Showing papers by "Douglas G. Martinson published in 1984"

Comments on "An Inverse Approach to Signal Correlation"

Abstract: In a recent paper by Martinson et al. [1982], a quantitative method for recovering the mapping function which makes a distorted data series resemble a reference series using a parameter estimation technique (not an inverse approach) was proposed and illustrated. The mapping function describes the stretching and compressing of one data set with respect to another and may be of considerable geologic interest; for example, the sedimentation rate is the mapping function obtained by correlating a stratigraphic variable with some known measure of time. Since the application of the method has aroused some interest, especially among paleoclimatologists, it seems appropriate to comment on some limitations and omissions in the algorithm of Martinson et al. [1982]. Specifically, we show that (1) their use of maximization techniques on a highly nonlinear functional of the data, the coherence, requires considerably more attention to numerical stabilization than the simpler minimization of a nonlinear penalty function, (2) their use of a Fourier series can, under very general circumstances, produce artificial, high-frequency fluctuations in the mapping function, (3) their failure to constrain the derivative of the mapping function to be nonnegative can yield nonphysical results in many cases of geologic interest, especially for a large number of degrees of freedom in the model, and (4) it is difficult to add additional equality constraints on the mapping function at known tie points using their approach. The nonnegativity constraint is not overly restrictive for most interesting applications ince, for example, the stratigrapher often cannot differentiate between zero and negative sedimentation rates. The equality constraint can be quite useful; the paleontologist can include the results of radiometric and magnetostratigraphic studies in the mapping function calculation. To accommodate points 3 and 4 and resolve 1 and 2, we choose to recast the problem into one of linearized optimization with linear inequality constraints and to parameterize the mapping function in terms of splines rather than a continuous orthogonal basis like sinusoids. To illustrate the advantages of these constraints and the spline basis, we correlate two oxygen isotope stratigraphies and give examples of both nonphysical mapping functions and artificial wiggliness introduced by a Fourier series representation. We are given a reference signal R (t) and a data series