Parameter nonuniqueness in velocity versus depth functions

Abstract

Except for hypothetical cases that are never encountered in practice, the solution for the parameters of analytic velocity versus depth functions is inherently nonunique. This nonuniqueness means that there is no particular parameter combination that represents “the” solution. The average discrepancy between the observed velocity‐depth (or time‐depth) curve and the calculated function curve gives a measure of the degree of fit between the two curves. In the parameter space, within a given degree of fit that may be considered as the margin of tolerance for the particular problem in hand, every parameter combination provides a valid solution. The solution trough is defined as the region in the parameter space containing these solutions. The shape of the solution trough depends on the relationship between the parameters and, therefore, on the form of the function being used. If the parameters are related linearly or if the relation is linearized then the axis of the solution trough is a straight line. The slope of the axis corresponds to a point close to the average depth of the interval to which the observed data pertain. The intercept corresponds to a velocity value that is quite close to the average velocity over the interval. These properties are extremely useful for understanding the nature of the solution trough in parameter space. Hence, composite displays of solution troughs pertaining to a number of wells can be interpreted to provide information on whether the unit to which the function corresponds has undergone any lateral changes. Such displays also illustrate how current practices that assume the existence of a unique or “true” solution need to be critically revised.