Smith Wessel 1990 Gridding With Continuous Curvature Splines in Tension

8/2/2019 Smith Wessel 1990 Gridding With Continuous Curvature Splines in Tension

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GEOPHYSICS, VOL. 55, NO.3 (MARCH 1990);P. 293-305, 7 FIGS.

Gridding with continuous curvature splines in tension

W. H. F. Smith* and P. Wessel:t:

ABSTRACTA gridding method commonly called minimum curvature is widely used in the earth sciences. The

method interpolates the data to be gridded with asurface having continuous second derivatives and minimal total squared curvature. The minimum-curvaturesurface has an analogy in elastic plate flexure andapproximates the shape adopted by a thin plate flexedto pass through the data points. Minimum-curvaturesurfaces may have large oscillations and extraneousinflection points which make them unsuitable for gridding in many of the applications where they arecommonly used. These extraneous inflection pointscan be eliminated by adding tension to the elastic-plateflexure equation. It is straightforward to generalizeminimum-curvature gridding algorithms to include atension parameter; the same system of equations mustbe solved in either case and only the relative weightsof the coefficients change. Therefore, solutions undertension require no more computational effort thanminimum-curvature solutions, and any algorithmwhich can solve the minimum-curvature equations cansolve the more general system. We give commongeologic examples where minimum-curvature griddingproduces erroneous results but gridding with tensionyields a good solution. We also outline how to improvethe convergence of an iterative method of solution forthe gridding equations.

INTRODUCTION

A wide variety of numerical procedures in the earthsciences require data on a regularly spaced lattice, includingFourier analysis and many map-drawing algorithms. In contrast, most geologic data are acquired at individual observation points or along traverses. It is therefore necessary toconstruct estimates ofthe value ofa function on a grid, given

observations of the value of the function at arbitrary locations in the x, y plane. This operation is called gridding.There are three areas of concern in evaluating a gridding

algorithm. The relative importance of each depends on theintended application. The first concern involves the globalproperties of the solution. Some procedures construct aninterpolant of a priori known functional form. The secondconcern involves honoring data constraints; e.g. decidingwhether the data are fit exactly or approximately. The thirdconcern involves the method of interpolation or extrapolation in poorly constrained regions. It is in this last area thatgridding algorithms differ most and where global propertiesstrongly affect the solution.In general, all gridding algorithms share these underlying

assumptions: (a) the function to be gridded is single-valuedat any point; (b) the function is continuous within the regionto be gridded; and (c) the function is positively autocorrelated over some length scale at least as large as the typicalspacing between observation points (Harbaugh et al., 1977;Davis, 1986). Nearly all methods est imate values at gridnodes from weighted averages of nearby data points, aprocedure justified by assumption (c) in particular. Althoughrarely pointed out, some methods also require that thecontrol data not contain information at wavelengths shorterthan twice the grid spacing in order that spatial aliasing willnot occur. Later in this paper we discuss prefiltering the datato avoid this problem.Weighted-average schemes differ in how they assignweights to the constraining values. The simplest methodsuse a polynomial or power law in distance; most othermethods use some sort of minimum-norm principle (Wegman and Wright, 1983). We divide these into two groupswhich we call statistical methods and integral methods.Statistical methods minimize the variance of the grid-valueestimator by selecting weights based on the data autocorrelation. The kriging methods used in economic geology (Olea,1974;Clark, 1979) belong to this class. An advantage of thesemethods is that they can yield confidence limits for the gridvalues; a possible disadvantage is that global properties of

Manuscript received by the EditorMarch 7, 1989; revised manuscript received August I, 1989.*Lamont-Doherty Geological Observatory of Columbia University, Route 9W, Palisades, NY 10964.*Formerly Lamont-Doherty Geological Observatory of Columbia University; presently Hawaii Instituteof Geophysics, 2525 CorreaRoad,Honolulu, HI 96822. 1990 Society of Exploration Geophysicists. All rights reserved.

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294 Smith and Wesselthe surface such as high-order continuity cannot be assureda priori. Integral methods begin with the requirement thatthe surface should minimize some global norm over some setof functions of the data; the weights are then determined tosatisfy this constraint while fitting the data. The advantage ofthese methods is that they assure a solution with the desiredproperties; their disadvantage is that they do not easily yieldconfidence limits.The method we present here is a generalization of apopular integral method called "minimum curvature." In the

minimum-curvature method, an interpolant with continuoussecond derivatives is cons tructed such that the squaredcurvature integrated over the entire surface is minimized.Briggs (1974) derived the equat ions and suggested theirsolution by iteration of finite-difference expressions; Swain(1976) gave a Fortran algorithm based on Briggs' method;and Sandwell (1987) solved the same equations by a matrixmethod. Variations on the Swain algorithm (e.g., the U.S.Navy's SuperMISP) are in widespread use in the earthscience community. For example. the gravity and magneticanomaly maps of North America (GSA Map Commit tees,1987), and the Digital Bathymetr ic Data Base of the U.S.Navy (Van Wyckhouse, 1973;NGDC, 1988)are prepared bythe minimum-curvature method. The heavy solid lines inFigures la and Ib are profiles through minimum-curvaturesurfaces. They honor the data at constrained points, buthave large oscillations between these points. This behaviorin unconstrained regions may be undesirable in some applications.In one dimension, the function with continuous second

derivatives that interpolates the data constraints exactly andminimizes total curvature is called the interpolating naturalcubic spline (cf., de Boor, 1978; Lancaster and Salkauskas,1986). It may have the large oscillations between constraintsshown in Figure la. Two modifications to this spline havebeen used to avoid these oscillations. The first (type I)modification relaxes the requirement that the data be interpolated exactly; a compromise is made between the misfit to

the data and the curvature of the solution. Solutions of typeI are called smoothing splines (cf., de Boor, 1978;Lancasterand Salkauskas, 1986). The second (type 2) modification, incontrast to the first, interpolates the data exactly but relaxesthe constraint that the total curvature must be minimized.One class of type 2 solutions is splines in tension (Schweikert, 1966).The oscillation of the natural cubic spline canresult in extraneous inflection points; Schweiker t (1966)showed that a spline in tension eliminates these inflections.Note in Figure 1b that the control data can be interpolatedwith a function which is everywhere concave down (e.g., thethin solid line), yet the minimum-curvature solution (heavysolid line) changes concavity. Spath (1973) has givenSchweikert's (1966) equations in a strictly diagonally dominant tridiagonal matrix form, and Cline (1974) has adaptedSchweikert's spline to curves in the (x, y) plane.In two dimensions, the minimum-curvature interpolant is

the natural bicubic spline which can have the same oscillations and extraneous inflection points as in the one-dimensional (I-D) case. Again the same methods may be used tosuppress these features. Inoue (1986) has given a type 1modification in which the damping of first and secondderivatives is traded off against the data misfit under aleast-squares norm. This is essentially a two-dimensional(2-D)smoothing spline which does not fit the data exactly. Inthis paper we present a type 2 modification. We show howthe minimum-curvature gridding method (Briggs, 1974;Swain, 1976; Sandwell, 1987) can be generalized to includetension in the interior and boundary equations, and we showcommon geologic examples where minimum curvature produces undesirable results but the introduct ion of tensionsignificantly improves the solution. Our method produces asuite of surfaces with con tinuous second derivatives ofwhich the minimum-curvature surface is one end member.Increasing the tension parameter relaxes the global minimum-curvature constraint by moving toward a solution withcurvature localized at the control data points; at the sametime, the surface fits the data exact ly. Adjustable tens ion

a) b)

:L800 -,- ----, -700~ - - + -800150000

50o + - - - ~ - - - ' -

-200... -300'i -4ooJJ-500,~ , ''-:'-,, >.,

Distance (km)FIG. 1. (a) Cross-sections through surfaces produced with splines in tension. The black squares are data constraints.The heavy line is the minimum-curvature end member, the thin line is the harmonic end member, and the dashedline is an intermediate case using some tension. Note that all solutions honor the data points. (b) Cross-sect ionthrough a continental she lf and slope. The black squares represen t the intersection be tween the measuredbathymetry (dashed line) and 100m isobath contours. These intersections (contour coordinates) were then griddedusing minimum curvature (heavy solid line) and some tension (thin solid line). The minimum-curvature methodintroduces an extraneous inflection point and exceeds the -100 m level, although we know that bathymetry in thisregion is bounded by the -200 m and -100 m levels. The surface produced with tension gives a much bet te rapproximation since it suppresses local maxima and minima between data constraints.

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Gridding with Splines in Tension 295

and

Minimum-curvature gridding algorithms use the norm

(9)

(7)

where TB is a tension parameter for the boundary which alsovaries between 0 and I. The free-edge condition corresponds

zero, and the boundary conditions represent zero bendingmoment on the edges [equation (3)], zero vertical shearstress on the edges [equation (4)], and zero twisting momentat the corners [equation (5)] (Timoshenko and WoinowskyKrieger, 1968). Physically, thejj represent the strengths qlDof point loads on the elastic plate; mathematically, they arecoefficients in a solution which is composed of a linearcombination of Green's functions for plate flexure due tounit point loads.The total stored elastic strain energy in the flexed plate isapproximately proportional to the curvature (1); of all twicedifferentiable surfaces interpolating the data, the minimumcurvature surface stores the least strain energy. If oneimaginesbending an elastic plate to interpolate the data, thenextra work must be done on the plate to create any interpolant other than the minimum-curvature solution. The minimum-curvature solution may seem to be the "natural" wayto estimate poorly constrained grid values. I t is clear fromFigure I, however, that the minimum-curvature constraintmay create extraneous oscillations.We derived our equation by investigating the role of auniform isotropic tensile stress T in equation (6). Supposethat Txx = Tvy = Tand Txy = O. Then equation (6) becomes

where TI is a tension parameter and the I subscript indicatesinternal tension. (We specify tension on the boundariesindependently of TI .) Now we may vary TI from 0 to I, withoand I givingthe end-member cases shown as solid lines inFigure la. When TI = 0, equation (8)reduces to equation (2);and therefore the minimum-curvature solution is one endmember case of equation (8). When TI = I, the first term inequation (8) vanishes; and the solution is harmonic betweenconstraining points. In the elastic analogy, TI = 1representsinfinite tension; since infinite tension is not physically meaningful, one may prefer to think of this end member asrepresenting the steady-state temperature field in a conducting plate with heat sources or sinks at the data points. Animportant property of this solution is that it cannot havelocal maxima or minima except at constraining data points(this follows from the interior mean value theorem forharmonic functions; e.g., Berg and McGregor, 1966). Notethat for any TI in 0 :s TI < I, equation (8) gives a solutionwith continuous curvature, although it does not minimizeequation (1) except when TI = O.We implemented boundary tension with conditions (4)and(5) above but replacing condition (3) by

( ) a2Z az1 - TB - + TB - = 0,an 2 an

When T = 0, equation (7) is equivalent to equation (2); butfor arbitrarily large T, the solution is dominated by thesecond term. Here T has units of force per unit length andthe T required to adjust the solution scales withD and q; weavoided this by writing

(3)

(1)

(4)

(2)2(V 2Z) = 2: j j 8(x - Xi, Y - Yi),i

c = II (V2Z)2 dx dy.Equation (1) is a valid approximation for the total curvatureof Zwhen IVzIis small. Briggs (1974) showed that minimizingequation (1) leads to the differential equation

"2("2) [a 2z a2z a2z]v v z - Tx x -2 + 2Txy -- + Tyy -2 = q (6)ax ax ay ay(Love, 1927). The minimum-curvature gridding equation (2)is a special case of equation (6) when horizontal forces are

where (Xi' Yi' z;) are constraining data. Thejj must be chosensuch that z z, as (x, y) (Xi' Yi), and the boundaryconditions are

A=o (5)ax ayat the corners. Equations (3), (4), and (5) are called free-edgeconditions; and with these conditions, equation (2) has aunique solution with continuous second derivatives calledthe natural bicubic spline. The nomenclature comes from ananalogy with elastic-plate flexure. Small displacements z of athin elastic plate of constant flexural rigidity D, subject to avertical normal stress q and constant horizontal forces perunit vertical length of r.; r.; and Tyy , approximatelysatisfy

permits the gridding of data of varying roughness and allowsthe user to satisfy his own criteriafor a good solution. All thelines in Figure Ia and the solid lines in Figure Ib are profilesthrough surfaces produced by our method.Swain (1976) gave a method for iterative solution of thefinite-difference equations of minimum curvature (Briggs,1974). We show that including tension in these equationsresults merely in a change in coefficients, and therefore anyminimum-curvature algorithm based on Swain's method caneasily be modified to solve our equation. We also discusshow the convergence can be improved to achieve an order ofmagnitude reduction in run time of the original Swainalgorithm. A C language program which incorporates all thefeatures of our method and the sample data gridded in theexamples in this paper are available from the authors onrequest.

Gridding equations and physical analogs

MINIMUM CURVATURE, ELASTIC PLATEFLEXURE, ANDTENSION

along edges, where a/an indicates a derivative normal to anedge, and



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296 Smith and Wesselto TB = 0: TB = I forces the solut ion to flatten at the edge(see Figure I). This flattening is sometimes desired, as whengridding potential anomalies which should decay toward aregional background field away from the source region.Tension as a weighted minimum-norm solutionRayleigh's theorem


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Gridding with Splines in Tension 297to this node. If we have at most one data point nearest eachgrid node, we can fit these data exactly; but if more than onedata value constra ins a given node, the surface is locallyoverdetermined and some sort of smoothing or data decimation is required. Swain (1976) uses the datum nearest thenode and ignores the others. We do not recommend thisprocedure, since it can alias information at wavelengthsshorter than the Nyquist wavelength of the grid. To makeuse of all the data, one might solve equat ion (8) separatelyfor each datum in turn and then average the solutions.However , because the sys tem is linear, it is equivalent tosolve equation (8) with one representative constrainingvalue, which is an average of the original data. This methodrequires only one fourth-order solution for each node; and ifit isdone outside the gridding process, it reduces the numberof data points that need to be stored in the gridding program's memory. Because spatial filtering procedures aregenerally useful apart from gridding and the best filteringmethod is application-dependent, we have decoupled theaveraging process from the gridding process . That is, ouralgorithm does not include any provision for smoothing oraveraging data at overdetermined nodes; we expect that thedata have been preprocessed to give only one filtered valueper grid node, which we then interpolate exactly. Often ourpreprocessing consists simply of finding the mean or median

.. .. -, .

0(_.2 ()) e(_lj) .()oj .(l()) 0(2())

-2)

FIG. 2. When either equation (2)or equation (8) is expressedin central finite differences among the grid nodes, the estimate at one node (the black square) is given by a weightedaverage of the values at 12nearby nodes (circles). Increasingthe tension in equation (8) increases the weight of the shadedcircles relative to the unshaded ones, producing a more localsolution. In any case the same linear system must be solvedand only the weights change: thus any minimum-curvaturealgorithm can be easily modified to include tension. Dataconstraints are assigned to their nearest grid node; a datuminside the dashed box is used to constrain the grid value atthe square. Subscripts in parentheses illustrate the indexingscheme used in the difference equations in the Appendix.

value at the mean or median position in each block of areanearest each node.Briggs (1974)gave an approximation for V2z at a grid node

in terms of other nearby grid values and one off-grid dataconstraint. The expression uses a second-order finite-difference Taylor series expansion to predict the value of theinterpolating surface away from grid nodes. Since bothequations (8) and (2)are equations inV2z, we can implementdata constraints in equation (8) by modifying Briggs' method(see the Appendix). Using this approach, the constrainingdatum enters the local difference equation through the Taylor expansion; and we do not need to solve for Ii explicitly.If the difference equations were to converge exactly, thenthe surface would fit the data exact ly , in the sense that theTaylor series expansion would match the data constraintswith zero prediction error. Because the solution is found tofinite precision, the fit is not perfect; the user enters atolerance for numerical convergence, and the predictionerror is of this order. We have found empirically that themean prediction error is always nearly zero (thus the methodisunbiased). and convergence to maximum absolute error ofone part in 104 can be achieved in short run times (themeaning of "short" is relative to the number of nodes in thelattice). One important feature of the Taylor series methodfor fitting the data is that it honors a datum exactly when thatda tum falls on the lattice; if the (x, y) coordinates of thedatum match those of a grid node, the value of that grid nodeis set equal to the datum value.ConvergenceThe gridding equations (4), (5), (8), and (9) have a unique

solut ion which we may call the true solution. Because wesolve these equations iteratively with finite precision, we donot reach the true solution; and our result depends not onlyon the convergence limit of the iterations but also on the pathtaken toward the solution. Optimization of this path isimportant not only to achieve convergence in only a fewiterations, but also because optimization yields a solutioncloser to the true solution. Convergence in computation isnot the same as convergence in mathematics . We considerour iterations "converged" to limit E when the maximumabsolute change at any node during one iteration is less thanE. This does not mean that the result is within E of the truesolut ion; it means that fur ther improvements in the resultwill be smaller than E for each iteration and are therefore notworth the effort.Details of our solution strategy are given in the Appendix.We generally follow the method of Swain (1976), but weinclude the tension parameters 1', and TB and an additionalparameter for grid anisotropy a. Users of gridding algorithms often grid data in map coordinates; at high latitudesthe anisotropy in distance on a latitude-longitude grid can besignificant. We have included an aspect ratio a in ouralgori thm. where the grid dimensions are such that dy =dxkx, and an nth difference in y is scaled by a" to accommodate the anisotropy. If x = longitude and y = latitude.then 0. = cosine (latitude). We also generalized the regionalgrid strategy of Swain (1976) and included successive overrelaxation to accelerate convergence. With these improvements, our algorithm solves the isotropic minimum-curva-



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298 Smith and Wesselture problem in one-tenth the time required by Swain'salgorithm; applications using tension converge even morerapidly because of the more local nature of the solution intension.

GEOLOGIC EXAMPLES AND THE USE OF TENSIONThe "questionable dipole" exampleFor this example we use shipboard gravity measurementsfrom offshore Mauritania which are in the Lamont-Doherty

marine geophysical data base. The data were cross-overerror corrected (Wessel and Watts, 1988;Wessel, 1989) andthen 5 by 5minute block mean values were computed. Thesemean values were input to our gridding algorithm. In Figure3 we show two contour maps prepared from these data.Figure 3a was prepared using T[ = 0 in equation (8),corresponding to the minimum-curvature method. Figure 3bshows the same datagridded with T[ = 0.3 in equation (8);i.e., with some tension in the gridding surface. The locationsof the input data values are shown as squares. Both maps

c)20' 30'

20' .1011' .102 . - - - ~ : : : : = = , " " " ' ' ' ' - - = = = : ! !22

1.\' .10

a)

d)

" 2.1 .10'

.


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Gridding with Splines in Tension 299

FIG. 4. (a) Map of U.S. Navy DBDB-5 bathymetry contoured at 1000 m intervals. Regions shallower than 15 mbelow sea level are shaded, revealing"shelf-break bulges."Heavy line indicates ship track ofR!V Vema cruise 17-11. (b)Profile along the Vema 17-11 track. Thin line: actual bathymetry observed by R!V Vema. Heavy line: DBDB-5 data setsampled along the Vema 17-11 track.

distance between constraining isopleths. When two contoursare separated by a large distance, we know that the averagegradient in that region is probably small; certainly, thesurface is of bounded variation on that interval. However,the gridding algorithm only sees an unconstrained region.The minimum-curvature solution is clearly wrong in thisapplication. If the surface had a bulge as shown by the heavysolid line in Figure lb, then there would have been another100m contour value somewhere as the bulge turned down tothe continental slope. Including tension in the solution worksvery well (thin solid line). There are two reasons for thissuccess. The first is that the bulge results from an extraneousinflection point, and tension has been shown to eliminatethese inflections (Schweikert, 1966). The second is thatincreasing the tension moves the solution toward the harmonic end member, which can have no local maxima orminima between data points. This feature is well suited togridding isopleth data, of course.Shelf-break bulges are common features in the U.S. Navy's Digital Bathymetric Data Base (VanWyckhouse, 1973;NGDe, 1988). In Figure 4a we have contoured this data setand shaded regions with elevations above -1 5 m, Le.,shallower than 15m below sea level. The shading reveals

00

.' 00

. ' 00

11)11

. SO

40000' 00

200 300Di stance (km)

100

100

. . / VII- - _:: 100

b)

The "shelf-break bulge"Figure Ib contains a vertical cross-section through anactual bathymetric data profile over a continental shelf and

slope (dashed line) and two attempts to reproduce thisbathymetric surface by gridding the coordinates of isobaths(squares). The heavy solid line is a section through theminimum-curvature solution, and the thin solid line is asection through a solution produced with some tension. Theheavy line displays a "shelf-break bulge" which occurswhere an unconstrained area lies between two areas constrained to have different gradients, a situation very similarto the one that produced the "questionable dipole." However, while the dipole mayor may not be real, the shelfbreak bulge is a clear case of extraneous inflection points.In this example we did not use the actual ship's bathymetry (dashed line) to constrain the gridding; instead, wefound the locations of 100m contours of the ship data andused the coordinates of these contours as the controllingdata points. This situation is quite different from the "questionable dipole" example. What we are illustrating here isthat attempts to grid surfaces using the coordinates ofisopleths of the data suffer from a peculiar lack of information. The shelf-break bulge occurs where there is a large

show a feature with a 175 mGal high which was observedalong the northeasterly ship track. Other tracks near thisfeature recorded gravity values near zero. The most significant difference between the two maps is at the unconstrainedlow to the south of the constrained high. The minimumcurvature map suggests a dipole anomaly; there is a - 25mGal low to the south of the 175 mGal high in Figure 3a.Gridding with tension reduces this low to less than 10mGal(Figure 3b). Note that there are no data at all inthe region ofthe dipole low; we cannot say whether this low exists or not.By increasing TI in equation (8), this low can be made todisappear. The user must decide whether he wants this lowin the map or not. For example, if the data in Figure 3 weremagnetic-intensity measurements, a high-low dipole mightbe expected; while if the data were bathymetric soundings,the low might be considered spurious; and in gravity data aflexuralmoat around a seamount high is sometimes reported(e.g., Watts, 1978). The point is that with our method theresult may be adjusted as is geologically appropriate.In Figures 3c and 3d we show perspective views of theshaded portions of the gridding surfaces of Figures 3a and3b. In these views we have "cut away" the data east andnorth of the center of the high and are looking at theremaining region from a vantage point above and northeastof the high. These views illustrate the oscillatory flexure ofthe surface obtained by minimum-curvature gridding. Notein Figure 3d that tension results in a much sharper transitionin the unconstrained area between the constrained high andthe constrained flat regions.In this "questionable dipole" example, it is not obviousthat minimum-curvature gridding has done anything wrong;the validity of the low anomaly is a subjective decision. Alesson to be drawn from Figure 3 is that we should alwaysplot the locations of constraining data on our contour maps.In the next two examples, we show that the minimumcurvature surface produces clearly undesired results.



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300 Smith andWesselprominent bulges in the continental shelf offshore SouthAmerica. Figure 4b shows a profile along the line in Figure4a. The thin line in Figure 4b is the actual bathymetryobserved by R/V Vema cruise 17-11; the heavy line showsthe DBDB-5 depths along the Vema track. We believe thatthe Navy has gridded contours of the original data, resultingin bulgeswhich then have been truncated ad hoc to a -1 0 mlevel. In the next example we grid bathymetry data directly;minimumcurvature produces a bulge in this example as well.Bathymetric map of Broken Ridge

Figure 5ais a bathymetric contour map ofBroken Ridge inthe southern Indian Ocean (Driscoll et aI., 1989). This mapwas hand contoured by a marine geologist at Lamont-

Doherty using bathymetric soundings from the Lamont database and additional data from the Defense Mapping Agencywhich are not in our digital data base. In Figures 5b and 5cwe have tried to approximate the hand-drawn map bymachine gridding and contouring the Lamont data only.Figure 5b is made with minimum curvature; and Figure 5c,with T[ = 0.75.While Figure 5c cannot match the hand-drawn map exactly, the major features are quite similar. There are localdifferences, and we do not know how much the geologistwasinfluenced by the DMA data which are not included in ourmap. The differences between Figures 5a and 5c and theminimum-curvature map (Figure 5b) are quite clear. In theunconstrained areas near the boundaries of the map, the

a)b)

31

30

954 3 96 1 l" ' " - - "" '''I''ir:::::;:=:;;;;::::;;:;::r''!....,---=r::;;=:::;:;==1/ 29 92

c)

FIG. 5. Bathymetric maps of Broken Ridge in the southern Indian Ocean. (a) This map was hand contoured by amarine geologist using bathymetry from the Lamont-Doherty data base and additional data not available to us(Driscoll et al., 1989). (b) and (c) are generated by machine gridding of only the data in the Lamont-Doherty database (locations shown as black squares). Both machine-drawn maps are in general agreement with the hand-drawnmap in the regions well constrained by data. The major differences occur in regions of interpolation andextrapolation. The minimum-curvature solution (b) has large oscillations and a "false island" rising 900 m abovesea level (gray shaded area). A solution using T[ = 0.75 (c) shows good agreement with the human interpretation(a) in the unconstrained regions.



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Gridding withSplines in Tension 301minimum-curvature solution has large undulations and actually rises above sea level for a considerable portion (shadedarea in Figure 5b). This "island" is certainly an unacceptable feature of this map.

DISCUSSIONIt should now be clear that minimum curvature is not the

ideal gridding method for all applications, something onemight have expected from the start. Gridding with tension isan improvement because it adds a degree of freedom andrelaxes the minimum-curvature constraint. We realize thatthere is no physical reason for using a plate-flexure equationto grid data such as gravity or topography measurements.Integral gridding methods can be designed for the physics ofthe par ticular application, such as the norms on harmonicsplines used by Shure et al. (1982) for magnetic data.However, it is commonfor scientists to become familiar withone method of solution and use it in a variety of applications.Also, in the interpretation of potential field anomalies, thegeophysicist must usually relate the anomaly to the topography of the source region; and he or she often wants to treatthe potential field data and the topography data with thesame procedure. A general gridding procedure is therefore apractical necessity.The minimum-curvature method was advocated by Briggs

(1974) for its smoothness propert ies, and Briggs (1974),Swain (1976), and Sandwell (1987) have all used it forpotential-field data, which are expected to define relativelysmooth analytic functions. The smoothest possible twicedifferentiable surface should not approximate topographyvery well; our examples show that minimum curvaturemakes very poor bathymetric maps. It may be that kriging isa bet te r approach , since topography data seem to be wellsuited to stochastic descriptions such as fractals and ARIMAmodels (e.g., Malinverno and Gilbert, 1989). However, thecomplete spatial autocorrelation of data given only alongship tracks is very difficult to compute and include in asimple kriging procedure. We feel that gridding with tensionis an acceptable method for topography data: it is a morelocal scheme than minimum curvature and better reflects thenature of the autocovariance of topography data.Isopleth data are a challenge for any algorithm. Obviously,one should not grid isopleth coordinates if the original dataare available; but frequently gridded values are needed, anda contour map is the only published form of the data.Sometimes it is necessary to grid data which are intrinsicallyisopleths, such as when one makes a gridded age data setfrom the locat ions of magnetic isochrons of the sea floor.Isopleth coordinates contain some information about thelocations of values of a function, but, more important, theydefine regions of bounded variation. We have shown thatgridding with tension works well for isopleth data becausethe end member T[ = 1 in equat ion (8) cannot have localmaxima or minima between constrained points. We do notknow how kriging would work on isopleth data. The notionof the autocovariance of sets of equal values seems ratherproblematic.The minimum-curvature gridding method has been widely

applied to bathymetry and other data for which it is not wellsuited. We have shown that in some applications minimum

curvature gives undesired results, and that including tensionin the solution overcomes the se difficulties. Since it isstraightforward to modify a minimum-curvature algorithm toinclude variable tension, we suggest that earth scientists whoalready use minimum-curvature algorithms should includethis feature. We do not claim that our method is ideal in allapplications. However, we feel that continuous curvaturegridding with adjustable tension provides the flexibility tohandle many cases which arise in the earth sciences.

ACKNOWLEDGMENTSWe wish to thank N. W. Driscollfor allowing us to use his

map of Broken Ridge (Figure 5a). W. F. Haxby and A. B.Watts brought to our attention the importance of regionalfields and solutions to the related homogeneous equation. W.Menke suggested the temperature field analogy for the caseT[ = 1. A. Lerner-Lam, A. Malinverno, P. G. Richards, andthe reviewers and editors of GEOPHYSICS provided manyhelpful comments. This work was supported in part by Officeof Naval Research con tr ac t TO-204 scope I and Nationa lScience Foundation contract no. OCE-86-14958.Lamont-Doherty Geological Observatory contribution

number 4555.REFERENCES

Ahlberg J. H., Nilson,E. N., andWalsh,J. L., 1967, The theory ofsplinesand their applications: Academic Press Inc.Berg, P.W., andMcGregor, J. L., 1966, Elementarypartial differentialequations: Holden-Day Inc.Bracewell, R. N., 1978, TheFourier transformand its applications:McGraw-Hill International.Brandt,A., 1984, Multigrid techniques: 1984 guidewithapplicationsto fluid dynamics: Gesellschaft fur Mathematik und DatenverarbeitungmbHBonn.Briggs, 1. C., 1974, Machine contouring usingminimum curvature:Geophysics, 39, 39-48.Clark, 1., 1979, Practical geostatistics: Applied Science Publ. Ltd.Cline,A., 1974, Scalarand planar-valued curve fitting usingsplinesunder tension: Comm. ACM, 17,218-223.Committee for the gravity map of North America, 1987, Gravityanomaly map of North America: Geol. Soc. Am.Committee for the magnetic mapof NorthAmerica, 1987, Magneticanomaly map of North America: Geol. Soc. Am.Davis,J. C., 1986, Statistics anddata analysisin geology: 2nd. ed.,JohnWiley & Sons, Inc.deBoor,C., 1978,Apracticalguideto splines: Springer-VerlagNewYork, Inc.Driscoll, N. W., Karner, G. D., Weisse!, J. K., and the ShipboardScientific Party, 1989, Stratigraphic and tectonic evolution ofBroken Ridge from seismic stratigraphy and Leg 121 drilling:InitialRep. OceanDrilling Prog., 121, 71-91.Fulton, S. R., Ciesielski, P. E., and Schubert, W. H., 1986,Multigrid methodsfor ellipticproblems: a review:Am.Meteorol.Soc. Monthly WeatherRev., 114,943-959.Hackbush,W., andTrottenberg,U., Eds., 1982, Multigrid methods:Lecture notes in mathematics, 960,Springer-Verlag.Harbaugh,J. W.,Doveton,J. H., andDavis,J. c., 1977, Probabilitymethods in oil exploration: JohnWiley & Sons, Inc.Inoue, H., 1986, A least-squares smooth fitting for irregularlyspaced data: Finite-element approach using the cubic B splinebasis: Geophysics, 51, 2051-2066.Lancaster, P., and Salkauskas, K., 1986, Curveand surface fitting:Academic Press Inc.Love, A. E. H., 1927, A treatise on the mathematical theory ofelasticity,4th ed.: Dover Publ. Inc.Malinverno, A., andGilbert,L. E., 1989, A stochasticmodelfor thecreationof abyssal hill topography at a slow spreading center: J.Geophys.Res., 94, 1665-1675.National Geophysical Data Center, 1988, ETOPO-5 Bathymetry!Topography Data, Data Announcement 88-MGG-02: NationalOceanic and Atmospheric Administration, U.S. Dept. Commerce.



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302 Smith and WesselOlea, R. A. , 1974,Optimal contour mapping using universal kriging:J. Geophys. Res., 79, 696--702.Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling,W. T., 1986, Numerical recipes: Cambridge Univ. Press.Richardson, L. F., 1910,The approximate arithmetical solution byfinite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam: Phil.Trans. R. Soc. London, Ser. A, 210, 307-357.Roache, P. J., 1982, Computational fluiddynamics: Hermosa Publ.Sandwell, D. T., 1987, Biharmonic spline interpolation of GEOS-3and SEASAT altimeter data: Geophys. Res. Lett. , 14, 139-142.Schweikert, D. G., 1966, An interpolating curve using a spline intension: J. Math. Physics, 45, 312-317.Shure, L., Parker, R. L., and Backus, G. E., 1982, Harmonicsplines for geomagnetic modelling: Phys. Earth Plan. Int. , 28,215-229.Spath, H., 1973, Spline-Algorithmen zur Konstruktion glatter Kurven and Flachen: R. Oldenbourg Verlag; English translation by

Hoskins, W. D., and Sager, H. W., 1974, Spline algorithms forcurves and surfaces: Utilitas Mathematica Publ.Swain, C. J., 1976, A FORTRAN IV program for interpolatingirregularly spaced data using the difference equations for minimum curvature: Computers and Geosciences, 1, 231-240.Timoshenko, S., and Woinowsky-Krieger, S., 1968, Theory ofplates and shells: 2nd ed., McGraw-Hill Book Co.Van Wyckhouse, R., 1973, SYNBAPS, Tech. Rep. TR-233: U.S.National Oceanographic Office.Watts, A. B., 1978,An analysis of isostasy in the world's oceans, 1.Hawaiian-Emperor seamount chain: J. Geophys. Res., 83, 59896004.Wegman, E. J., and Wright, I. W., 1983,Splines in statistics: J. Am.Stat. Assn., 78, 351-365.Wessel, P., 1989, XOVER: A cross-over error detector for trackdata: Computers and Geosciences, 15, 333-346.Wessel, P., and Watts, A. B., 1988, On the accuracy of marinegravity measurements: J. Geophys. Res., 93, 393-413.Young, D., 1954, Iterative methods for solving partial differenceequations of elliptic type: Trans. Am. Math. Soc., 76, 92-111.

APPENDIXSOLUTION BY ITERATION OF FINITE-DIFFERENCE EQUATIONS

Difference expression for the homogeneous equationOur notation is shown in Figure 2. For convenience weuse subscripts to refer to the relative position of a grid nodewith respect to a local origin. Thus Zoo refers to the current

zij, and ZI_I appearing in an equation with Zoo refers toZi+lj-I. We approximate derivatives by central finite differ-

Using the finite-difference approximations (A-I) and (A-2),the homogeneous equation(A-3)

may be solved for zoo:

(A-4)

ences, e.g.,aZZ ZIO - 2z oo +Z - 10ax z= (.


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Griddlng withSplines in Tension 303We do this at five distinct points ( ~ "k), k = 1,5. Then wemultiply each expansion by a real number bk and sum thefive expressions:

The above matrix expression is easily solved to yield2(1 + ( 2)bs = ,( + a,,)(1 + + a,,)

b 4 = 1 - !b s ( + e),b az 1 2 a2z+ L k"k - + - L b k ~ -1ay 2 ax-a2z 1 2 a2z+ L b k ~ k " k --+- L bk"k -2ax ay 2 ay

I f the b, are chosen such thatL b k ~ = L bk"k = L b k ~ k " k = 0 andL bk ~ = L bk"l= 2

b3 = a,,(l + ~ ) b s - 2b4 , (A-6)bz = a,,(1 + ~ ) b 5 - b3 , andb, = $s + b4 - b2

For a constraining datum in other quadrants, analogousexpressions may be obtained.The constraint is implemented by substituting equation(A-5) into equation (A-3) and solving for zoo:

L .;. .J

~ - - - - - - - * - - ~I E II II I:11 IIIII

(A-5)then

[-: - - -: ~ ] [::] = [ ],- 1 0 0 - 1 ~ a b4 01 0 1 1 a 2,, 2 bs 2a 2

where we have used the fact that the grid dimensions havebeen normalized by Ax and a is the anisotropy; here and a"represent fractional distances on the grid,

While any five points which yield a nonsingular expressionfor the bk may be chosen, it is convenient to use four nearbygrid node values and one off-grid constraint. For example,suppose that zoo is the location at the square in Figure A-I,and that we wish to implement the datum at E in Figure A-Ias a constraint. Let us assign k = 1, 4 to other points on thegrid (A-D in Figure A-I), and k = 5 to the point E. Then weseek bk satisfying

(XE -xoo)~ = - - -J1x

or

FIG. A-I. The grid node indicated by the black square is to beconstrained by the datum at E. A Taylor series expansion ismade from the node to the five points (A-E). Since four ofthese are "known" (they are other points on the grid whichare solved in separate steps), they are used to eliminateterms in the Taylor series, leading to an expression for theLaplacian at the node which includes point E as a constraint[equation (A-5)]. When point E is in the first quadrant, A-Dmay be chosen as shown to yield equation (A-6). For E inanother quadrant, A-D may be chosen by rotating this figureappropriately; this willmodify the system for equation (A-6).



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304 Smith andWessel

Difference expressions for boundary conditions

The second outside points may then be set using boundarycondition equation (4):

(A-lO)Z ~ 2 o = Z 2 o + a 2 ( Z I I +ZI- I -Z-I I -Z- I - I )

We solve equations (A-4), (A-7), (A-8), (A-9), and (A-IO)iteratively. One iteration consists of one application of theappropr iate equat ion at each posi tion zi j to update eachvalue. The updating is done immediately so that typicallyhalf of the points in the equation are "new" values and halfare "old" values (Gauss-Seidel method; e.g., Press et aI.,1986). For programming convenience, we visit these pointsin sequence in loops over the i and j indices. A nonunityanisotropy factor a effectively "couples" the equationsmore strongly in one dimension, and the convergence ismore efficient if the loops are nested so tha t the stronglycoupled direction is looped first.The coefficients on the 12 points of Figure 2 are determined from equat ion (A-4) or equat ion (A-7) and are constant during the iteration process; only the zi j change. Wetherefore compute arrays of these coefficients once, prior toentering the iteration loop. This prior step includes sortingthe constraining data into the order in which they will beneeded in the interact ion loop, and computing and storingthe bk used with each constraint.In the iteration loop itself. we use an overrelaxationparameter I < W < 2 to accelerate convergence. Thedifference equat ion is used to compute a new value for zij 'and the change in zi j is increased by the overrelaxat ionfactor:

Solution by successive overrelaxation

(A-9)

If the constraining datum (E in Figure A-I) lies exact ly onthe grid node, then ~ and TIE are both zero and the abovematrix is singular. However, in this case the grid node valueZoo may simply be replaced by Zt: without using equation(A-7). In this way, the gridded surface always interpolatesthe constraining datum to second order in the Taylor series.

After equation (A-8) has been applied, we use boundarycondition equation (5) to set the auxiliary "corner" point:

Application of equations (A-4) and (A-7) throughout thedesired domain of (x, y) requires two additional outside rowsor columns of auxiliary points (Figure A-2). We express theboundary conditions at an x edge here; the expression for they edges are analogous.We set the first outside points using boundary conditionequation (9):

new "' - (I ) old newZi j '" - w Zu + WZij .- -- ---e-----e----

0 --- --6.--- - -----e--- e---0 -- ---6.----- - - - - - - - - - - -

This method is called successive overrelaxation or simultaneous overrelaxation and is well known (Richardson, 1910;Young, 1954; Roache, 1982; Press et aI., 1986). Spectralanalysis of the iteration operator may, in theory, yield anoptimal value for w. In our application the best W depends onthe tension used; we have determined empirically thatW = 1.4 works well for T = 0, and w may be increased as Tis increased. The system is considered "converged" to thelimit e when

I _new _ oldmax "-ij zi j I < f.- - - E - - - - - - - ~ - - - - - ~ - - - Multiple grid strategy

-----0 ----- 0 -- -0---FIG. A-2. Implementing boundary conditions. Points in thelower left corner of the desired grid are represented by theblack circles. The array containing the desired grid is augmented by two addi tional rows and columns surroundingeach boundary. These exterior points allow application ofequa tion (A-4) or equa tion (A-7) at every interior point.During each iteration, equation (A-8) is used to set the valuesof the white triangles, (A-9) the corner point (circle and crosssymbol), and (A-lO) the white diamonds. The grey trianglesand diamonds on the y boundaries are set using analogousexpressions but including the anisotropy factor a. Only onecorner of the array is shown here but the ent ire boundary isset in a similar manner.

We use a system of grids of various mesh sizes to enhancethe efficiency of convergence of the system (A-4) and (A-7).We derived our method by a generalization of a technique inthe minimum-curvature algori thm of Swain (1976). Ourmethod shares some similarities with the multigrid methodsdeveloped for the second-order equations of fluid dynamics(Hackbush and Trottenberg, 1982: Brandt, 1984; Fulton etaI., 1986).In the iterative solution, the array Zstarts with some initialvalues which are then changed by an amount ilz when

convergence is achieved. From equations (A-4) and (A-7)and Figure 2, it can be seen that in each iteration the newvalue computed for each zi j is a weighted average of twelveneighboring values. The iteration operator is a local smoothing process, and as a consequence short-wavelength compo-



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Gridding with Splines in Tension 305nents of ~ are found quickly. Conversely, iteration does notefficiently propagate the effects of data constraints to longwavelengths. For this reason our algorithm does not beginiteration on the array which is ultimately desired; instead,we first find a long-wavelength solution on a coarser meshconsisting of every Nth point in the x and y dimensions of thefinal (desired) array. We begin with the largest N whichdivides both grid dimensions (and leaves at least four pointsin each direction so that some work needs to be done). Theabove system is solved to convergence on this sparse lattice.Then we divide N by its largest prime factor, exposing newnodes in a finer mesh. These new points are initialized byinterpolation from the previous mesh, and then the system ofequations on this new mesh is again iterated to convergence.We continue this cycle until N = I and the full system hasbeen solved.Multigrid techniques (Hackbush and Trottenberg, 1982;Brandt, 1984; Fulton et al., 1986) include both coarseningand finingmesh transfers in sequences called V-cycles. Weuse the simpler approach of starting with the coarsestconvenient grid and successively fining (one half of oneV-cycle). It can be shown (Ahlberg et al., 1967) that acoarse-mesh spline is the best estimator of a fine-meshspline; in this sense we are starting with optimal initial valuesat each successive stage, since only short-wavelength perturbations to the solution need to be found. These localcorrections are exactly what iteration of equations (A-4) and(A-7) performs efficiently. The computing time spent on thecoarse meshes is small because the number of points in eachlattice is only l/N 2 of the final number of points; the use ofa series of meshes results in fewer iterations on the final(N = 1) stage and less total run time than if the solution hadbegun directly on the final grid. The coarse stages run so fastthat we actually use EIN as the convergence limit at eachstage, where E is the convergence limit set by the user for the

final stage. This allows the user to choose a reasonable limitto be used on the final grid when the iterations are slow, butmakes a better approximation of the long-wavelength components without much increase in total run time.Equation (A-7) is constructed from the condition that thegrid must interpolate the data constraints exactly (to secondorder in the Taylor series), and thus the prediction error ofthe surface would be zero ifthe equations could converge toE = O. In practice, we have observed that this sequence ofcoarse grids with division of N by its largest prime factor ateach stage results ina smaller prediction error than any othersolution strategies we tried. The minimum-curvature solverof Swain (1976) uses a similar sequence of coarse grids,except that his sequence of grid mesh N values is limited topowers of two and the initial N must be chosen by the user.Our algorithm allows any N and finds the initial N automatically. In map applications using a latitude-longitude mesh inminutes of arc, the grid dimensions commonly have factorsof 3 and 5 as well as 2; and in these cases our mesh systemgoes through more intermediate stages than Swain's algorithm. Wefind that these extra states result infaster total runtime, smaller prediction error, and better overall visualquality of the surface.With the use of multiple grids, each lattice is initialized byan interpolation from the previous stage, and therefore onlythe first (coarsest) lattice needs to be seeded with initialvalues prior to iteration. In Swain's (1976) algorithm, this isdone by a weighted average of points inside a user-specifiedradius. We have retained this feature as an option to be usedwhen the grid dimensions have few common factors and thecoarseness factor N starts near 1. However, because wehave removed a planar trend from the data prior to iteration,we find that in most cases involving several regional gridstages it is adequate to begin with the coarse-lattice valuesinitialized to zero.

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Smith Wessel 1990 Gridding With Continuous Curvature Splines in Tension

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