C1-Approximationof Seafloor Surfaces With Large Variations Christian Gout' and Dimitri Komatitsch2 Department of Applied Mathematics, IJniversit6 de Pau, E.R.S. 2055-CNRS, 64000 Pau, France, Christian.gout Quniv-pau.fr 2Department of Earth and Planetary Sciences, Harvard IJniversity, Cambridge, Massachusetts 02138, IJSA, komat itsQseismology.harvard. edu
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In many problems of geophysical interest, when trying t o describe surfaces,one has t o deal with data t h a t exhibit rapid variations. This occurs for instance when describing t h e topography of mountain ranges, volcanoes, seafloor surfaces (bathymetry maps), islands, or t h e shape of geological entities, t h a t can present large and rapid variations due for instance to the presence of faults in t h e structure. T h e correc,t description of such geological surfaces, by a fitting process from a given set of points, is therefore of great importance. Usual methods give good results in the case of curve fitting, but less accurate results in the case of surface fitting. T h e new method we propose here uses scale transformations (spline under tension), and is applied without any particular a priori knowledge of t h e data. We first propose a construction of these scale transformations families, and then show t h e efficiency of this innovative approach by applying it to seafloor surfaces around t h e Big Island in Hawaii in order t o get a regular surface with at least continuity of t h e first derivatives. DESCRIPTION O F T H E METHOD T h e method we propose uses two scale transforniations, namely (Pd for t h e pre-processing and ?)d for the post-processing. T h e first one, ( P d , is used to transform t h e z-values representing t h e height of t h e unknown surface f into values ( U ; ) , regularly distributed in a n interval chosen by t h e user. T h e preprocessing function (Pd is such t h a t the transformed data do not exhibit large local variations, and therefore a usual spline operator T d can subsequently b e applied without generating significant oscillations. T h e second scale transformation ?)d is then applied t o the approximated values t o map them back and obtain t h e approximated values of z. It is important t o underline that t h e proposed scale transformations d o not 0-7803-6359-O/OO/$lO.OO0 2000 IEEE
create spurious oscillations. Moreover, this method is applied without any particular knowledge of t h e location of t h e large variations in t h e dataset. Let us consider a dataset (xt,~ t ) : = ~ , , , , ~ indexed (~) with a real d , such t h a t when ti tends t o 0, the number of d a t a points N ( d ) tends to infinity. For t h e purpose of a theoretical study of t h e convergence of t h e approximation, we introduce a function f : R -+ [a,b] , such d t h a t t h e dataset becomes ( X : ; , Z ; = f ( ~ ? ) ) ~,,,, = N, ( d ) . T h e functions introduced above have t h e following expression, for m E IN: (Pd : [a,b]
------)
- Td : ( ( P d o f ) H'" (R, [a,PI>
[O,P]
c R,
E H'"(R,[O,P])
+
Td((PdOf)
E
7
- $d
(Td((Pd f ) )
9
where the preprocessing (Pd iind t h e post-processing ?)d are continuous scale transformations families, where T dis a n approximation operator, for instance a spline, and where H"(R, .) denotes t h e usual Sobolev space. More precisely, we introduce a bounded non empty connected set R with a Lipschitz-continuous boundary of R2, and an unknown function f E H'"' (a,[a,b ] ) t h a t we want t o approximate. We also introduce t h e set 2: of N = N ( d ) real numbers such that
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Vxt E Ad,f(x:f) E Z,", and t h e sequence 2; of p ( d ) distinct z-values obtained from t h e ordering of Z,", Vi$ cf Z,d,z = 1,...,p ( d ) ,
a = 5: < 5;
I REFERENCES
[ 11 R. Arcangkli, “Some applications of discrete D”splines”, in Lyche T. and Schumaker, L. I,., eds.,Mathematical Methods in Computer Aided Geometric Design, Academic Press, New York, p. 35-44, 1989. [2] M. Bouchon, C. A. Schultz, and M. N. Toksoz, “Effect of three-dimensional topography on seismic motion”, Jour. Geophys. Res.,v. 101, pp. 5835-5846, 1996. Figure 2 - 3D view of t h e approxiniant
approximant after post-processing, evaluated on a n evenly spaced grid comprising 200 x 200 points. To compare this approximant to t h e original dataset more precisely, in Fig. 3,
[3] A. Frankel, and W . Leith, “Evaluation of topographic effects on P and S waves of explosions at the northern Novaya Zeinlya test site using 3D numerical simulations”, Geophys. Res. Lett., V. 19, p. 1887-1890, 1992.
[4] C. Gout, and D. Komatitsch, “Surface fitting of rapidly varying data using rank coding: application t o geophysical surfaces”, Mathematical Geology, in press (to appear Fall 2000). [5] K. Ishihara, M. Iguchi, and K. Kamo, “Numerical simulation of lava flows on some volcanoes in Japan”, in Fink, J. H., ed.,Lava flows and domes: Springer-Verlag, Berlin, p. 174-207, 1990.
[6] E. H. Issaks, and R. M. Srivastava, “An Introduction to Applied Geostatistics” , Oxford University Press, Oxford, 560 p., 1989. X
[7] D. Komatitsch, and J. P. Vilotte, “The Spectral Element method: a n efficient tool t o simulate the seismic response of 2D and 3D geological structures”, Bull. Seis. Soc. Am., v. 88, p. 368-392, 1998.
Figure 3 - 2D view of t h e approximant
we present a top view of t h e approximated values, with isocontours representing the height every 0.2 km, in addition t o t h e same plot for t h e original dataset, as in Fig.1 . It is clear from these plots that the approximant is very close te the original data, with local variations smoothed as expected. More detailed studies of the approximation error, and evidence that the rate of convergence is higher in this method than in usual approaches with no preprocessing, such as splines under tension or thin plate splines, can be found in [4].
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[8] D. Komatitsch, J. P. Vilotte, R. Vai, 1. M. Castillo-Covarrubias, and 17. J. SBnchez-Sesma, 1999, “The Spectral Element method for elastic wave equations: application to 2D and 3D seismic problems”, Int. Jour. Numer. Meth. Engng., V. 45, p. 1139-1164, 1999.
