Abstract
Nonlinear, nonlocal and adaptive optimization algorithms, now readily available, as applied to parameter estimation problems, require that the data to be inverted should not be very noisy. If they are so, the algorithm tends to fit them, rather than smoothening the noise component out. Here, use of Bernstein polynomials is proposed to prefilter noise out, before inversion with the help of a sophisticated optimization algorithm. Their properties are described. Inversion of gravity and magnetic data for basement depth estimation, singly and jointly, and without and after Bernstein-preprocessing is conducted to illustrate that the inversion of Bernstein-preprocessed gravity data alone may be slightly superior to the joint inversion of gravity and magnetic data.
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References
Bernstein S 1912/1913 Demonstration du theoreme de Weierstrass, fondee sur le calcul des probabilites;Comm. Soc. Math. Kharkow 13 2 1–2
Bernstein S N 1952 On the distribution of zeroes of the polynomials tending to a continuous function positive on a given interval; InComplete Works, Vol. I, Akad. Nauk USSR, 443–451
Cal J D L and Luquin F 1992 A note on limiting properties of some Bernstein-type operators;J. Approx. Theory 68 322–329
Chang G Z and Davis P J 1984 The convexity of Bernstein polynomials over triangles;J. Approx. Theory 40 11–28
Dennis J E Jr 1977 Nonlinear least squares, In:State of the art in numerical analysis (ed.) D Jacobs (Academic Press) 269–312
Ditzian Z 1987 Rate of convergence for Bernstein polynomials revisitedJ. Approx. Theory 50 40–48
Ditzian Z and Ivanov K 1989 Bernstein-type operators and their derivatives:J. Approx. Theory 56 72–90
Dobrin M B and Savit C H 1988 Introduction to geophysical prospecting (New York: McGraw-Hill) 525
Garland G D 1965The Earth’s shape and gravity (Oxford: Pergamon Press) 72
Goel A K 1979 Oct., Some preprocessing operators for geophysical interpretation: M. Tech. Dissertation, Dept. of Earth Sciences, University of Roorkee, Roorkee, 72
Hirschman I I and Widder D V 1955 The convolution transform; (Princeton: Princeton University Press) 268
Karlin S 1968Total positivity (Stanford: Stanford University Press)
Karlin S and Ziegler Z 1970 Iteration of positive approximation operators;J. Approx. Theory 3 310–339
Krasnosel’skii M A 1964Positive solutions of operator equations (Trans. Flaherty R. E., ed. Boron L. F.) (Groningen: P. Noordhoff Ltd.) 381
Lapidot E 1984 Convexity preserving and predicting by Bernstein polynomials;J. Approx. Theory 40 298–300
Lorentz G C 1953Bernstein polynomials (Toronto: University of Toronto Press)
Marquardt D 1963 An algorithm for least-squares estimation of nonlinear parameters;J. SIAM J. Appl. Math. 11 431–441
More J J 1977 The Levenberg-Marquardt algorithm: implementation and theory, in Watson, G. A. (ed.), (Springer-Verlag: Numerical analysis) 105–116
Radhakrishna Murthy I V and Mishra D C 1989 Interpretation of gravity and magnetic anomalies in space and frequency domains; AEG, Hyderabad, 78–89
Ramakrishna R S 1978 Some iterative techniques in digital image restoration: Ph. D. Thesis, Department of Electrical Engineering, Indian Institute of Technology, Kanpur
Ramakrishna R S, Mullick, S K, Rathore R K S and Subramanian R 1979 Orthogonalization, Bernstein polynomials and image restoration;Applied Optics 18 464–468
Sahai A and Prasad G 1984 Sharp estimates of approximation by some positive linear operators;Bull. Austral. Math. Soc. 29 13–18
Schoenberg I J 1947 On totally positive functions, Laplace integrals and entire functions of the Laguerre-Polya-Schur type;Proc. Nat. Acad. Sci. 33 11–17
Schoenberg I J 1948 On variation-diminishing integral operators of the convolution type;Proc. Nat. Acad. Sci. 34 164–169
Schoenberg I J 1958 On variation diminishing approximation methods;Proc. Symp. on Numerical Approximation (Madison: University of Wisconsin Press) 249–274
Schurer F and Steutel F W 1977 The degree of local approximation of function in C1 [0,1] by Bernstein polynomials;J. Approx. Theory 19 69–82
Tikhonov A N and Arsenin V Ya 1974 Methods for solving improperly posed problems; Nauka, Moscow, 223
Varshney O P and Singh S P 1982 On degree of approximation by positive linear operators;Rend. Mat. 7 2, 219–225
Wójcik A P 1991 Some remarks on Bernstein’s theorems;J. Approx. Theory 67 252–269
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Srinivas, S., Moharir, P.S. & Maru, V.M. Bernstein preprocessing for nonlinear least squares inversion of gravity and magnetic data. Proc. Indian Acad. Sci. (Earth Planet Sci.) 108, 269–275 (1999). https://doi.org/10.1007/BF02840504
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DOI: https://doi.org/10.1007/BF02840504