AL-CHALABI,

A rigorous proof is presented to show that over a horizontally layered ground the rms velocity cannot exceed the stacking velocity. The proof helps to illustrate the difference between stacking and rms velocities in a quantitative manner. The series of Taner and Koehler (1969) is used for the purpose. Convergence of this series is tested. Including more terms will not necessarily improve the convergence. Although the series is rapidly convergent when the spread length/depth ratio is small, strong oscillations are observed when this ratio is high.

Polygonal models which are of uniform density or magnetization contrast and which contain no cavities are sufficient to ensure the uniqueness of solutions in gravity and magnetic problems. Nonuniqueness is then attributed to a number of other factors which are discussed in detail. We study the problem of nonuniqueness using mainly a parameter hyperspace in which ambiguity takes the form of a scatter of local minima or a continuous domain bounded by a contour whose value is determined by the amplitude of observational errors. The possible solutions to the problem being examined are contained in a region which would have contained the unique solution under exact conditions and which decreases in t Manuscript received by the Editor June 30,197O; revised manuscript received March 8,197l. 836 Al-Chalabi but each emphasizing a certain aspect of the density (or magnetization) contrast with a unianomalous feature. form surrounding medium. A further related point is the lack of exact adherence to the conditions assumed by the model. A familiar example is the use of two-dimensional models to interpret anomalies which are only approximately two-dimensional. 3. Any line vertical to the x axis will not meet the bounding surface more than twice. The absence of cavities is an importanl implication of this condition. IV) Observational errors resulting from measurement, reduction, etc., are always present on field anomalies. This factor is of prime importance because it determines the limit within which a proposed solution must satisfy the observed anomaly. V) Instability in the inverse solution may be introduced if the widths of individual parts of the model are small compared with the depth. A large number of highly oscillating models may then produce anomalies which closely agree with the observed anomaly Another cioseiy related factor is the decrease of the resolving power of gravity and magnetic methods with depth. When a large number of sides are used to represent a polygonal model, the two factors, i.e., small ratios of width to depth and decrease of resolving power, can often become significant, so that the problem becomes ill-conditioned. The two factors have been amply described in the literature in various forms (e.g., Skeels, 1947; Bullard and Cooper, 1948;

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