FREQUENCY DOMAIN ANALYSIS OF LEAST SQUARES POLYNOMIAL SURFACES WITH APPLICATION TO GRAVITY DATA IN THE PEDREGOSA BASIN AREA (ARIZONA, MEXICO).
Abstract
The fact that least squares polynomial surfaces can be used to effect a form of low pass filtering (Coons et al., 1967, 1963) is well known, and they are frequently incorporated in anomaly separtation procedures for gravity studies (Simpson, 1954). A detailed analysis of the precise filtering effects (amplitude and phase response) of least squares polynomials is provided for the special case of regularly spaced data. Response properties are also considered for polynomial residuals (high pass filters) and polynomial differences (bandpass filters).
The analysis is based on a simplified form of the normal equations which arises from the use of orthogonal polynomials. A convolution-like form is derived for the least squares solution which is interpreted to represent the application of shift-varying filters to the original data. Thus, a separate filter response function is associated with each observation location.
An analysis of each of the response functions for a given data set and a given degree polynomial confirms the essential low pass character of least squares polynomials. The response functions also reveal that a bandpass nature exists near the edge of the data set. The phase response is zero at the center of the data set and remains near zero and approximately linear within the passband for all other data locations. The filtering effects are summarized by considering an average bassband width computed from the center three-fourths of the data set. Various combinations of the degree of polynomial and the number of observations are used to generate a family of curves representing average passband widths.
The role of weighting various observations is seen to be related to the concept of data sampling. Zero weights which do not alter the regular spacing of the data can be viewed within the framework of the filtering analogy. Arbitrary weighting schemes are not described in terms of the filter response functions.
The least squares polynomial method is applied to gravity data from the Pedregosa basin area of the southwestern United States and northern Mexico. A suite of surfaces of various degrees is used to show the filtering properties produced. A suite of weighted surfaces is provided to demonstrate the resulting variations in the polynomial surfaces. The weighting scheme employed is successful in reducing the effects of near surface geology (Basin and Range features) so that the weighted regional trend is more indicative of older and deeper geologic structures.
Subject Area
Geophysics
Recommended Citation
LANCE, JAMES ODELL, "FREQUENCY DOMAIN ANALYSIS OF LEAST SQUARES POLYNOMIAL SURFACES WITH APPLICATION TO GRAVITY DATA IN THE PEDREGOSA BASIN AREA (ARIZONA, MEXICO)." (1982). ETD Collection for University of Texas, El Paso. AAI8303574.
https://scholarworks.utep.edu/dissertations/AAI8303574