetization 1 10 100 magnetic field mT CODICA 5 0 reference manual 8 How does CODICA work The method used by CODICA is based on three fundamental properties that characterize any magnetization curve 1 The curve is monotonic it always either increases or decreases over the entire range of fields 2 The curve has a horizontal asymptote that is approached but not reached at
126. with one SGG function The residuals curve has 5 zero crosses in the range considered for fitting and GECA assumes 6 degrees of freedom for the y distribution The confidence limits of x l are 0 27 and 2 1 while y 1 8 8 for that model which is rejected In Fig 7b two SGG functions are used for the component analysis Now l 12 and the confi dence limits of 1 are 0 44 and 1 8 With y 1 1 5 this model is acceptable With four SGG functions Fig 7c 18 and the confidence limits of 1 are 0 52 and 1 6 while 7 1 0 37 for that model which is rejected GECA 2 1 reference manual 14 A program example In 1 lt lt Utilities Geca Load the program GECA v 2 1 for Mathematica 5 0 and later versions Distributed with the package MAG MIX release 1 04 04 2005 Copyright 2005 by Ramon Egli All rights reserved In 2 Geca Start the program Data from file Enter file name C users ramon papers fitting WDKarm slog Checking the coercivity distribution Confidence limits of the coercivity distribution Plot the distribution Total magnetization 1 47 1 5 Coercivity distribution is significant between 0 5 and 2 474 Fitting is performed in the range between 0 1957 and 2 396 Set the fitting range GECA 2 1 reference manual 15 Enter initial parameters 1 Initial distribution parameters 1 al 1 5 1 mi 1 7 sl 0 6 qi 0 5 ol 2 2 Is 0 O 0 O Optimizing the distribution pa
127. xed holder both during field application and measurement 5 Time effects Time effects are a very important and often disregarded error source during the measurement of magnetization curves All samples regardless of their composition have a time dependent magnetization known as magnetic viscosity and a time dependent coercivity which is related to N el s concept of fluctuation field Furthermore the effectiveness of the AF demagnetization depends on the decay rate of the alternating field Thus time effects influence the shape of a coercivity distribution Fig 7 Hence a precise and constant timing of both the applied field and the interval between field application and measurement is necessary to obtain CODICA 5 0 reference manual 15 good magnetization curves Precise timing is generally fulfilled by automated measurements If you perform manual measurements you should always apply the field for let say 2 seconds and wait a precise amount of time before measuring Demagnetization curves need an additional consideration Immediately after the magnetization is acquired there is often a significant vis cous decay We recommend waiting long enough before starting with demagnetization measu rements since the viscous decay of the sample increases the initial gradient of the demagneti zation curve which is erroneously interpreted as a low coercivity component _ v Fig 7 Calculated effects of thermal activation effects on the
128. y small steps are required at small fields and large steps at large fields However the ideal choice of field values depend also on the error affecting the applied field Generally a precise control of the field coil become increasingly difficult at large fields Accor dingly more measurement are required at high fields to compensate for this effect Many auto mated measurement system use a constant field increment which compensates for the larger error at high fields If the distribution of field values becomes too irregular when plotted after the scale transformation defined by Pym large holes may result in the scaled magnetization curve To avoid this situation CODICA compares the scale transformation with the distribution of field values and finds a compromise between both The results of the operations described above is summarized in a table that reports the mean misfit between the scaled magnetization curve and a reference hyperbolic function used to eva luate the symmetry The misfit is expressed as of the amplitude of the magnetization curve for 1 the original field scale 2 the scale for the most equally spaced fields 3 the scale for the most symmetric curve 4 the scale suggested by CODICA which is a compromise between 2 and 3 CODICA uses the suggested field scale to fit the magnetization curve with an asymmetric hyper bolic function see section 6 3 for more details The shape of this function is controlled by
129. zation curve Before analyzing the magnetization curve CODICA evaluates its general properties such as the initial and the final magnetization the median field and the 75 quantiles These parameters are necessary for further processing of the data CODICA prints a summary of these properties The information reported in this summary does not affect the final result of the program gt Digital rounding Digital rounding effects occur when the digitalization of measurements by the acquisition software produces errors that are larger than the measurement error of the mag netometer The resulting magnetization curve contains characteristic steps that are recognized by CODICA Fig 5 A warning message is displayed together with a list of points where digital rounding has been detected At places where steps have been identified CODICA reorganizes the measurements in order to minimize the negative consequences of digital rounding This is done by taking the arithmetic mean of the two measurements across a step produced by digital rounding and by discarding all the other measurements between two or more such steps By doing so the maximum error introduced by rounding to the n th digit is reduced from 10 to the half In case of large amounts of data produced by an automatic measurement system some small jumps in the magnetization curve can occur by chance CODICA might confuse them with digital rounding effects without negative consequences on t
130. zation curves coercivity distributions and component analysis MAG MIX will be updated periodically A profound revision of GECA that takes into account recent progess in understanding coercivity distributions is already plan ned Other programs for the automated analysis of a large number of similar sam ples such as those collected from a sediment profile will be added in the future Your feedback is important to improve the MAG MIX programs If you encounter problems in using the programs or if you have suggestions please write me eglixO0O7 umn edu Many suggestions of early users have been already taken into consideration in the present version of CODICA Ramon Egli Minneapolis april 4 2005 CODICA COercivity Distribution CAlculator Version 5 0 User s manual CODICA 5 0 reference manual 1 Preface to CODICA 5 0 The previous versions of CODICA have been used to analyze magnetization curves of the most different materials measured with various instruments This program was first conceived in 2000 as a particular filter for the analysis of natural sediment samples By that time the knowledge related to the measurement of magnetization curves for the purpose of component analysis was based on the seminal papers of Robertson and France 1994 and Kruiver et al 2001 Since then a few hundred sediment samples have been measured in detail and analyzed with CODICA The experience collected led to interesting conc
131. zation or demagnetization curves but not hysteresis loops Ideally the magnetization curve starts at a zero field and the field increases decreases monotonically to a maximum mini mum value Some automated measurement systems merge different sets of data e g an IRM acquisition curve followed by a DC demagnetization CODICA selects all data that define the first magnetization curve encountered in the data file If you want to analyze the successive magnetization curves you have to save them in a separated file The field increment steps of automated systems may be smaller than the error of the applied field in such cases the fields are not strictly monotonic but have nevertheless a monotonic trend CODICA stops to read the data as soon as the increasing decreasing trend of the applied field is inverted Because of measu rement errors and other instrumental effects initial values of the field may have the wrong sign e g small negative fields instead of zero if the nominal applied field is positive CODICA skip these fields automatically A magnetization curve must contain at least 9 measurements in order to be analyzed by CODICA and at least 20 measurements are recommended If the checking pro cedure fails a critical error message is displayed and the program is stopped Otherwise the measured magnetization curve is plotted CODICA 5 0 reference manual 31 Multiple measurements Multiple measurements are sequences of measurements cha

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