A Gauss Elimination Method for estimating locations of extrema in gridded data: Applications for Potential Field Data

Abstract. Extrema in gravity measurements can be used to locate geological structures of interest and the boundaries of such structures can be associated with the maxima in the gradients of the gravitational field strength. Finding the extrema of measured geophysical fields measured on the Earth's surface when the data is sparse is challenging. The inferred positions of such extrema are highly model dependent. Polynomial functions of two variables can be fitted to the data. Higher order polynomials typically give more accurate determination of the extrema, but the maximum order of the polynomial is limited by the number of data points. Difficulties are accentuated in the vicinity of boundaries of the existing data. The maximum horizontal gradient method has often been applied in this context. But in that particular construction, quadratic functions are developed in each dimension. Although the magnitudes of the extracted coefficients are obtained from three points related by their positions on orthogonal straight lines, off axis information should be included as well. The present paper introduces a modification of the maximum horizontal gradient method to overcome these difficulties. A Function f of the two variables x and y: f_(x,y) = a₁x² + a₂y² + a₃x²y² + a₄x²y + a₅xy² + a₆xy + a₇x + a₈y + a₉ is established by Gaussian elimination method base on a 3x3 neighborhood data grid. An extract creates a 4-dimensional space based on 4 specific cases of function f, including x = 0, y = 0, y = −x and y = x, they are four functions of one variable. The extreme points position are detected from these functions of one variable. To prove the proposed theoretical basis, as well as the built computer program, the paper presents two numerical models. The obtained results shown that the new approach has more maxima points than the traditional approach. Beside advantages of new approach, some disadvantages is also discussed in this paper. Moreover, we conclude with the application of our new approach to gravitational data in the East Vietnam Sea and demonstrate that we thereby disclose the existence of a gravity trench undetectable in the traditional method.