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1. 50 Appendix K Second vertical and horizontal xy gradient 2 D Fourier transform analysis 2 D Fourier transform analysis Inverse data Inverse data 1830 1830 1820 1820 gt 1810 gt 1810 1800 1800 1790 1790 2840 2850 2860 2870 2880 2840 2850 2860 2870 2880 A xX 15 6 4 13 5 2 2 6 Field Field Appendix L Horizontal gradient and total gradient analytical signal 2 D Fourier transform analysis 2 D Fourier transform analysis Inverse data Inverse data 1830 1830 1820 1820 gt 1810 e 1810 1800 1800 1790 1790 2840 2850 2860 2870 2880 2840 2850 2860 2870 2880 Le x 0 5 10 15 0 6 11 16 Field Field 51 Appendix M Tilt derivative and theta map 2 D Fourier transform analysis 2 D Fourier transform analysis Inverse data Inverse data 1830 1830 1820 1820 gt 1810 gt 1810 1800 1800 1790 1790 2840 2850 2860 2870 2880 2840 2850 2860 2870 2880 A x 89 29 30 90 0 30 60 90 Field Field Appendix N Gravity potential and pseudo magnetic field k 0 01 SI Ag 0 2 g cm and To 52000 nT 2 D Fourier transform analysis 2 D Fourier transform analysis Inverse data Inverse data 1830 1830 1820 1820 gt 1810 gt 1810 1800 1800 1790 1790 2840 2850 2860 2870 2880 2840 2850 2860 2870 2880 A 248 67 115 296 458 161 136 433 Field Field 52 Appendix O Sun shading and general derivative Elev 75 deg and Azim 10 deg 2 D Fourier transform analysis Inverse data 1830 1820 gt 1810 1
2. d H Ze Ki Bi Pi H Ca Wi ei k G ess PELI Pid aset Figure 2 Schematic view of a cut off rings and b sectors of 2 D filters and the smooth bell shape filter function of c low pass d high pass and e band pass filters 4 6 1 Low pass and high pass filtering In low pass filtering all data outside the outer ring high frequencies are nulled and all data inside the inner ring are preserved see Fig 2a and 2c On the contrary in high pass filtering all data inside the inner ring low frequencies are nulled and all data outside the outer ring are preserved see Fig 2d The radii of the rings are defined as wave length 1 which is the inverse of radial frequency k ky thy where k and ky are the axes of the 2 D frequency graph Note The x and y axes of the 2 D FFT plot are not scaled If sampling dx and dy 1s different in the x and y directions the Nyquist frequencies 1 e the minimum and maximum values of the axes are different In the present version of FOURPOT the frequency filtering uses the same wave length both in x and y directions The lengths of the k and k axes however are the same As a consequence circular filter rings will appear as ellipses in the 2 D FFT plot 26 4 6 2 Directional filtering Directional filtering is equal to the fk filtering commonly used in seismic and GPR data processing Elongated linear features in the data appear as linear features also in the FFT spectrum
3. 1800 1800 1790 1790 2840 2850 2860 2870 2880 2840 2850 2860 2870 2880 X X st 9 35 61 4 5 1 5 1 4 4 4 Field Field 48 Appendix G Direction filtered frequency spectrum 2 D Fourier transform analysis Transformed data Generic data Dimension 64x 64 Spacing 0 01562 x 0 01562 1 km Nyquist 0 50000 0 50000 1 km Wave length 2 00 2 00 km Direction filtered Automatic low pass filtering 0 4 0 2 0 0 0 2 0 4 Kx 2 0 0 3 1 5 3 2 log10 F Appendix H Direction filtered data and difference with the original data 2 D Fourier transform analysis 2 D Fourier transform analysis Inverse data Difference 1830 1830 1820 1820 gt 1810 gt 1810 1800 1800 1790 1790 2840 2850 2860 2870 2880 2840 2850 2860 2870 2880 X X 9 11 32 52 27 11 4 20 Field Field 49 Appendix I Upward and downward continued data h 2 km 2 D Fourier transform analysis Inverse data 1830 1820 gt 1810 1800 1790 2840 2850 2860 2870 2880 X 27 11 29 47 Field 2 D Fourier transform analysis Inverse data 1830 1820 gt 1810 1800 1790 2840 2850 2860 2870 2880 X Aii 227 56 139 Field Appendix J First vertical and horizontal x gradient 2 D Fourier transform analysis Inverse data 1830 1820 gt 1810 1800 1790 2840 2850 2860 2870 2880 az 15 5 4 14 Field 2 D Fourier transform analysis Inverse data 1830 1820 gt 1810 1800 1790 2840 2850 2860 2870 2880 X 13 5 2 10 Field
4. The slopes are reciprocal so that horizontal features parallel to x axis in the data are shown close to the vertical k axis in FFT spectrum Directional filter removes linear features from the data by nulling frequency content inside given sector see Fig 2b The directional filter is defined by two angles that correspond either a to the upper and lower limit of the sector interactive mode or b to the center and the width of the sector manual mode The angles are defined in degrees in counter clockwise direction taken from the positive E axis Because of the symmetry of the frequency spectrum the filter will be automatically mirrored around the origin Thus the user needs to define a single sector either above the k axis or left to the k axis In addition the width of the cut off range must be defined by providing a third angle 4 6 3 Interactive and manual filtering Frequency filtering can be done either a interactively using the mouse or b manually by providing the numerical values of the filter cut off ranges The selection is made when the corresponding menu item is chosen in the Edit menu When interactive mode is chosen most of the other program widgets become inactive and the mouse cursor changes from normal arrow into a cross hair above the graph area note only above the graph area The 2 D frequency spectrum is shown together with auxiliary polar coordinate grid The user is given instructions to provide first the i
5. Blakely 1995 F k eT f x dx and f x a e FE dk 2 E 1 Considering discrete 2 D data fx f x y the governing equations become N M pn 1 N IM 1 E EK A ia and reet E 2 k NM n 0 m 0 where N and M are the number of data values in x and y directions The transform is complex which means that it consists of both amplitude and phase spectrum The discrete 2 D Fourier transform is computed using fast Fourier transform FFT algorithm Claerbout 1976 More information about FFT algorithm can be found from Press et al 1988 for example The Fourier transform represents a sum of sine and cosine terms having different spatial frequencies or wave numbers ky and k that are based on data sampling d and d in x and y directions The highest spatial frequency is the so called Nyquist frequency eg max k 0 5 d The lowest frequency is based on the data coverage e g min k 0 5 max x min x Considering that the reciprocal of the spatial frequency represents wave length A 1 k zero frequency means infinite wave length 1 e constant level of the data Because of the properties of the Fourier transform symmetry linearity shift and derivative properties several computational operations can be performed in Fourier transformed frequency kx ky domain more efficiently than in the spatial x y domain For more detailed information about Fourier transform methods in potential field analysi
6. and MES k 2F U 6 and r k kyF U F z k k F U and r ee ky k F U 7 where n 1 or 2 is the degree of the gradient and i is the imaginary number e 1 The above mentioned basic gradients are also used to compute additional gravity tensor related components namely the first and second rotational invariants see Pedersen amp Rasmussen 1990 and the 2 nd Falcon gravity gradient component gx 8yy 2 15 3 3 4 Derived gradients The derived gradient operations include horizontal gradient H total gradient A sometimes also called analytical signal tilt gradient 7 maximum horizontal gradient amplitude MH also called total horizontal derivative horizontal gradient of tilt gradient A and theta map TM Blakely 1995 and Cooper amp Cowan 2006 du 2 du H JI vU 8 du du du 2 A JI dE dU 1 ES T tan SE 10 dU MH Re ka EN 11 d T d t Di EE ES TM tant IR 13 duU az The derived gradients are used to enhance small amplitude features in the data to delineate and locate geological targets and contacts and derive information about structural geology Examples of derived gradient operations are shown in Appendix L and M 3 3 5 Potentials and pseudo fields Potential fields are defined as the negative gradient of corresponding scalar potential F V dd dx d dy d dz In Fourier domain the gradients of the field were computed by simple multip
7. by the Academy of Finland The original idea was to utilize the depth analysis methods of Ruotoistenm ki 1987 After I become lecturer of applied geophysics at the University of Oulu in 2005 I started to modify FOURPOT for educational purposes Since then I ve used it teaching gravity and magnetic data processing Every now and then I ve added new features and fixed bugs to make it more and more useful The Fourier transform is based on the FFT algorithm FORK by Jon Claerbout 1976 The bi linear interpolation algorithm is adapted from Numerical Recipes Press et al 1988 Most of the Fourier operations have been documented in Blakely 1995 and Cooper and Cowan 2011 The principal directions are computed using RIDGEORIENT algorithm for fingerprint detection by Raymond Thai amp Peter Kovesi http www csse uwa edu au pk research matlabfns FingerPrints ridgeorient m Thanks to Richard Stuart for his comments on smooth frequency filtering FOURPOT is written in Fortran 90 and compiled with Intel Visual Fortran 11 1 The graphical user interface is based on the DISLIN graphics library version 10 2 http www dislin de by Helmut Michels In principle FOURPOT could be compiled and run on other operating systems Linux Mac OSX without major modifications provided that DISLIN is also installed At the moment however the source code is not made available and I do not provide active support for the software Nonetheless if you find the
8. gradient and the last column contains whatever processed data is currently active When profile data is saved the program asks if the data are saved with or without the GRABODS MAGBODS header information discussed later 4 9 2 Prism model generation Once profile data has been saved to disk one could start model based inversion to estimate for example the density contrast or magnetic susceptibility depth extent and dip angle of the geological targets giving rise to potential field anomalies The tilt gradient however contains useful information that can be used a to locate the number and the center position of the anomalous targets and b to divide the data into sub anomalies Le sections of data the inversion should use per each model prism FOURPOT uses the tilt gradient data to generate a semi automatic model for GRABODS and MAGBODS modeling and inversion software by M Pirttij rvi These programs which are still under development and have not been put to public distribution can be used to interpret single profile data using multiple dipping prism models 32 To create the initial model the user must first define the regional field 1 e the base anomaly for the profile This is made pressing Edit base button Once the interactive mode is entered the user needs to press the left mouse button over the response graph to define the knot points through which the base anomaly should go The editing mode is ended with a single click of r
9. not contain coordinates it is important to know the order in which the data are stored By default the data is stored in column wise fashion from bottom to top and from left to right The origin is always located in the bottom left corner Using generic programming notation we would have do w Ee do J 1 mp read or write f 1i j end do end do The default order can be changed so that for each column the rows will be read by giving a negative value for the number of points in x direction NP NP This works with data stored using following do loop do gt 1 mp do Ly np read or write f 1i j end do end do Furthermore if MP lt 0 the data are stored in multiple columns If NP gt 0 and MP lt 0 each line is considered to be one y columns of the mapped data as in the default case If NP lt 0 and MP lt 0 the lines are considered to be data rows as in the ASC file Using generic programming notation we would have in this case do j 1 mp read i i y Se EE end do 39 Note Because the matrix format does not contain coordinates the sampling frequencies and corresponding wave lengths shown by FOURPOT are computed after the user provides the necessary values in the dialog window that appears on the screen The input dialog is used to define the x and y coordinates of the origin and the x and y step between the matrix elements Note that if the x and or y steps are negative the axis gets reversed Moreover the dat
10. padded before FFT When Step number mode is active the user provides the number of grid nodes Nx Ny in x and y directions integer values When Step distance mode is active the user provides the distance between two grid nodes dx dy in x and y directions floating point values 11 The remaining items in the Edit menu are related to frequency filtering operations that can be performed either interactively with the mouse or manually by providing the numeric values via GUI Low pass and high pass filtering are the two most typical operations Considering 2 D FFT low pass filtering removes nullifies values outside certain radius from the origin of the amplitude spectrum see Appendices C E and F On the contrary high pass filtering removes low frequency data around and near to the origin of the spectrum Combined use of both low and high pass filtering is equivalent to band pass filtering The direction filtering can be used to remove linear features in the original data see Appendices G and H The frequency filtering will be discussed more in the chapter 4 6 3 3 Process menu The Process menu contains the items for inverse processing tasks Most of these operations can also be accessed using a quick tool button in the control panel next to the graph area Upward continuation performs upward continuation of potential field data Downward continuat Pole reduction Equator reduction dF dz gradient dF dx gradient dF dy gradi
11. peaks of the tilt gradient data The tilt gradient varies between 90 and 90 degrees Usually positive potential field anomalies are reflected as positive values of tilt gradient and its sign becomes negative between two anomalies The only input parameter needed for model generation is a threshold value which can be used to ignore small anomalies The threshold value depends on the tilt gradient data but usually the default value of 0 0 is a safe choice Reducing the threshold below zero allows smaller anomalies to be added to model generation Vice versa increasing the value rejects small anomalies and builds model only below the strongest anomalies When the model has been generated a vertical cross section of the model will appear below the response graph see Figure 4 The model generation uses the minima around the positive peaks of the tilt gradient data to define the sub areas for the inversion dotted lines through the top of the prisms in Figure 4 The width of the sub areas is then used to define the initial value for the thickness of the prism models and the anti symmetry of the sub areas with respect to the center of the bodies is used to define the initial dip angle The strike extent width of the prism models in a direction perpendicular to the profile and depth extent are based on length of the profile The density or magnetic susceptibility is based on the amplitude of the potential field data at the center location The mod
12. results erroneous or if you have suggestions for improvements please inform me 43 7 References Bhattacharyya B K 1967 Some general properties of potential fields in space and frequency domain a review Geoexploration 5 127 143 Blakely R J 1995 Potential theory in gravity and magnetic applications Cambridge Univ Press 441 p Claerbout J 1976 Fundamentals of geophysical data processing With applications to petroleum prospecting McGraw Hill Book Co Cooper G R J amp Cowan D R 2006 Enhancing potential field data using fuilters based on the local phase Computers amp Geosciences 1585 1591 Cooper G R J amp Cowan D R 2011 A generalized derivative operator for potential field data Geoph Prospecting 59 188 194 Pedersen L B and Rasmussen T M 1990 The gradient tensor of potential field anomalies some implications on data collection and data processing of maps Geophysics 55 1558 1566 Press W H Flannery B P Teukolsky S A amp Vetterling W T 1988 Numerical recipes in C The art of scientific computing Cambridge Univ Press 735 p Ruotoistenm ki T 1987 Estimation of depth to potential field sources using the Fourier amplitude spectrum Bulletin 340 Geological Survey of Finland PhD thesis 44 8 Terms of use and disclaimer The FOURPOT software is free for personal and academic use It should not be used for commercial purposes without permission and must
13. the Pick prof button the gridded and frequency filtered data is interpolated bilinear method along the profile and shown in the graph area An example of the profile response is shown in Figure 3 Unless some processing operation has already been made the profile graph shows only the unprocessed frequency filtered inverse data against the y axis on the left Processed data e g vertical gradient in Fig 3 are shown against the auxiliary y axis on the right The profile output thus allows detailed comparison of the effects of various processing operations If the user changes the view form profile graph to 2D map 1 e using Plot data Plot inverse or Plot FFT buttons one must use the Show prof button to see the profile response again 31 JUSIPeIS ZP AP 20 Unprocessed data 30 0 5 10 15 20 Distance km Figure 3 Example of the profile response The solid and dashed curve correspond to the unprocessed frequency filtered data left y axis and the first vertical derivative right y axis interpolated along the profile Once the profile has been generated the data along it can be saved into column formatted text file using File Save profile data menu item The file will contain lots of information columns 1 4 contain xyz coordinates and running profile coordinate distance columns 5 12 contain the base level discussed later unprocessed data the first xyz gradients horizontal gradient total gradient tilt
14. 2970x2100 pixels landscape A4 The fifth parameter defines the aspect ratio of the graph area for widescreen mode Aspect ratio refers to normal 4 3 screen and values less to zero indicate widescreen displays The remaining parameters are reserved for future use e The 4 th line defines the initial values for the magnetic and gravity field components used in pole reduction and pseudo field computations These parameters include 1 the inclination and 2 declination in degrees from horizontal plane and true north and 3 intensity in nano Teslas of the magnetic field 4 the density contrast in grams per cubic centimeter and 5 susceptibility dimensionless in SI units The extra parameters are reserved for future use e The following block of text lines define various text labels used in the graphs The maximum length of each text label 1s 70 characters e The last block of text lines define the labels of the frequency domain operations shown in the graph max 70 characters The format of the DIS file is likely to change in the future Also note that the character caret is used to define superscripts exponents the _ character underscore is used to generate subscripts and the character dollar is used to move the text back to the baseline for example Log 10 F 42 6 Additional information FOURPOT got started in 2003 when I worked at the Geological Survey of Finland for the 3 D crustal model project funded
15. 6SL06GETOL 4 35469 TOSO GLO O SeS5eooLETOL E ZOO sO EE EE sO E The header line at the first line defines the number of lines NOP and the index number of the columns that refer to the x and y coordinates ICO1 ICO2 and the data ICO3 The file can contain multiple columns of which only one will be read FOURPOT ignores empty lines and lines starting with 1 L or characters Thus provided that NOP is large enough and ICO1 ICO2 and ICO3 refer to correct column indices FOURPOT can also read Geosoft XY Z files X YZ illustrated below 5000 3 2 4 Test data LINE 1 OVO O00 On6 O65 06RFOL On 33459 LOSU 000 SO EE EPUL Gradara 205 90 0 00 S OVS365255ER 501 0 63474 LINE 2 0 00 EE EE O 81L6STUGEPOL O4e7543 9 LOr0i0 EE bk EREM N54 SCH HE ER lh 69255E401 0 55474 57 If the first line starts with a comment character either or the user must provide the relevant parameters NOP ICO1 ICO2 ICO3 interactively using an input dialog that appears The example below illustrates the generic column file format x y zl z2 O00 0 00 O 86651T0GET01L 0 33469 10 00 0 00 O 8558651E 01 0 44134 Z0 200 0200 OsS3205Z55E70L U 63474 5 2 ASC grid files The ArcInfo ASCII grid file ASC format a k a ESRI grid is supported because it is quite common and Golden Software Surfer supports it directly The example below illustrates the ASC format Please see the Wikipedia article http en wikipedia org wiki Esri
16. 800 1790 2840 2850 2860 2870 2880 e 3 0 0 8 1 5 3 7 Field 2 D Fourier transform analysis Inverse data 1830 1820 gt 1810 1800 1790 2840 2850 2860 2870 2880 A 1 0 0 3 0 3 1 0 Field Appendix P Sun shading and general derivative Elev 75 deg and Azim 10 deg shown transparently on the interpolated low pass filtered data 2 D Fourier transform analysis Inverse data 1830 1820 gt 1810 1800 1790 2840 2850 2860 2870 2880 X 17 9 35 61 Field 2 D Fourier transform analysis Inverse data 1830 1820 gt 1810 1800 1790 2840 2850 2860 2870 2880 A 17 9 35 61 Field 53 Appendix Q FOURPOT GUI Sun shading Elev 20 deg and Azim 145 deg shown transparently on the low pass filtered data using slightly altered color scale B 2 D Fourier anal by M Pirttij rvi 2012 File Edit Process Exit Plot data Shortcuts Upward Reset Make mat Downed Padding PoleRed Plot FFT _EquaRied Plot inverse __ Zod Plat diff asi Step x N 52 __Y grad Step p M 48 d dzdz Directions d dxdx Height units 0 d dydy Incl Elev deg 45 d dxdy Decl Azim deg 135 d dxdz Filter l d dydz ane units 0 HoriGr Ring2 units 0 TotalG Iw Activate rotation z Rotation angle deg 45 0 M MaxHGr Rotate Bes etal Reset rotation theta Iw Transparent shading SE Colors GenDeri Range Undo Al L 1 25 Center __Update ER ri Reset 0 15 an Ge Z
17. F i Im The FFT graph however shows the 10 base logarithm logio of the amplitude spectrum A F Re F Im FI The graph is centered so that the origin k k 0 is located at the center and the horizontal and vertical axes range between the wave numbers Ev kny related to the negative and positive Nyquist frequencies fx fry the values of which are shown in the information text next to the graph The lowest frequencies largest wave lengths widest data variations are located in the middle of the graph and the highest frequencies shortest wave lengths are located far from the origin Continuous linear features appear as linear features in the FFT spectrum as well the angle of slope is inversed though 4 5 Inverse FFT and difference Once the FFT has been computed and the 2 D frequency spectrum is available one can perform non automatic frequency filtering and the various frequency domain processing tasks The Plot inverse button will display the current inverse results The Plot diff button will display the difference between the original gridded data and the inverse data If automatic low pass filtering is not active the Plot inverse will show the inverse of the original FFT and the difference will be very close to zero If automatic low pass filtering 1s active the inverse results show a smoothed version of the original map and the difference will show the high pass filtered parts of the data When processing ta
18. FOURPOT Processing and analysis of potential field data using 2 D Fourier transform User s guide to version 1 3 G Markku Pirttij rvi 2012 University of Oulu Department of Physics Copyright 2012 M Pirttyarvi Table of contents EE rege ele EE 5 Z Fabtech RE EE 7 E EE 8 Falket Ler 0 Ol dL Renee eee Nee Pome ee vat ae noc ne Oe ets ee a ee ee eve ee ee 8 EE 9 Di i POCESS INCU EE 12 3 3 1 Upward and downward continuation cccccesseesseccececceeeeesseeeceeeeesaaeeeseeeeeees 13 535 2 Pole Ald COUALOL TECUC HONS E 14 EE 15 3 3 4 Deri Ved Ee tS resres aaa 16 WE Poena ANG e GE Te ge 16 I OFM TN Eh hs WO E 17 3 3 1 Sun shading and generalized Ceri Vati VC waveacccucdcndsadaveadectnicetilaveacedectindaaavaateads 18 SE ONS IMS NU sheer cei a E dee saaa tues nmteatnccnicepuaetedsersanadtunnaamtentncestanemns 19 Die KAU TIC MU EE 20 AU SIS ANSTO SPAIN E 21 Abe IRCAGING I We RE 21 AED cc MUMS pV Veh UN OL niet ls cat A E cso end ealeiea tele Ren Neate abe 22 dor AACA and Tape EE 22 4A FOUE EE 24 4 5 Inverse Fourier transform and difference ccc eeeeeeseseseeeeeececeececceceeeaaaeeeaeeseeeeeees 24 EO e ee d EEN 29 4 6 1 Low pass and high pass filtering enoeesssooeenesssssssseeerssssssssseeresssssssseerrssssss 26 4 022 Direcional EE pa 4 6 3 Interactive and manual filtering cc eeecccecccceeceeeeeesesecceeeeeeaaeeeeseeeeeeeeaaas 27 EE A tomatic low pass Atern EE 28 4 Ts ROt
19. FOURPOT folder to a preferred place eg C TOOLS To be able to start the program from a shortcut that locates in a different directory or from the desktop you should move or copy the DISDLL DLL file into the WINDOWS SYSTEM or SYSTEM32 folder or someplace else defined in the system PATH statement Control Panel System Advanced Environment parameters If you replace the EXE files of a previous version please remember also to replace existing DISLL DLL and FOURPOT DIS files 3 Menu items 3 1 File menu The File menu contains following items Read data dat read in irregularly spaced data in default data format Read grid asc read in regularly spaced data in ASC ascii grid format Read matrix mat read in regularly spaced data in matrix format Save current data dat save the data of current view in general xyz column format Save current grid asc save the data of current view in a ASC grid format Save current matrix mat save the data of current view in a matrix format Save ridge data save the discrete data points derived from ridge analysis Save profile data save the data interpolated along a user defined profile Save model file save the initial GRABODS MAGBODS model file Read BNA map read an overlay map file in Atlas BNA file format Read disp read new graph parameters from a DIS file Save Graph as PS save the current graph in Adobe s Postscript format Save Graph as EPS save the current graph in Adobe s Encapsula
20. _grid for the details of the file format nceols 141 nrows 241 xllcorner 3097500 yllcorner 6597500 cellsize 5000 nodata value 1 7014100000000001E 038 54 J804520061 652583 51 9764 496971 S171 49 685173416514031 AG ws OD TZOUZ O04 SEO rr e Note The ASC format requires square grid cells dx dy However FOURPOT saves the data in ASC file even if the x and y steps are not equal In this case cellsize in the header corresponds to grid step in x direction and the user needs to manipulate the y coordinates to get them right The correct y coordinates are computed as y y yo dy t yo where yo is the y coordinate of the origin yllcorner and dy dy dx is the ratio of the true y step with respect to the x step Note also that the data are stored as rows from top to bottom but the header defines the x and y coordinates of the lower left corner 5 3 Regularly gridded matrix files The matrix format MAT assumes that the data that are already regularly sampled and stored without x and y coordinates The matrix format is provided for better functionality with 38 programs such as Matlab and Maple The format of a matrix file DAT is illustrated in the example below 64 64 e 4 3 LOU BRO A433 20EF09 SUAZ 7009 The header line defines the number of data points in x and y directions NP N and MP M The following NOP NP x MP lines contain the actual data values is a single column Since the matrix file does
21. a are mirrored in x or y direction if the last two parameters in the dialog are set to one instead of zero Because normally the origin is considered to be in the lower left corner the negative steps and mirroring allow redefining the order in which the data are stored 5 4 Graph parameters Several graph parameters and text strings can be changed by manually editing the FOURPOT DIS file with a normal text editor This allows the users to translate localize the graphs to suit one s needs better Note that the format of the FOURPOT DIS file is important If it becomes corrupted FOURPOT is likely to crash when started In this case one should delete the file and a new one with default values will be generated automatically next time the program is started The file format is illustrated below Fourpot ver 1 20 parameter file 36 32 20 0 0 0 2 1 1 0 0 0 O 350 500 0790 0790 0480 0 00 0 0 0 0 Sg EEN EN Et 102500 DEER E E 2 D Fourier transform analysis Original data Interpolated data Padded data Fourier transform Inverse data Difference Radial ave spectre Ky Log 105 F Kar Log F Log F C Magnetic data 40 Gravity data Undefined data Dimension EST Nyquist Wave length Rad spectre depths Tensor rotation Rotation Pectin orations Elev amp Azimuth Distance Depth Low Pass filtered High Pass filtered Direction filtered Upward continued Downward continued Reduced to pole
22. accustomed with interactive filtering one should learn manual number based filtering which is more accurate than interactive filtering Manual filtering is made by providing the wave lengths in units of dimension of the filter rings low pass and high pass filtering in the input dialog that appears after the filtering is initiated from the Edit menu 4 6 4 Automatic low pass filtering The results of most frequency domain filtering operations especially the second degree gradients are strongly affected by the smoothness and continuity of the data To circumvent computational artifacts in the inverse transformed data it is often necessary to perform low pass filtering before any other processing operations Automatic LP filter item enables disables automatic low pass filtering To activate automatic low pass filtering the user needs to provide the radii of the filter rings manually Note that in this case the inner and outer radii are defined as a frequencies normalized by the maximum Nyquist frequency 1 e a value between 0 and 1 in both x and y directions not as wave lengths Thus low pass filter with default values 0 5 and 0 67 removes frequencies greater than two_ thirds of Nyquist frequency 0 67 0 5 dx or wave lengths less than 3 dx also in y direction if dx lt dy Automatic values are used if both values are zero The automatic low pass filtering is disabled if the input values are omitted totally 1 e a blank line is given a
23. ards southern hemisphere Declination D 180 is taken from the direction of positive y axis geographical north and it is positive in clock wise orientation See chapter 4 7 for information about rotating the equator reduced data 14 3 3 3 Basic gradients Vertical dU dz and horizontal gradients dU dx and dU dy are computed automatically after FFT and kept in compute memory so that other derived gradient components can be computed faster Also second degree gradients are computed automatically and kept in memory so that they can be rotated horizontally faster Here x and y directions represents the horizontal axis East and vertical axis North of the mapped data The vertical gradients dU dz for magnetic data and d U dz for gravity data are particularly useful in sharpening the anomalies below the anomalous sources The less common gradient components d U dxdy d U dxdz d U dydz correspond to the off diagonal components of the gravity tensor 8xy 2x and gyz provided that they are computed from the gravity potential Examples of gradient operations are shown in Appendices J and K In frequency domain the Fourier transforms of the basic gradients F d U dx are computed by multiplying the Fourier transform of the field F U with wave numbers kx ky and k The gradient operations can be formulated as Blakely 1995 WE TI ik F U F ikyF U and F 2 k F U 5 r 4 eee NES Z kyF U
24. ata by a constant height level above the surface of the earth or the plane of measurements It is used to estimate the large scale or regional low frequency or long wave length trends of the data To some degree upward continuation is equivalent to low pass filtering since it removes high frequency contents of the data On the contrary downward continuation shifts the data below the plane of measurements e g inside the earth It is used to enhance the high frequency content of the data and to estimate the depth to the top of the targets The upward and downward continuation can be formulated as Blakely 1995 F U F U x e 7 and F U4 F U xen 3 where F U F U and F U4 are the Fourier transforms of the potential field U upward continued field U and downward continued field U4 Az gt O is the elevation difference and k ky thy 1 is the radial wave number The transformed fields are then computed by taking the inverse Fourier transform of F U and F Ug Note that downward continuation is not a stable operation If the plane of continuation is located below the actual potential field sources the results become erratic Laplace s equation does not hold anymore Examples of upward and downward continuation are shown in Appendix I 13 Note The height difference or elevation Az used in upward or downward continuation is read from the Height text field in the control panel left to the graph area The value of
25. ation ANG transparent a EE 29 4 8 Color mapping range center and level 30 B29 Prone Mode re TEE 31 AD een le EE 31 AO 2 Prism Odell Ee EE a2 4 10 Radial amplitude SEET 34 E ces Ve Ts cere meee E eet weer eta eer renner ene eee E ee T E eee eer ree rece ee Del Columa lomaned Cala TNS vocectitat ed vehureasnssicaatets E eae ILAS CTI E UE 5 5 Regularly Cridded Matrix Ee eebe Stee orca KENE EE 6 Additonal EE ER ME BEINC E E A E A E E E A A E TT Se Torms Of USE and EE 9 Contact IMNTOPMANON EE Appendices A HR Keywords Potential fields Fourier transformation Frequency filtering Data processing List of Appendices Appendix A FOURPOT GUI after data input and default interpolation Appendix B Data map after padding with shift and tapering Appendix C 2 D frequency amplitude spectrum Appendix D Difference between original data and inverse transformed data Appendix E Low pass filtered frequency spectrum Appendix F Low pass filtered data and difference with the original data Appendix G Direction filtered frequency spectrum Appendix H Direction filtered data and difference with the original data Appendix I Upward and downward continued data Appendix J First vertical and horizontal x gradient Appendix K Second vertical and horizontal xy gradient Appendix L Horizontal gradient and total gradient Appendix M Tilt derivative and theta map Appendix N Gravity potential and pseudo magnetic field Appendi
26. dF dz gradient dE dx gradient dF dy gradient aQ 25F dAz 2S gradient AS 25F Ox 2S gradient G Z2SP ay 25 Gradient d 2SF dxdy gradient d 2SF dydz gradient d 2SF dydz gradient Horizontal gradient Total gradient Tilt gradient Pseudo gravity Pseudo magnetic Gravity potential Magnetic potential Theta map Fee invariant l Rots invariant 2 Reduced to equator d ZS PAY e 2R PZ Maxe hOra zontal gradient Hilbert transform Sun shading General derivative Unprocessed data e The first line is used as a header to identify the file and version number e The 1 st relevant line defines three character height values 1 main title 2 axes labels and 3 the plot information text The other two values are reserved for future use e The 2 nd line defines some integer valued options 1 the screen mode 0 1 between the normal 4 3 aspect ratio and wide screen mode 2 the color scale 0 6 3 padding mode 0 3 4 tapering shift modes 0 1 5 contour vs image map 0 1 6 automatic low pass 41 filtering 0 1 and 7 step number vs step distance 0 1 The extra parameters are reserved for future use e The 3 rd line defines the x horizontal and y vertical position of the origin of the main graph in pixels from the bottom left corner of the page integers and the length of the x and y axis relative to the size of the origin subtracted width and height of the plot area The total size of the plot area is always
27. dded values are made such that they prevent rapid amplitude changes at the borders of the data area This means that the data and their derivative are made more or less continuous across the original data area and padded margins Together padding and tapering effectively remove so called Gibbs phenomenon as well as ringing and other artifacts in the inverse transformed data Padding will be discussed more in chapter 4 3 FOURPOT does not care what kind of data it operates on Strictly speaking pole reduction and pseudo gravity for example should be applied to magnetic data only For teaching and testing purposes the Data type menu item allows defining the actual data type undefined magnetic or gravity In practice this has effect only on the information text next to the graph the scaling of the radial amplitude spectrum see chapter 4 9 and the proper selection of the output model files GRABODS vs MAGBODS The Miscellaneous submenu contains following items Zoom Clip area select GUI directed or manual a rectangular area from the data Contour Image change between contour map and image pixel map modes Show Hide points show hide the original data points m original data view Meters Kilometers swap dimension labels between meters and kilometers Unit conversion scale distances units with user defined values Data conversion scale data values with user defined value Step number distance swap gridding between steps number or step di
28. de item in Edit menu The padding modes are a Zeros which actually uses for padding the median of few selected data points instead of zeros b Linear extrapolation which will extend with the outmost data point c Mean value based which will use the nearest points inside some increasing search radius and d Gradient based which uses the mean value and the derivative of the field to extend the data to the padding zone The Shift yes no option is used to define if padding is added only to the top and to the right of original data area or if padding is made all around the data area which looks like the original data area were shifted Note The best results are usually obtained using gradient based padding with shift The other padding options are preserved primarily for testing and teaching purposes so that the users can see the effect and importance of padding and tapering 23 4 4 Fast Fourier transform After sampling interpolation and padding the user must press the Plot FFT button to compute the FFT and to see the 2 D frequency spectrum see Appendix C When Plot FFT button is pressed if valid inverse results do not exist the automatic low pass filtering if activated is also made and the inverse FFT is computed and inverse results and the difference between original interpolated data and inverse transformed data are computed Note that the true frequency spectrum F is complex having both real and imaginary parts F Re
29. ed to find the direction or directions that enhance wanted features in the data Remember however that some gradient components and derived gradients are invariant to horizontal rotation Also note that some inverse results pole and equator reduction potentials and pseudo fields and Hilbert transform are not shown above the original data transparently Instead the inverse results are shown transparently on top of themselves Although this may sound unnecessary it often improves the visibility of data maxima or minima depending on the current color scale 4 8 Color mapping range center and levels Range and Center scale widgets can be used to change the color mapping of the data maps and 2 D frequency spectrum Levels widget is used to increase or decrease the number of contour levels default is 21 By default the color scale e g rainbow or reverse rainbow is evenly distributed between the minimum and the maximum data values of the current map Range scale widget changes the width of the color scale between 5 and 125 of the original minimum and maximum values Decreasing the value is useful if the maximum data value of the map is larger than the mean data level due to outlier for example Increasing its value is useful if the maps get saturated at their minimum or maximum limit values Center scale widget changes the middle point of the color scale between 85 and 85 of the current data range Increasing the center value
30. el can be saved into MAGBODS or GRABODS input file format using File Save model file menu item For this purpose the data type needs to be set accordingly using Edit Data type sub menu The user should also take care to define the spatial dimension correctly using Miscellaneous Meters Kilometers menu item When the model is saved the user is given an option to save the profile data as discussed in chapter 4 9 1 4 10 Radial amplitude spectrum The items in the Tools Radial spectrum submenu become operational when the 2D Fourier spectrum is the active graph Show spectrum item shows the amplitude spectrum as a function of radial wave number k The radial ampl7itude is computed as the mean of the 2 D Fourier amplitude spectrum A IEF Rei Fr fm Fi g along rings with radius k ky k 1 S centered on the origin k k 0 The amplitude spectrum has traditionally been used to 34 estimate the depth to the top of the potential field sources This is accomplished by fitting linear lines to the decaying amplitude curve on a semi logarithmic scale An example is shown in Figure 5 Dimension 64x 64 5 Spacing 1 000 x 1 000 km Rad spectre depths Slope 2 437 km 4 Slope 0 746 km log F Kr Figure 5 Radial amplitude spectrum with two manually fitted lines The slopes of the lines give estimates for the depth of the potential field sources The spectrum is related to the gravity data shown in the Appendices In frequenc
31. ent d F d gradient d Flax gradient Fiche gradient d F dxdy gradient d F dxdz gradient d F dydz gradient el F dx d F d 2 Rot invariant Rot invariant 2 Hori gradient Total gradient Tilt gradient performs downward continuation performs reduction to magnetic pole performs reduction to magnetic equator vertical gradient of the data horizontal x gradient of the data horizontal y gradient of the data second vertical gradient of the data second horizontal x gradient of the data second horizontal y gradient of the data xy gradient of the data xz gradient of the data yz gradient of the data tensor component measured by Falcon gravity gradient system first rotational invariant gravity tensor second rotational invariant gravity tensor horizontal gradient total gradient analytic signal tilt gradient 12 Max hori grad maximum horizontal gradient amplitude Pseudo gravity pseudo gravity field from magnetic data Pseudo magnetic pseudo magnetic field from gravity data Gravity potential gravity potential from gravity field Bouguer data Magnetic potential magnetic potential from total field TMI data Hilbert transform Hilbert transform phase shift Sun shading sun shading General derivative generalized derivative operator by G F Cooper Undo process reverts back to unprocessed but frequency filtered data 3 3 1 Upward and downward continuation Upward continuation is an operation that shifts the d
32. form however the program will ask the user if the results are given as the absolute of the complex value which will yield the so called envelope function of the original function 3 3 7 Sun shading and generalized derivative Sun shading and generalized derivative are Fourier operations that can be used to enhance small and linear features in the data This is accomplished by altering the direction of illumination or gradient direction defined by two angles elevation from the vertical axis zenith and azimuth from the positive x axis They are defined by equations Cooper amp Cowan 2011 1 p sin8 q cos cos p sin Pr opta aad CDE J acot coset psie 19 J1 p2 q2 1 p q y where po cos tan g go sin tang p dU dx and q dU dy and and are angles that define the azimuth and elevation of the illumination or gradient direction and A x y is the total gradient defined in Eq 9 Examples of gravity sun shading and generalized derivative are shown in Appendix O Note The elevation and the azimuth angles in degrees of the sun shading and generalized derivative are read from the same two text field as the inclination and declination Elevation 0 180 is taken from the zenith out of the plane of the mapped data and azimuth 180 is taken from positive x axis east and it is positive in clock wise orientation See chapter 4 7 for more information about rotating the azimuth direction 18 Undo processing me
33. ft XYZ formatted files can be read as normal DAT files provided that the file header is suitable Also note any missing data values inside ASC files are replaced for data processing by the median of the special points and blanked out by white color in the map graphs See chapter 5 for more information about file formats After successful data input the program activates Plot data and Reset buttons and plots a contour or image map as seen in Appendix A If the data were read from a DAT file it is assumed to be irregularly sampled and the Make grid button is active If the data were read from an ASC file the data are already regularly sampled and Make grid is inactive until Reset button is pressed The initial grid sampling is based on the number of data values and the spacing between the first two data points in the file Depending of the current status of Step number distance option in Edit Miscellaneous submenu the number of grid nodes ON and M or the distance between nodes dx and dy will appear in Step x and Step y text fields in the left control panel 21 4 2 Interpolation All input data read from DAT files are assumed to be irregularly sampled Even if the original data were already evenly discretized it will be re interpolated before it s passed to FFT Therefore after reading in the data the user must first provide suitable values for the number of grid points N and M or grid stepping dx and dy in the Step x and Step y text f
34. height difference is always positive 3 3 2 Pole and equator reductions Unlike the gravity field the static magnetic field of a symmetric body e g vertical prism exhibits non symmetric anomaly shape because of the inclined direction of the inducing magnetic field The behavior of Earth s magnetic field can be estimated using the IGRF model international geomagnetic reference field Pole reduction transforms magnetic field data as if it were measured on a magnetic pole where inclination is vertical This helps estimating the true dip and strike directions of the targets Since pole reduction is an unstable operation at low latitudes Eguator reduction can be performed instead These operations involve a phase transformation and they can be formulated as Blakely 1995 Din Diir Om Of F U F U x 4 Where and are functions of the direction vectors of the original magnetization m m m m and external magnetic field f f f f and O m and Dr are functions of the direction vectors of the transformed magnetization and field m and TL Unless remanent magnetization is taken into consideration m f and m f Note The inclination and the declination angles in degrees of the original magnetic field and magnetization are read from the Inc and Decl text fields in the left control panel Inclination J 90 is taken from horizontal plane and it is positive downwards northern hemisphere and negative upw
35. ields and then press the Make grid button to perform the required interpolation Note that the grid spacing dx and dy does not need to be equal in x and y directions After interpolation Make grid button becomes inactive and Plot data button is used to visualize the original automatically interpolated data To define new grid sampling one must first revert back to original automatically computed sampling using the Reset button edit new values for the Step x and Step y text fields and press the newly active Make grid button Note that normally Step x and Step y text fields show the number of nodes ON and M Only when looking at the original data Plot data button and when Step distance option Edit Miscellaneous 1s active they can show the grid node step distances dx and dy Note The interpolation discussed above is performed using the DISLIN subroutine GETMAT see DISLIN user manual This subroutine works best if the data are already regularly sampled and the new grid coincides with the original grid If the grid sampling is too dense compared to data sampling the interpolation does not give good results and a blank white map may be shown In this case the user should provide larger step values Alternatively highly irregular data should be interpolated on a regular grid using more advanced interpolation algorithm e g minimum curvature of third party software e g Golden Software Surfer 4 3 Padding and tapering Paddi
36. ight mouse button If more than two base knot points are defined cubic splines are used to compute the regional trend see Figure 4 If only one or two knot points are defined the base anomaly will be a horizontal or dipping line Existing base anomaly can be removed pressing the left mouse button once outside to the left or to the right of the graph area New knots can be added to existing base anomaly pressing the left mouse button once either above or below the profile graph and then once at the new knot location over the graph Likewise knots can be removed from the base anomaly pressing the left mouse button once over the graphs close to the selected point and then once above or below the response graph z 38 5 el Kal 18 ya 2 e 2 S 22 S S 42 62 82 ai d i F 3 S S ba eh W o 4 A 8 0 5 10 15 20 Distance km Figure 4 Example of the prism model generated using tilt gradient data The solid and dashed curves are the unprocessed frequency filtered and the tilt gradient interpolated along the profile The dotted line is the base level fitted using splines through the three base level knots The dotted horizontal line at the top of prisms is the width of the sub areas to be used in the GRABODS inversion algorithm 33 Once the base anomaly has been defined the Make mod button for the automatic prism model becomes active The number and location of the prism models is based on the positive
37. ization and field will be rotated towards north m f 0 1 0 based to the given value of declination However the direction of the transformed field reduced to equator can be rotated Thus for example the reduced field can be computed in EW direction using an angle of 90 m f 1 0 0 When performing sun shading and generalized derivative the elevation and azimuth angles are read from the same two text fields as the inclination and declination used in pole and equator reduction and pseudo field computations Instead of editing the azimuth angle manually the Rotate button allows fast rotation of the direction of sun shading and generalized derivative Resetting makes the azimuth zero along x axis Most of the other gradient operators as well as sun shading and generalized derivative are more informative when displayed using greyscale colors Activating the Transparent shading check box allows displaying the inverse results as greyscale colors transparently below the 29 original unprocessed but gridded data that uses the current user selected color scale When activated the program asks the user for the strength of transparency Transparency ranges between O fully transparent and 1 fully opaque Negative value of transparency will apply inverse greyscale Examples of transparent color shading are shown in Appendix P for sun shading and generalized derivative Together with transparent shading the rotation can be us
38. lication Eqs 5 7 Thus the potential can be obtained from the vertical component of the field by dividing its Fourier transform by the radial wave number k In case of the magnetic potential the direction of the inducing field must be taken into account Furthermore the gravity and magnetic potentials and hence the fields of an anomalous target with the same dimensions but different petrophysical properties density and magnetic susceptibility are interrelated via Poisson s relation Thus it is possible to compute so called 16 pseudo gravity field from the measured anomalous magnetic data and anomalous pseudo magnetic field from gravity data The pseudo gravity and magnetic fields g and ATs and the gravity and magnetic potentials and V are computed from following equations Blakely 1995 F 9ps x Elan and F ATps F On 0 x Flg 04 Fld 7 x Flo and F V SR x F AT 15 where AT is the anomalous magnetic field reduced to the pole g is the vertical component of the gravity field and A is a function that depends on the density contrast and the intensity of magnetization which in turn is a function of magnetic susceptibility and intensity of the inducing magnetic field The values of inclination and declination defined in GUI window are used for the direction of magnetization and magnetic field 0 The program asks the user for the values of density contrast g cm gt susceptibility SI and i
39. mit values by clicking the right mouse button without any left mouse button clicks or by giving a blank line in the manual input dialog The information text next to the graph may show the dimensions incorrectly unless the Meters Kilometers menu item is utilized Although knowledge about the spatial dimensions is usually not required the pseudo gravity and pseudo magnetic field need to know the real dimensions for the amplitude of the transformed field to be more or less truthful Unit conversion can be used for example to scale all distances from miles into kilometers or from meters to kilometers Likewise Data conversion can be used for example to scale the data from mgal m into E tv s mgal km It can also be used to reverse the sign of the data e g the negative gradient of the potential Both in unit and data conversion the program shows an input dialog where the user provides a single character for a mathematical operation addition subtraction or x multiplication and or division and the numerical value For example string 100 will multiply all data by one hundred The operator has to be the first character on the given line An empty line or zero value cancels the operation Step number distance is a mode switch for grid sampling that changes the meaning of Step x and Step y text fields in the left control panel The original data are considered to be irregularly sampled and must me gridded and
40. ng and tapering are essential parts of successful Fourier transform processing The Padding button will automatically add extra columns and rows around the interpolated data matrix so that the grid dimensions N and M will become even powers of two e g 64 128 256 512 etc that is required by the FFT algorithm The padding can be performed with or without tapering Tapering means that artificial values are added to the padded area so that the 22 continuity of the data is preserved at the edges See Fig 1 and Appendix B for examples of padding a normal padding b shifted padding b padding tapering padded values tapered values padded values original data original data area area original data area padded values tapered values 0 0 N N 00 N OO N Figure 1 Schematic view of a padding without shift b padding with shift and c shifted padding with tapering The new dimensions N and M are powers of two Sometimes the original grid dimension may already be so close to some power of two that the tapering will not ensure enough data continuity To further increase the padded dimension from 64 to 128 for example one needs to change the current values of Step X or Step Y to some number between 65 and 128 and then press the Padding button again Note that the dimensions of the grid can be decreased only using re interpolation Make grid button Padding and tapering depend on the selection made with the Padding mo
41. nner and then the outer ring by pressing the left mouse button above the graph To validate the last value and to end the editing mode one must press the right mouse button The distance from the origin determines the radius of the filter rings Likewise in directional filtering the user is instructed to provide the upper and lower angle of the filter sector and the width of the cut off range The width can be defined as an angular difference from the lower or upper sector and the filter cut off range will be adjusted symmetrically at both sides automatically Note that the ring and sector values can be given in reverse order in which case they will be automatically arranged properly Values can be omitted by pressing the right mouse button 21 instead of the left one In low pass and high pass filtering omitting the first value cancels the whole interactive editing mode Omitting the second value will make the two rings the same which means that the sine cosine bell function is replaced by a box car filter In directional filtering the upper and lower angle must always be provided but the cut off range can be omitted if the right mouse button is pressed when its value is asked for Note The filter rings and sectors will be shown by dotted lines on the frequency spectrum In addition the wave lengths corresponding to inner and outer ring and the limit angles of the sector are shown on the information text of the graph After the user gets
42. not be distributed onwards for profit If you decide to publish results computed with FOURPOT please make a reference to this user manual and FOURPOT homepage at https wiki oulu fi display mpi oulu fi 2D Fourier domain operations If you find FOURPOT useful please send me a postcard The FOURPOT software is provided as is The author MP and the University of Oulu disclaim all warranties expressed or implied with regard to this software In no event shall the author or the University of Oulu be liable for any indirect or consequential damages or any damages whatsoever resulting from loss of use data or profits arising out of or in connection with the use or performance of this software 9 Contact information Markku Pirttijarvi Department of Physics Tel 358 8 5531409 P O Box 3000 E mail markku pirttyarvi at oulu fi FIN 90014 University of Oulu URL http www egf oulu fi mpi Finland 45 Appendix A FOURPOT GUI after data input and default interpolation 5 2 D Fourier analysis ver File Edit Process Tools Exit Plot data Shortcuts eset TF LDewnwd Step AN Step AM Directions Height units Incl Eley deg Decl Azim deg 6 6 Lp ctuale rotatior ddyd Rotation angle deg EE G Colors Range ThetaM Al E SunShed 1 00 CH Center Pete 2 D Fourier transform analysis Original data 1830 1820 1810 1800 1790 2840 2850 2860 2870 2880 X Gra
43. ntensity of the inducing magnetic field nT Examples of gravity potential and pseudo magnetic field are shown in Appendix N 3 3 6 Hilbert transform Hilbert transform is a phase shift operation which is related to the concept of analytic signal It is usually performed on magnetic data because the analytic signal does not depend on the direction of magnetization Hilbert transform is closely related to pole reduction although it is much easier to compute For a 3 D case analytic signal is defined as a vector field Blakely 1995 dU dU dU Ay yz nito tiaz 17 where the real and imaginary parts are a Hilbert transform pair Note that the norm A IAI is equal to the total gradient in Eq 9 According to Blakely 1995 the Fourier transform of 1 D analytic signal can be computed as F A F U 1 sgn k 18 17 where sgn k 1 if k lt 0 and sgn k 1 if k gt 0 In practice the 2 D Hilbert transform is accomplished by multiplying the spectrum with two and nulling all items in the lower left quarter of the centered amplitude spectrum kx lt 0 and ky lt 0 When initiated the program asks the user if the transform is made for 2 D map data or 1 D trace data In the latter case the program also asks if the transform is made along x axis half spectrum kx lt 0 is nulled or y axis half spectrum ky lt 0 is nulled Note that in FOURPOT the results are returned as the real part of the inverse FFT In case of Hilbert trans
44. nu item can be used to revert back to the unprocessed status Possible low pass high pass or directional filtering operations however will not be unmade 3 4 Tools menu The Tools menu contains following items Median filter 3x3 smooth the data using a median of a moving 3x3 window Ridge tools pick continuous ridges fingerprint directions and remove them Profile Model tools pick a profile and create initial 2 5D prism model automatically Radial spectrum show the radial spectrum used for depth analysis Median filter 3x3 can be used to smooth gridded data and to remove outliers spikes from it Median filter can often greatly improve the results of many Fourier operations which assume that the data and its derivatives are continuous The Pick ridges 3x3 and Pick ridges 5x5 menu items utilize an experimental algorithm that investigates if the grid point at the center of a moving 3x3 or 5x5 window represents a continuous ridge like anomaly peak Ridge is defined by three coordinate pairs the first indicates the center of the ridge the second and the third indicate where the ridge comes from and where it goes to The ridges can be straight e g W E or SE NW or bent up to 90 degrees eq W SE or SE NE Ridge points can be saved into a column formatted text file using File Save ridge data menu item The ridge data can also be saved in Atlas BNA format Hint Ridge picking is usually applied on horizontal gradient data to locate geological c
45. ontacts When the operation is re applied on upward continued data one can estimate the dip directions of the contacts To do this one should save the results of the ridge picking made on horizontal gradient of original data in BNA format perform upward continuation apply Inverse data menu item perform the ridge picking on the upward continued data and read in the previous BNA file for comparison Pick directions is somewhat similar operation that computes the orientation or principal axis directions of the data based on the x and y gradients of the data Directions are saved as angles in degrees taken from the horizontal x axis east towards positive y axis north Both 19 ridge picking and computation of principal directions ask the user for a threshold value in percent that will ignore smaller features when increased The items in the Profile Model tools and Radial spectrum sub menus are discussed in chapters 4 9 and 4 10 3 5 Exit menu Exit menu has two items The first one restarts the whole program GUI window so that it suits better either normal 4 3 or widescreen displays When changing to widescreen mode the program asks the user for a scaling value for the aspect ratio Value 1 0 is used for 4 3 displays and values 0 6 0 8 should be used for widescreen displays OK to Exit item will close the program without any additional warnings Therefore the user should take care to save the results before exiting Errors tha
46. oom 35 SH 2 D 1830 1820 1810 1800 1790 2840 2850 A 21 12 44 77 Field Fourier transform analysis Inverse data 2860 2870 2880 Dimension 64x 64 Spacing 1 000x 1 000 km Low Pass filtered 4 00 3 00 Sun shading Elev amp Azimuth 45 amp 135 deg Appendix R Ridge picking 3x3 made on max horizontal gradient and principal directions shown on top of horizontal gradient 1830 1820 e 1810 1800 1790 2840 2 D Fourier transform analysis Inverse data 2850 2860 2870 2880 Dimension 64x 64 Spacing 1 000x 1 000 km Low Pass filtered 4 00 3 00 Max horizontal gradient 54 2 D Fourier transform analysis Inverse data 1830 1820 1810 1800 1790 2840 2850 2860 2870 2880 X Dimension 64x 64 Spacing 1 000x 1 000 km WW Low Pass filtered 4 00 3 00 Horizontal gradient 0 5 10 15 Field
47. played on the information text on the right side of the graph Multiple lines can be added to the same graph The Delete last line item removes the most recent line Note Because of the cut off range the radial spectrum of low pass and high pass filtered data may look quite different from the spectrum of unfiltered data Also note that the horizontal axis starts from zero and ends either at the minimum Nyquist frequency or the outer ring of low pass filtering In other words low pass filtered part of the spectrum is not shown at all In this case a vertical dotted line indicates the position of inner cut off rings of low pass filter or high pass filter if applied 36 5 File formats Data can be read and saved in three file formats 1 column formatted data file DAT and 2 ArcInfo ASCII grid file format ASC and 3 gridded data matrix MAT When saving data into file the contents of the current graph will be saved original data cannot be saved again Thus either a the interpolated data together with the inverse transformed and processed data and their difference or b the padded and interpolated data or c the centered Fourier transform the will be saved In the first case the program asks the user if the data will be interpolated bilinear interpolation at original data locations or not 5 1 Column formatted data files The format of a DAT file is illustrated below 4096 1 2 4 H EEE O200 SOSC6
48. ring High pass filter perform GUI guided or manual high pass filtering Direction filter perform GUI guided or manual direction filtering Inverse gt data menu item makes the current inverse results the new interpolated input data for successive frequency domain operations For example tilt gradient can be computed after upward continuation or reduction to the pole without saving the UC or RTP results first in a data file Subtract inverse item allows computing the residual field as the difference between the original interpolated data and the upward continued or low pass filtered data Similarly Add inverse item can be used to add the current inverse results to the original data Although this item is not very useful it could be used to enhance high frequency content of the data for example The items in Color scale submenu are quite self explaining they change the color scale of the image and contour maps Comparable colors item imposes the same limit values on the color scales of the original and the processed data and thus makes their comparison easier Padding mode submenu is used to demonstrate different padding and tapering modes Padding is an operation where artificial or null values are added around the original data area see Appendix B The purpose of padding is to extend the data ON and M to an even power of two e g 64 128 256 512 etc as required by the FFT algorithm Tapering is an operation where the pa
49. s please see Blakely 1995 2 Installing FOURPOT The FOURPOT distribution file contains both 32 bit and 64 bit versions that can be run on a PC under Windows XP Vista 7 system and a graphics display with at least 1280x1024 pixels in size FOURPOT uses dynamic memory allocation and reserves memory for several auxiliary data matrices derived from the interpolated data Thus memory requirements may become critical when working with very large data sets Unlike the 32 bit version which is limited to 1 GB of continuous memory the 64 bit version can use all available system memory In case of large arrays however the CPU becomes critical factor because computation of the 2D FFT of matrices larger than 4096 by 4096 takes lots of time The program has simple graphical user interface GUI that is used to handle file input and output to initiate the computational operations and to visualize the data and the results The user interface and the data visualization are based on DISLIN graphics library The program requires either FOURPOT EXE and DISDLL DLL the 32 bit executable file and dynamic link library for DISLIN or FOURPOT64 EXE the standalone executable file for 64 bit Windows The distribution file FOURPOT ZIP also contains a description file _ README TXT this user s manual FOURPOT_MANU PDF and some example data files DAT To install the program simply decompress Winzip or 7Zip the distribution file and move the resulting
50. s input 28 4 7 Rotation and transparent shading Horizontal rotation around z axis can be applied to basic gradient components as well as reduction to equator sun shading and generalized derivative operations to enhance linear features of varying direction in the potential field data The Activate rotation check box must first be activated and a non zero value provided for the Rotation angle text field The Rotate button is then used to add one rotation step to the current value of total rotation Rotation is not made if the current view is not one of the abovementioned inverse results Depending on the current view the information text on the graph shows either the total rotation of 1 st degree gradient or 2 nd degree tensor component rotated field direction of equator reduction or the azimuthal direction of sun shading or generalized derivative The Reset rotation button resets the total rotation to zero When basic gradients 1 st and 2 nd degree gradients are rotated their values are updated in computer memory so that all derived components can be computed faster However except for the Gxx Gyy 2 Falcon response none of the derived gradient components depend on horizontal rotation After resetting the x and y gradients are again directed along x and y axes of the data In pole reduction the transformed field becomes vertical m f 0 0 1 and declination does not play any role Normally in equator reduction the transformed magnet
51. sks are made Process menu or quick task buttons the inverse results will be shown automatically Le without pressing Plot inverse button Once the gridding has been made the Plot data button will show the original interpolated Successive utilization of Plot inverse and Plot data buttons can be used to visualize the changes that are made to the data due to the frequency domain operations As an alternative 24 the Plot diff button can be used to visualize the difference between the original interpolated data and the inverse transformed data Note that pressing the Reset or Padding buttons will invalidate the FFT and current inverse results and the user needs to perform the frequency filtering operations again Note also that after interpolation the Make grid button becomes inactive because the N and M are affected by the padding to the next power of two To be able to re grid the original data one needs to apply the Reset button Note The Fourier operations are made on the interpolated and padded data matrix Sequential Fourier operations e g first reduction to pole and then vertical gradient can be performed using the Edit Inverse gt Data menu item In this case the data are already defined on a regular grid so the process can be continued from padding step and the Plot data button reverts back to the original data Alternatively one can save the current inverse results and read the file as new input data 4 6 Frequency fil
52. stance The minimum and maximum coordinate limits of the data matrix are based on the original data coverage The Zoom Clip area item can be used to redefine and round the coordinate limits and place the grid nodes nicely and or to select a smaller area from large map The Operation can be made either manually giving the x and y coordinates of the borders or interactively with the mouse In the former case the program shows an input dialog where the user gives the x and y limit coordinates of the area In the latter case the program enters interactive editing mode and the user selects the corner points of the data area using the mouse as described below After providing new limit values the program redraws the data map and the gridding see chapter 4 2 of the data can be made using new data limits 10 Note All interactive editing operations including the zoom clip are made in FOURPOT in following manner When editing is initiated the mouse cursor becomes a cross hair cursor above the graph area and most of the GUI widgets become inactive so that the user cannot escape the editing mode accidentally The user then defines the location of the point or points by pressing the left mouse button and exits the editing mode by pressing the right mouse button Depending on the task the program shows instructions on the screen on the first time Also note that in case of Zoom Clip area operation the user can revert back to the original li
53. t are encountered before the GUI starts up can be found from the FOURPOT ERR file When operating in GUI mode run time errors arising from improper parameter values for example are shown on the screen 20 4 Using the program When the program is started it reads graph parameters and some additional settings from the FOURPOT DIS file see chapter 5 4 for more information If this file does not exist default parameters will be assigned and the file is created automatically The program then builds up the GUI shown in Appendix A None of the program controls widgets however are active before data has been read in The data processing and analysis is performed in few successive steps that are discussed next The preliminary step is always to check that the input data file is in correct format so that it can be read in see chapter 5 on file formats After reading in the data one should apply a Make grid b Padding and c Plot FFT buttons to perform interpolation and padding and to compute the 2 D FFT and its inverse Only after the FFT has been computed one can perform frequency filtering Edit menu and other frequency domain data processing Process menu and take a look at the inverse Plot inverse and difference Plot difference 4 1 Reading in the data The first thing to do after starting up the program is to read in the data using the Read data DAT Read grid ASC or Read matrix MAT menu items Note that Geoso
54. ted PS format Save Graph as PDF save the current graph in Adobe s PDF format Save Graph as WMF save the current graph in Windows metafile format Save Graph as GIF save the current graph in GIF file format These menu items bring up the standard file selection dialog of the operating system that is used to provide the filename for I O operations All data files DAT ASC MAT are in text format see chapter 5 DISLIN cannot be used for direct output to a printer but the graphs can be saved into image files PS EPS PDF WMF GIF for printing The graphs are saved as they appear on the screen in A4 size and landscape mode see Appendix A The preferred output format is PDF or PS because their resolution is much better than that of GIF and WMF files The PDF format is particularly useful because Adobe s Acrobat Reader can be used to take bitmap snapshots at desired resolution 3 2 Edit menu The Edit menu contains following items Inverse data set current results of the inverse FFT as new initial data Subtract inverse subtract the inverse results from the original data Add inverse add the inverse results to the original data Color scale change the color scale rainbow gray scale etc Padding mode change the method used to pad and taper data Data type define type of potential field data gravity vs magnetic Miscellaneous sub menu for some additional settings Low pass filter perform GUI guided or manual low pass filte
55. tering Frequency filtering affects the current FFT spectrum on which gradients components and all other processing functions are computed This means that frequency filtered data are always passed to further processing operations without the use of nverse data menu item As a matter of fact low pass filtering is often an essential preliminary step for successful operation of the gradient filters Frequency filtering means that some parts of the spectrum are nulled Rapid changes in the spectrum however can cause oscillations in the inverse transformed data This is known as the Gibbs phenomenon Therefore instead of using rectangular or box car shaped filter functions bell shaped smoothing is applied over the cut off range of the filter In low pass and high pass filtering this means in practice that instead of a single filter ring the user must provide the radii of the inner and the outer ring of the cut off range Between the two rings half sine or half cosine function is used to define the smoothing In directional filtering the user needs to provide the direction angles of the upper and lower limit of the notch filters and the width angle of the cut off range The filter rings and sectors and the cut off ranges are illustrated in Figure 2 25 a 2 D low pass filter b 2 D directional filter nulled EE cut off values es ranges EECH preserved values Ke upper and lower sector outer ring Ge e band pass filter
56. vity data Dimension 52x 48 43 19 4 28 Field Appendix B Data map after padding with shift and tapering 2 D Fourier transform analysis Interpolated data 1840 1830 1820 2870 2880 2890 Generic data Dimension 64x 64 Spacing 1 000x 1 000 km gt 1810 1800 1790 1780 2840 2850 2860 X E 9 35 46 Appendix C 2 D frequency amplitude spectrum 2 D Fourier transform analysis Transformed data Generic data Dimension 64x 64 Spacing 0 01562 x 0 01562 1 km Nyquist 0 50000 0 50000 1 km Wave length 2 00 2 00 km 0 4 0 2 0 0 0 2 0 4 Kx 1 3 0 2 17 3 2 log10 F Appendix D Difference between original data and inverse transformed data 2 D Fourier transform analysis Difference 1830 1820 gt 1810 1800 1790 2840 2850 2860 2870 2880 X Generic data Dimension 64x 64 Unprocessed inverse data 0 015 0 006 0 003 0 011 Field x 103 Appendix E Low pass filtered frequency spectrum 2 D Fourier transform analysis Transformed data Generic data Dimension 64x 64 Spacing 0 01562 x 0 01562 1 km Nyquist 0 50000 0 50000 1 km Wave length 2 00 2 00 km Automatic low pass filtering 0 4 0 2 0 0 0 2 0 4 Kx 2 4 0 5 3 2 log10 F Appendix F Low pass filtered data and difference with the original data 2 D Fourier transform analysis 2 D Fourier transform analysis y Difference Inverse data 1820 1820 gt 1810 gt 1810
57. will emphasize small data values and increases the amount of colors at the beginning part of the color scale Decreasing the center value will emphasize large data values and increases the amount of colors at the end of the color scale 30 Note After changing any of the abovementioned scales the user must press the Update quick button to make the changes effective Because the scale widgets give slightly different results for different graphs one can reset the color mapping parameters to their defaults using the Reset quick button Note that FOURPOT was not designed to be used as a plotting program The user is advised to save the mapped results into text files and use more advanced programs like Golden Software s Surfer for publication quality plotting 4 9 Profile Model tools The six push buttons below the Levels scale widget at the bottom of the left control panel are used a to see the results of various filtering operations as a line graph along a straight user defined profile and b to create an initial model based on 3 D dipping prisms for the interpretation of the potential field data along the picked profile These tasks are also available through the Tools Profile Model tools sub menu 4 9 1 Profile plotting Pick prof button is used to define the x and y coordinates of the profile start and end This can be done either interactively with the mouse or more precisely giving the coordinates in an input dialog After applying
58. x O Sun shading and general derivative Appendix P Transparency amp sun shading and general derivative Appendix Q Sun shading shown transparently on the low pass filtered data Appendix R Ridge picking and principal directions 1 Introduction FOURPOT is a computer program for frequency domain processing and analysis of two dimensional 2 D potential field data arising particularly from geophysical gravity and magnetic field measurements The data can be regularly or irregularly sampled In the latter case the data must be first interpolated on a regular grid The frequency domain operations include Frequency filtering low pass high pass amp directed filtering Upward and downward continuation Reduction to magnetic pole and equator with rotation Let vertical and horizontal gradients d dz d dx amp d dy 2 nd degree gradients d dx d dy d dz d dxdy d dxdz amp d dydz Rotational invariants of gravity tensor and Falcon response Horiz gradient Total gradient Tilt gradient Theta map amp Max horiz gradient Pseudo magnetic field and pseudo gravimetric field ee Ch oe ee ee ek E Gravity potential and magnetic potential 10 Hilbert transform 1D or 2D 11 Sun shading and generalized derivative operator with rotation Considering a continuous function f x with continuous first derivatives the Fourier transform F k and the inverse Fourier transform can be written as
59. y domain the Fourier transform of a potential field can be formulated as F ste giving log F C h k Thus the depth A to the top of the source is related to the tangent of the line that is fitted to the linear parts of the amplitude spectrum on semi logarithmic scale The coefficient C depends on the selected data type Edit Data type for generic undefined data C 1 for gravity data C 1 k kyk and for magnetic data C 1 k k Depending on the selected data type and the data the tangents give different depth estimates that may or may not be realistic In such cases one should prefer to the un normalized spectrum C 1 Please be noted that the method used for depth determination has not been verified Refer to Bhattacharyya 1967 and Ruotoistenmaki 1987 for more information about the use of radial amplitude spectrum in depth determination Interactive line fitting is initiated using the Add new line item in the Tools Radial spectrum submenu The normal mouse pointer arrow will change into a cross hair cursor over the graph area and most of the GUI widgets become inactive The user should then select the start and end points of the line over the radial amplitude spectrum by pressing the left mouse 35 button twice In case of multiple mouse clicks the last two points are used The editing is ended by pressing the right mouse button once The graph will be redrawn with the new line on it and the slope of the line will be dis

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