TR-226 TECHNICAL REPORT SPLINE INTERPOLATION ALGORITHMS FOR TRACK-TYPE SURVEY DATA WITH APPLICATION TO THE COMPUTATION OF MEAN GRAVITY ANOMALIES DECEMBER 1970) = f W HOTS "DOCUMENT } ‘ COLLECTION , Sey ee bee oa) 3 NAVAL OCEANOGRAPHIC OFFICE WASHINGTON, D. C. 20390 hb.TR- iNy Price $1.00 AVE o> hon AUG Cubic spline interpolation is a mathematical procedure which is an analog of the draftsman’s plastic spline. The advantage ofthis | interpolation procedure over the more commonly used methods such as Lagrange lies in its ability to not only fit each given data value exactly but to maintain continuity of the first and second de- rivatives. The relative accuracy of the cubic spline interpolation procedure for generating gridded data values and estimates of mean gravity anomalies from track-type geophysical surveys is shown to be ex- cellent when applied to properly designed surveys. Techniques for interpreting the two-dimensional Fourier transform in terms of track spacing, track orientation, and down-track sampling rate are pre- sented to demonstrate the effect of these parameters on interpolation accuracy. A procedure for utilizing closed form integration of the bicubic spline surface to produce mean gravity anomalies is shown to yield accuracies comparable to the method of averaging cubic spline grid values. THOMAS M. DAVIS ANGELO L. KONTIS _ Earth Physics Branch Hydrographic Development Division Ocean Engineering Department FOREWORD The U. S. Naval Oceanographic Office, as part of its mission to provide deflection of the vertical information in ocean areas, is currently engaged in studies concerning the accuracy of computing deflections of the vertical from gravity survey data. The deflection of the vertical calculation is the last step in a process involving many stages; consequently, the overall accuracy will be dependent on the accuracy of each individual stage. This report is concerned with the interim step of determining mean anomalies for use in the Vening Meinesz formulation. Utilization of the numerical method presented in this report is expected to result in a significant improvement in the accuracy of interpolating mean anomalies from original survey data. ra Pea LZ fee ine Oe oS F, L. SLATTERY Captain, U. S. Navy Commander U. S. Naval Oceanographic Office i Hmmm ATMA Wisealy i TABLE OF CONTENTS Page INTRODUGTIOING Prete ee steticniete oe eects Sedtott. tell. WevRterieW oil ebb of 'alRleterel eine). 1 DEVELOPMENT OF THE CUBIC SPLINE ALGORITHM FOR GRIDDING TRAGCKSINPESSURVEVEDATAI St Reet.) ats siecle tele at's ee os) 6 3 THE BICUBIC SPLINE ALGORITHM FOR THE GENERATION OF NAEVAIN}. -NNIQYMVANEITLESS am 6ho.o dad. oath Oecuomoeo sidcsics oi OEon cba eontuc lomo clined 10 GENERATION OF MODELS FOR TESTING THE ACCURACY OF INITHENPOI/AATIIOIN PROXCIEDIUINIES. 4 6-6 6 5 00 OG G10 OldmO OyONS Gooua lala c 14 A. The Two-Dimensional Fast Fourier Transform as a Tool fon simulatedssunvey, Design) ay au-i ie rettcme clon -tmear-Miell fle) tai 14 B. Development of a Point-Mass Model Field. .........--. 18 COMPARISON OF GRIDDING TECHNIQUES AND THE COMPUTATION OF MEAN ANOMALIES USING MODEL DATA ..............-- Ie /\5 (Gipictehints) CP SUNS? Wretiel bg Bla. 6 oo 66 06 boo 6 00 6 21 B. Mean Gravity Anomaly Calculations. ............-. 36 EONGEUSIONS Serotec. aa CPE ee Sete d 2 se) et ones Bolte 43 REREREINGES Ara omer ete, oo Site teis steer "atl ta’ ger fhe Tate toy Mole ste) Ney Me) oo ter Rey neta 49 APPENDIX A - FORTRAN PROGRAM FOR GRIDDING AND COMPUTATION OF MEAN ANOMALIES USING THE CUBIC SPLINE. . A-1 APPENDIX B - FORTRAN PROGRAM FOR GENERATING MEAN ANOMALIES USINGMRE BIECUBIEG SEINE Bins cr tik crepe ares cl B-1 APPENDIX C - FORTRAN PROGRAM FOR POINT-MASS MODEL DATA. C-1 APPENDIX D - FORTRAN PROGRAM FOR THE TWO-DIMENSIONAL FASI FOURIERMIRAINSEORM@s arate i ciematis@cccvene to tctle) fume fomiclitels. lo D-1 WZ. 13. FIGURES Page The Rs Grid of Data Values for Input into the Bicubic Spline Algorithm . 11 The: Functionig Geny)" 0 7a Mein: ouctsed ae cUbotod sir oulaitn, chcolhe Ve euy. FoMene 16 Geometry for Computing the Gravitational Potential Due toa MM IMCS GO 6 Oh Oua 0 oldeGlacc OO 05 OG DOO oOo oo 0 O06 18 Contour Chart of the Gravity Anomalies Generated by 16 Point Masses Wath) Contour Infenyalaonmoamalsiey clan. aemen nen me ain oie antennae 20 . The Two-Dimensional Fourier Transform of the Entire 80 x 80 Minute Model Field with One Data Interval Equal to One Minute and Gontourssin)|Nenmallized| Units eee) on eee tne nnn nme 22 Model 1 - Simulated Survey Tracks Based on the Fourier Transform of the 80 x 80 Minute Model Field. Nominal Track Spacing and Sampllingsinteral#Equallatoy Simmer ears semi n apenicn rarer aes 23 Contour Chart of Model-1 Point-Value Residuals (Spline Interpolation vs. True Value) with a Contour Interval of O.5mgl ..........-. 25 Histogram for Model-1 Spline Point-Value Residuals. .......... 26 . Contours of the 12° Least-Squares Polynomial Surface Generated from Model-1 Data. Contour Interval isS5mgls..........2.2.-. 27 Histogram for Model 2 Spline Point-Value Residuals. .......... 29 The Two-Dimensional Fourier Transform of the Model Field from Longitude 0' to 60' and Latitude 10' to 70' with One-Data Interval Equal to One Minute and Contours in Normalized Units. ........ 30 Simulated Survey Tracks Based on the Fourier Transform of the 60 x 60 Minute Model Field. Nominal Track Spacing and Sampling Interval Equal to 2 nities 3. a. gees aren sates See nee ne eee 31 Contour of Model-3 Point-Value Residuals (Spline Interpolation vs. iirve: Value)! with a Contour Intervaliof Os5) mal.) si memeienelenenene 32 vi 14, ioe 16. Ze 18. 19 20. Zi 22 23. 24. 2a FIGURES (CONTINUED) Histogram for Model-3 Spline Point-Value Residuals. ........ Amplitude Response of the Two-Dimensional High-Pass Filter Applied to the 80 x 80 Minute Model Field. Contours are in Normalized tUimiitsics vc. acrve as xe «so pcmereRTOneniiel lo ay ci Nerina, Pirives Ve ioubaalagee Contours of Point-Mass Anomaly Data After Application of High- Pees lAiliier, Conteue Inirenven TS @sU mglls go bbs 6 oh 6 6 a6 « Hand-Drawn Contours of Total-Magnetic Intensity from a Shipboard Magnetic Survey with E-W Tracks spaced 3 nm Apart and Contour Infervalkof>OkGammeasieet.) ao: ..c Seema omtetas Os etait Gee tencstreue Machine Drawn Contours of Total-Magnetic Intensity Generated by the Cubic Spline Algorithm from Original Survey Data with Contour IntenvaloreoOxGammases cis) 2: s) ..02) REM cuente). cute ae) bones ay eetatns Model 4 - Survey Plan Applied to Point-Mass Model Data to Simulate a Normal Shipboard Gravity Survey, i.e., Track Direction E-W, Track Spacing 3 nm, Sampling Interval 1/3 nm... . Histogram for Model-4 Spline Point-Value Residuals. ........ Contours of Mean-Anomaly Residuals Generated from Model-3 Survey by Utilization of a Cubic Spline Interpolation Procedure “niln Comer niece CF Osamelloccgcocdoponpoeoue ad Contours of Mean Anomaly Residuals Generated from the Model-3 Survey by Utilization of a Lagrange Interpolation Procedure with ContouraIntenalrof.Ocoimal .)ucuen eum enctrents soi seylspksiie sp eps Model 3 Histograms Showing Comparison of Accuracies of Spline and Lagrange Algorithms for Computing Mean Gravity Anomalies . . . Contours of Mean Anomaly Residuals Generated from Model-4 Survey by Utilization of a Cubic Spline Interpolation Procedure avin, Coluetr lnicinfelleh les) titel} o 6 of ogc oem plo lotovolgo ooo Contours of Mean Anomaly Residuals Generated from Model-4 Survey by Utilization of a Lagrange Interpolation Procedure ain (Conta inbtincihen Ons tniell 6 o 6 ele ales Hloc ao oeolo1d 0 6 vil 26. Nf « FIGURES (CONTINUED) Model-4 Histograms Showing Comparison of Accuracies of Spline and Lagrange Algorithms for Computing Mean Gravity Anomalies. . . Model-3 Histogram Showing Accuracy of the Bicubic Spline Algorithm for Computing Mean Gravity Anomalies. ........+-+2202- Vill Page 46 INTRODUCTION Many of the analytical techniques which are currently in use in geophysics involve some form of mathematical operation on digitized survey data. The two- dimensional character of some of these techniques requires that the survey data, usually collected along tracks, be reduced to values of the surveyed parameters on an equally spaced grid. Examples of these techniques include upward and downward continuation, derivative calculations, regional trend removal, two- dimensional trend filtering, model studies, deflection of the vertical and geoidal undulation calculations, and automated contouring techniques. The interpolation procedures which are discussed in this report are applicable to any measurements obtained by track-type surveys. Emphasis, however, is placed on the problem of gridding gravity survey data and the determination of mean values of the gravity field within specified areas. In general there are two approaches available for gridding track-type survey data. One approach is that of fitting a least-squares surface, usually of polynomial form, to all of the available survey data within a predetermined area and using the resulting formula to generate the required grid values. Examples of this procedure, applied to irregularly spaced data, may be found in Crain and Bhattacharyya (1967) and Krumbein (1959). The advantages of this least-squares method are that it tends to operate as a low-pass filter, which reduces the effects of noisy or erroneous data and it can be applied directly to unequally-spaced data points. The chief disadvantages are: 1. The wavelengths of the features which will be accurately fit is a function not only of the degree of the polynomial but also of the size of the area which is selected. This causes some difficulty in controlling the technique in a production type operation. 2. The method is quite sensitive to round-off errors and, except in cases where orthogonalization techniques are used, can become unstable for high degree surfaces. Recent work by Wampler (1969) discusses the problems associated with inverting ill-conditioned matrices using standard procedures. 3. The technique requires a rather large amount of computer time, although this problem can be minimized through the utilization of recursive linear-regression techniques (Fagin, 1964) when additional data points are added to the set. The second general approach which is used to grid survey data is based on interpolation techniques which fit each data value exactly. In this case, it is assumed that any large errors or noise in the data have been removed prior to application of the gridding process, and that the survey coverage is sufficient to minimize aliasing in the data. Historically, this approach has consisted of connecting each pair of data points in such a way as to partition the survey area into non-overlapping triangular sections (Rankin, 1963) and fitting a plane to each of these sections, or by partitioning the survey area into a preliminary rectangular grid and interpolating within each of these areas by an exact-fitting two-dimensional polynomial (Crain and Bhattacharyya, 1967). The principal disadvantage of these methods is that no conditions are imposed on the derivative of the interpolating functions. Except in very smooth areas, this results in the generation of non-realistic gradients. The problem is somewhat reduced by the use of a two-dimensional weighting function of the type developed by Shepard (1968). Shepard's technique generates the data value at a desired grid location by applying a weighting function based upon the distance and direction to the neighboring data points. In addition, estimates of the derivatives at each data point are included. This removes the requirement that the partial derivatives be zero at each data point thus allowing the interpolating function to have extrema anywhere and not just at the original data points. In order to overcome some of the difficulties associated with these existing techniques and to effectively utilize the computational efficiency afforded by track-type survey data, an interpolation procedure was developed by Bhattacharyya (1969) which makes use of cubic and bicubic splines. The advantage of the cubic spline method, which is a mathematical analog of the draftsman's plastic spline, lies in its ability to not only fit the observed data values but to maintain continuity of the first and second derivatives. This paper concerns the development and testing of a cubic spline technique as applied to the computation of mean gravity anomalies. By utilizing simulated survey data obtained from a model gravity field, it is shown that the spline technique will produce accurate estimates of the mean gravity anomalies provided the original survey is properly designed. The use of a two-dimensional Fourier transform for a solution to the survey design problem is also presented. DEVELOPMENT OF THE CUBIC SPLINE ALGORITHM FOR GRIDDING TRACK TYPE SURVEY DATA The algorithm presented here is based upon the technique developed by Bhattacharyya (1969) with some modifications to the spline formulation in order to use the boundary conditions and development presented by Pennington (1965). en where x & Xapr we first assume that the second derivative of f(x) is a linear Given data values ¥1%0°° Vm located at the points xq 1X function of x between any pair of adjacent data points. Thus, if Zy1Zy 220% 2 m are the values of the second derivative of f(x) at the given data points then for the interval x, Sx S% 41 We have i eis ee as = Ze) ae oe eee (1) dy k where qd. =e ~ X,- This is simply the equation for a straight line passing through the points Oger) and enor Integrating (1) once yields Pe en ae f (x) = OA . ar EOE Tera aR Ss (2) k k where Cy is the constant of integration. Integrating (1) a second time yields 3 3 vz, (| - x) z (x - x, ) Ss ks side 25 Skaaied Cart arch. (3) 6d). 6d, 1 2 F(x) With the requirement that f(x) match the observed data at the points and , the constants c, and c, may be evaluated from equation (3). aimee 1 2 : Thus, at the point Xr 2 aa Yb F( ) a Sj ate Sa, ate ty) (4) and at the point Xe]! 2 panera s ie sikh nik eq = ey) =e Sey Ene 2 Solving equations (4) and (5) for the values of Cy and Cor and substituting these into equation (3), yields the following formula for the cubic spline in the interval x ',| r\ | ',| ij i (@] ~ soo 00G000000000 (0) 00 03 ij : and @ = : mn Q ij a 30 33 ij Substituting the @ a elements into equation (19) produces a two-dimensional cubic polynomial formula representing the data within each of the R.. subareas covering R. The mean value of the field (C), within the Re rectangle, is computed from equation (19) by ij m xX. a 1 Si ne f ity ie is The bicubic spline computer program given in the appendix contains the option of evaluating this integral between arbitrary limits within each R.. rectangle. This feature allows computation of the mean anomalies at a grid spacing either equal to that of the U.. input data, or one-half or one-third the spacing of the input data. Setting the (x.y Y;-) origin equal to (0,0), 13 rare Sactaetal cya Sheil, f 3 @ w EO (pes < 2 (x x1) (y yy) dy dx. (21) and letting the limits of integration be C< x f(t) = = F(x) Speen (23) n=-@ 27A ¢ =e] where the band limits are +2nA and the sampling rate is 2A. See Papoulis (1962) for a proof of this theorem. Although it is theoretically impossible for a function of finite length to be band limited, a reasonably accurate estimate of A may be obtained for a small area by application of a two-dimensional Fourier transform. In one dimension, the direct Fourier transform of an arbitrary function, g(x), is defined as (09) Gif) = af g(x) ou nes dx, (24) with f, equal to the frequency. By means of this formula, it is possible to transform information from the space domain into the frequency domain. G(f,) is, in general, a complex number of the form a(f, ) + ib(f,) or equivalently 2 ely) 2ae b a +b We Sta a where ie ze |8) }) = |G) | is called the amplitude spectrum and arctan ae O(f,) is termed the phase spectrum. For our application of the Fourier transform, we will be primarily concerned with the amplitude spectrum. In the context in which it is used here, the term frequency refers to spatial frequency with the units being either cycles per unit distance or normalized to cycles per data interval . For a two-dimensional function g(x,y), equation (24) is expanded to Bradt p ~i2u(f x +f y) Gif aL) = f f a g(x by x" y” dxdy . (25) -@O -@ With g(x,y) consisting of digitized values on a uniform grid covering a finite area R, an estimate of G(F rf ) may be obtained for the normalized frequency range 0 < hank < 0.5 cycles/data interval, by use of the discrete two-dimensional fast Fourier transform (FFT) of Cooley and Tukey (1965). This discrete approxima- tion to Na must be interpreted to provide estimates of the three survey parameters; track spacing, sampling rate, and track orientation. The specifica- tion of track orientation requires a knowledge of what happens to trends or lineations in the (x,y) domain when they are mapped into the (Frf ) plane. It has been determined by Fuller (1967) that a trend in the (x,y) plane maps into a trend in the Co) plane in an orthogonal direction. Fuller shows this simply in the following way. Let g(x,0) be an arbitrary function of x defined for y =o. Using this definition, we can construct a function of both x and y, i.e., g(x,y), by shifting g(x,o) a distance Sy in the x direction, with S = ee as shown in Figure 2. FIGURE 2. THE FUNCTION g(x,y). 16 We will use what is usually called the "shift theorem" from transform theory which states that, if Gif) is the Fourier transform of g(x) then, from equation (24), eieal, x Gif) is the Fourier transform of g(x-A). With A = Sy, this theorem is used to integrate out the x dependence in equation (25) resulting in the equation inf SHE) Y iG) ll SGNe (26) = and, by symmetry, Ff) = 2G(F) fi os [2ny(F, s+] dy. (27) Integrating over y yields Sin [2nvir, S +) Ait) 280) saga x y FEA) = 2YG(E) Sine [2av (FS +f)] ; (28) Since the Sinc function has its maximum value when the argument is zero, equation (28) defines a function in the (fer f ) plane possessing a trend with -f Ss == . Thus we see that, since trends in a (x,y) plane are mapped into x trends in the (Ff ) plane in the orthogonal direction, survey track direction may be determined from the two-dimensional FFT by aligning the tracks parallel to the trends in the (fs e) plane. With this direction specified, the shape of the two-dimensional amplitude spectrum is used in conjunction with the Shannon sampling theorem to estimate the band limits of g(x,y) which, in turn, define the required track spacing and sampling rate for the survey. A general two-dimensional FFT computer program utilizing equally spaced gridded data and a fast Fourier 17 transform subroutine called NLOGN, written by Robinson (1967), is contained in the appendix. This program was used to design the simulated surveys for testing the cubic spline algorithms. B. Development of a Point-Mass Model Field To provide a reference surface for determining quantitative estimates of interpolation accuracy and to ensure that the data used in the interpolations are error free, gravity model data were generated from a random distribution of point masses as follows. Consider a point (P) on the surface of a sphere (Figure 3) with spherical coordinates (8, \), and a point mass located at depth d beneath the surface with coordinates (@', X'). The gravitational potential at P(®, X) due to the point mass is -km Vp =a (29) where m is mass, k is the universal gravitational constant, and r is the distance between the point mass and the point (P). P(®,)) FIGURE 3. GEOMETRY FOR COMPUTING THE GRAVITATIONAL POTENTIAL DUE TO A POINT MASS. For a sphere of radius R, the distance r is r= (Ro eR? - 2RR' cos Le where R'=R-d and cos W = cos® cos 9’ + Sin® Sin @' cos(X-dA). The radial (vertical component) Gp of the gravitational force at P is oN km Gp(9,4)= - R- = a (R-R' cos). For a finite number (N) of point masses situated at various depths and locations beneath the surface of the sphere, we can generate gravity anomalies on the sphere by summing the radial components due to the N point masses, i.e., Ne ne Gp (8A) = ZS gz (R-Ri cos p.). (30) i=l (F I A computer program to calculate the radial, easterly, and northerly components, in addition to the deflection of the vertical components with respect to a spherical earth of a set of point masses is given in the appendix. COMPARISON OF GRIDDING TECHNIQUES AND THE COMPUTATION OF MEAN ANOMALIES USING MODEL DATA The accuracy of the spline interpolation method, relative to several other techniques, is now considered in terms of gridding survey data and the computation of mean gravity anomalies by utilizing simulated survey data. A set of gravity anomalies, for use as a reference surface, were calculated from equation (30), at one-minute intervals, over an area of 80 x 80 square minutes. The anomalies, given in Figure 4, result from 16 point | FIGURE 4. CONTOUR CHART OF THE GRAVITY ANOMALIES GENERATED BY 16 POINT MASSES WITH CONTOUR INTERVAL OF 5 mals. masses located at depths ranging from 7 to 12 kilometers beneath the surface of a spherical earth. Amplitudes of the anomalies range from -17 mgls to +83 mgls. For purposes of comparing the various models and residual charts to follow, a transparency of Figure 4 is contained in a pocket attached to the rear cover of this report. A. Gridding of Survey Data To design a simulated survey for obtaining data to test the interpolation procedures, a two-dimensional FFT was computed using all of the model field data (Figure 4) available on a one-minute grid interval. Contours of the amplitude spectrum resulting from this computation are shown in Figure 5. Since only the shape of the spectrum is used to design the simulated survey, the contour values are arbitrarily normalized, and the frequency range is normalized to the units of cycles per data interval. Utilizing the result embodied in equation (28), Figure 5 shows that the trends in the model field are predominately in the NE-SW direction indicating that the preferred survey track direction is NW-SE. With the track direction specified, the flattening of the amplitude spectrum at a frequency of approximately 0.17 cycles per data interval in both the along-track and cross-track directions indicates that a track spacing and sampling interval of three nautical miles would be sufficient to minimize aliasing of the data. Figure 6 shows the simulated survey, designated as model 1, which was designed from this interpretation of the two-dimensional amplitude spectrum. By overlaying this survey with the transparent copy of Figure 4, it is apparent that no attempt was made to sample the peak values of the anomalies. Simulated deviation of the survey vehicle from the planned survey tracks was included, but measurement errors were excluded by computing the exact value of the gravity field at each survey point by using equation (30). The cubic spline algorithm was utilized with this survey data to compute inter- polated values of the gravity field at one minute grid intervals. The interpolated 21 .25 AMPLITUDE SPECTRUM OF POINT MASS ANOMALIES 81x 81 eh —— Fy IN CYCLES PER DATA INTERVAL ae .10 15 .20 .25 Fy IN CYCLES PER DATA INTERVAL FIGURE 5. THE TWO-DIMENSIONAL FOURIER TRANSFORM OF THE ENTIRE 80 X 80 MINUTE MODEL FIELD WITH ONE DATA INTERVAL EQUAL TO ONE MINUTE AND CONTOURS IN NORMALIZED UNITS. 22 FIGURE 6. MODEL 1 - SIMULATED SURVEY TRACKS BASED ON THE FOURIER TRANSFORM OF THE 80 X 80 MINUTE MODEL FIELD. NOMINAL TRACK SPACING AND SAMPLING INTERVAL EQUAL TO 3 nm. 23 value at each grid point was then subtracted from the true value obtained via equation (30) and the result was machine contoured at a contour interval of 0.5 mgl. This residual contour chart is shown in Figure 7. Figure 8 isa histogram of the model-1 errors with the standard deviation and range noted. In order to compare the accuracy of the spline interpolation to that which could be achieved through the use of a two-dimensional least-squares polynomial surface, a twelfth-degree surface was fitted to the simulated survey data obtained from model 1. The contoured field values generated by this surface are shown in Figure 9. Comparison of this surface with the true field values in Figure 4 indicates that the least-squares surface reproduces the general shape of the low-frequency components but does not depict the higher frequency features. As stated previously, the least-squares surface can be made to fit the high frequency components by reducing the size of the area over which the surface is computed but this is not considered to be an appropriate approach to interpolation when adequate survey data is available. By overlaying Figure 7 with the contour chart of the model field, it is readily apparent that the errors in the interpolated values are correlated with those regions of the model field which contain a large amount of energy at the higher frequencies. This correlation is interpreted as arising from the fact that the two-dimensional FFT is essentially a least-squares operation which produces an estimate of the average amplitude spectrum over the entire region in which the computation is carried out. This effect, coupled with the obvious non- stationarity (in space) of the model field, caused the amplitude spectrum to flatten out at an unrealistically low frequency (0.17 cycles/data interval). In turn, this led to an estimate of track spacing and sampling interval which was too large to adequately define the higher frequencies. In order to determine the effect of a shorter sampling interval, a second model, designated model 2, was constructed with track spacing and direction similar to model 1 but with a sample interval of 1 nm. The histogram of the errors resulting from the 24 +. an 00 30 60’ FIGURE 7. CONTOUR CHART OF MODEL-1 POINT-VALUE RESIDUALS (SPLINE INTERPOLATION VS. TRUE VALUE) WITH A CONTOUR INTERVAL OF 0.5 mgl. 25 80’ “STIVNGISAY INTVA-LNIOd ANI1dS L-17GOW YOs WVYDOLSIH °8 FNS! FIGURE 9. CONTOURS OF THE 12° LEAST-SQUARES POLYNOMIAL SURFACE GENERATED FROM MODEL-1 DATA. CONTOUR INTERVAL IS 5 mgls. (SURFACE COMPUTED FROM PROGRAM SUPPLIED BY GULF RESEARCH AND DEVELOPMENT COMPANY .) 27 interpolated values determined from this survey is shown in Figure 10. It is apparent that only a slight decrease in error was achieved by use of the increased sampling rate. In order to more effectively reduce this error, the FFT was computed for that anomalous portion of the model field contained within 10’ and 70' latitude and 0' and 60' longitude. Contours of the normalized amplitude spectrum for this part of the model field are shown in Figure 11. With an estimate of 0.25 cycles per data interval as the frequency at which the spectrum for this smaller area flattens out, a track spacing and sampling interval of 2 nm was selected. Although the lack of a definite trend in the high frequency components indicates that the track direction is not a particularly significant factor for this model field, the distinct trending of the low-frequency components indicates that some control of this factor is required. The model 3 simulated survey, based on the amplitude spectrum in Figure 11, is shown in Figure 12. Figure 13 isa contour of the differences between the spline interpolated data values obtained from the model-3 survey and the true values from the model field. The histogram of these errors, as shown in Figure 14, indicates that the model-3 survey produced a significant decrease in the overall errors in the interpolated data values. In order to determine whether the model-3 errors were caused by the cubic spline interpolation algorithm or by the simulated survey design, a two- dimensional, high-pass filter was applied to the one-minute grid values of the model field. This digital filter, which was designed by Fuller (1967) as a general high-pass filter, has the two-dimensional amplitude response shown in Figure 15. The weight function for this filter is symmetric, thus, no phase shifting is involved in the operation. The amplitude response is such that those frequencies which were not adequately sampled with the model-3 survey are retained while the lower frequencies are removed. The contoured result of this filtering operation is shown in Figure 16. By comparing this result with the residual contours in Figure 13, it is apparent that at least some of the errors in the interpolated values from the model-3 survey are generated by locally inadequate sampling. 28 “STIVNGISSY INIVA-LNICd ANI IdS @ THGOW YOdI WVADOL : WodiITll 6Z0= dS SANIVA LNIOd SSVYW LINIOd SNSY3A ANIIdS OML 1300W Fy IN CYCLES PER DATA INTERVAL -305 . AMPLITUDE SPECTRUM \ OF he POINT MASS ANOMALIES Fr 61x61 \ 20 Zl { Si : aes, aie —-l- i = - ) a \ 1S a, cok Se . 5 \ SS oS us Ss, lo Le oll} 20 25 30 Fy, IN CYCLES PER DATA INTERVAL FIGURE 11. THE TWO-DIMENSIONAL FOURIER TRANSFORM OF THE MODEL FIELD FROM LONGITUDE 0' TO 60' AND LATITUDE 10' TO 70' WITH ONE-DATA INTERVAL EQUAL TO ONE MINUTE AND CONTOURS IN NORMALIZED UNITS. 30 80’ i ceo ck ean Ns ° ». ~~ \ = 1 N + \ SS ~ N \ . 3 Sa. OS WS 4 a ~ = a . S \ " 4 ~ 4 * M, . “ . \ _ \ DS ; \, 7 s Nh * » \. ~ * Ne ie . a , . s . S » 607 -~—__— + ke x} ts - . SN \ . : . \ \ \ Nii st ( Se 4] ‘ ‘ ee DN 5. . . . Ok . } . . . . sr ise +t Xe i 4 — . tt -+: > 4 7 3 Yee : I . miles 5 BS 4 4 == + FIGURE 12, 30’ 60’ SIMULATED SURVEY TRACKS BASED ON THE FOURIER TRANSFORM OF THE 60 X 60 MINUTE MODEL FIELD. NOMINAL TRACK SPACING AND SAMPLING INTERVAL EQUAL TO 2 nm. 3] 80’ ~ Le a 4 a [L _| Ae % oe | a =| wh | yen | , @ =) - dle a aA iN my tS Oy a rie FIGURE 13. CONTOUR OF MODEL-3 POINT-VALUE RESIDUALS (SPLINE INTERPOLATION VS. TRUE VALUE) WITH A CONTOUR INTERVAL OF 0.5 mgl. 32 4IN3DuaAd (22 ie NON CS 1.1 Fy IN CYLES PER DATA INTERVAL 0 J of 38) 4 Fx IN CYLES PER DATA INTERVAL FIGURE 15. AMPLITUDE RESPONSE OF THE TWO-DIMENSIONAL HIGH-PASS FILTER APPLIED TO THE 80 X 80 MINUTE MODEL FIELD. CONTOURS ARE IN NORMALIZED UNITS. 34 b ot “oO on FIGURE 16. CONTOURS OF POINT-MASS ANOMALY DATA AFTER APPLICATION OF HIGH-PASS FILTER. CONTOUR INTERVAL IS 0.1 mgls. 39 These tests lead to the conclusion that, given an adequate track type survey design based on the frequency characteristics of the data, the cubic spline algorithm is a reliable interpolation technique. The computational efficiency of the algorithm is demonstrated by the fact that the 81 x 81 grid values interpolated from the model-1 survey required only one minute and ten seconds of computer time on a CDC-3800. As a qualitative test of the cubic spline gridding procedure applied to actual shipboard survey data, the algorithm was used to grid total magnetic intensity data obtained from a well-controlled marine magnetic survey consisting of E-W tracks spaced nominally 3 nm apart. Figure 17 is the hand-drawn contour chart of this area and Figure 18 is the machine contoured result using the cubic spline generated grid points. Aside from the fact that the cubic spline generated what is considered to be a more realistic position for the peaks of the anomalies, the two contour charts are nearly identical. The computer time required to generate the gridded data and plotter tape was less than three minutes on a CDC 3800 with an actual plotter time of twenty minutes. This is compared to an estimate of approximately 100 man-hours to plot the data values and produce the hand-drawn contour chart. B. Mean Gravity Anomaly Calculations In order to test the algorithms for computing mean gravity anomalies, a survey was designed which simulated a conventional shipboard gravity survey. This survey plan, designated model 4, consists of E-W tracks spaced 3 nm apart with a sampling interval of 1/3 nm. Figure 19 is a plot of this survey showing the random deviation from the planned track which was added to simulate a normal survey track. Figure 20 is a histogram of the errors in the interpolated data points resulting from the model-4 survey. Comparing this result with that obtained from the model-3 survey (Figure 14) indicates that the increased sampling rate (six times that of model 3) was sufficient to overcome the effect of increasing 36 “SWWWY9 0S JO TVAYSLNI YNOLNOD HLIM VLVG AJAINS IWNISIYO WOU WHLIYOOTV ANIT1dS DIGND JHL Ad GALVYINIO ALISNILNI DILINOWW-1VLOL JO SYNOLNOD NMYVUG ANIHDWW “81 AINSI 38 Me ae Saeaye — By FIGURE 19. MODEL 4 - SURVEY PLAN APPLIED TO POINT-MASS MODEL DATA TO SIMULATE A NORMAL SHIPBOARD GRAVITY SURVEY, i.e., TRACK DIRECTION E-W, TRACK SPACING 3 nm, SAMPLING INTERVAL 1/3 nm. 39 “STVNGISAY ANIVA-INIOd ANI1dS ¥-TAGOW YO WVADOLSIH “02 FNS! the track spacing and surveying at an angle to the trends in the data. Obviously this will not be true in the general case, since the effect of changing any of the survey design parameters will be controlled by the characteristics of the field being measured. Survey models 3 and 4 were used to produce data for testing three algorithms for computing mean gravity anomalies. The technique for generating mean anomalies using both a nine point averaging of the gridded data produced by the cubic spline and the two-dimensional integration of the bicubic spline surface have been discussed previously. In addition to these methods, a technique was tested which utilized a one-dimensional Lagrange interpolation (see e.g., Hamming, 1962). This procedure generated mean gravity anomalies on a uniform grid by averaging all of the survey data points falling within a grid interval to form a mean for that interval. Following this, a one-dimensional fourth-degree Lagrange interpolation was applied to these mean values to obtain an estimate of the mean gravity value for each of the grid cells for which no survey data was available. The results of these mean anomaly tests are shown in Figures 21-27. Figure 21 is a contour chart of the differences between the true mean anomalies computed for each one-minute square of the model field via equation (30), and the mean values computed by averaging the 1/2 minute grid values produced by applying the cubic spline procedure to the survey data from model 3. Figure 22 is a contour chart showing the differences between the true one-minute means and those generated by application of the Lagrange procedure to the model-3 data. The contour interval is the same in these two figures. The errors produced by the lack of control of the derivatives in the Lagrange interpolation method is apparent in Figure 22. It is probable that these errors could be reduced somewhat by using a third-degree Lagrange polynomial instead of the fourth-degree but, in light of the obvious advantages of the spline formulation, no additional testing of this approach was undertaken. Al 60° pels S 10 30° 50° FIGURE 21. CONTOURS OF MEAN-ANOMALY RESIDUALS GENERATED FROM MODEL-3 SURVEY BY UTILIZATION OF A CUBIC SPLINE INTERPOLATION PROCEDURE WITH CONTOUR INTERVAL OF 0.5 mgl. FIGURE 22. CONTOURS OF MEAN ANOMALY RESIDUALS GENERATED FROM THE MODEL-3 SURVEY BY UTILIZATION OF A LAGRANGE INTERPOLATION PROCEDURE WITH CONTOUR INTERVAL OF 0.5 mg}. 42 Figure 23 compares the histograms of the errors generated by the cubic spline and Lagrange algorithms. Figures 24-26 show that simular results are obtained with the cubic spline and Lagrange methods when they were applied to a con- ventional shipboard gravity survey simulated by the test model 4 previously described. In order to determine if the increased computing time necessary for producing mean anomalies by integrating the bicubic spline surface was justified, the algorithm was applied to the one-minute grid values generated by application of the cubic spline to the model-3 survey data. A histogram of the errors resulting from this technique is shown in Figure 27. A comparison of Figures 23 and 27 clearly indicates that the unique capabilities of the bicubic spline approach are wasted in so far as the computation of mean anomalies is concerned. CONCLUSIONS After evaluating the results obtained from this series of tests, the following conclusions may be drawn: 1. The cubic spline algorithm for gridding track-type survey data and for computing mean gravity anomalies is accurate and computationaly efficient in comparison with the other techniques which were tested. 2. The significant improvement achieved by the model-3 survey in comparison to the model-1 survey illustrates the concept of the sampling theorem embodied in equation (23), i.e., the accuracy of mathematical interpolation, and in particular the spline method, is highly dependent on the density of the survey data points relative to the frequency content of the anomalies. The results obtained by models 3 and 4 show that, if a survey is well designed, highly accurate interpolation may be anticipated by use of the spline procedure. Implicit in this conclusion is the requirement that the survey data itself be of high quality. 43 “SAITVWWONYV ALIAVASD NVAW ONILNdWOD YOsA SWHLIYOO1V JONVYOVI GNV INIIdS JO SAIDVINDIV JO NOSIYVdWOD ONIMOHS SWVYOOLSIH € TAGOW “EZ JUNO! STVOITIIW SNV4JW JONVYOVI SNV4W 4NIidS €ZZ Ol y8'8— JONVY JONVYOV) 891 OL 78 l— JONVA INIIdS 909 L= aS JONVYOVI O€'0=GS INIdS J3a¥HL 1AGOW 44 INIDa4d ce 50’ 2) Es, 0. 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ORIGINATING ACTIVITY (Corporate author) 2a. REPORT SECURITY CLASSIFICATION UNCLASSIFIED iain