LEAST SQUARES SMOOTHING OPERATORS

Abstract

Smoothing data to extract desired trends has been standard scientific and engineering practice for many years. Use of polynomials in a least squares sense to accomplish this end has also been conventional procedure, and it is well known that smoothing acts as a low‐pass filter. However, detailed analysis of filtering behavior is lacking in the literature and should be useful to geological and geophysical data processors. This paper has two objectives: to review least squares polynomial smoothing and to discuss some z‐transform properties of the convolution operator that implements the smoothing. These operators have real, symmetrical coefficients that lead to z‐polynomials having roots lying in unique patterns. Zeros occur in complex conjugate pairs that are also inverse points with respect to the unit circle. Polynomials of order 2M and 2M+1 produce identical operators; thus, no differences exist in smoothing between polynomials of order 0 or 1, 2 or 3, 4 or 5, . . , 2M or 2M+1. A result of fitting a polynomial of order 2M to n+1 data points in a least squares sense is that exactly n−2M roots lie on the unit circle, whereas 2M zeros have magnitudes other than unity. Of the 2M roots lying off the circle, polynomials of orders, 2, 6, 10, 14… have exactly two positive, real roots while those of orders 4, 8, 12, 16… have no roots that are positive and real. Zeros lying on the unit circle influence principally the passband, reject band, and reject level. Roots lying off the circle, on the other hand, mainly control the rolloff rate. Several figures illustrate how an interpreter can use this knowledge to help in choosing the number of points and orders of polynomials required to smooth data of various kinds: gravity, magnetics, electrical, well log, stratigraphic. The least squares weights also apply to array design.