The advantages and limitations of Prony's (1795) method as a scheme for model identification are explored. A description of the data requirements is followed by a presentation of a theory for bounds on the sample period for data collection. The validity of the existence of a sample period window is tested by Monte Carlo simulations. Many of the results presented in this paper are compared with those obtained with regression analysis techniques. Finally, using measured data, Prony'a method is applied to the identification of a dynamic model for the movement of toxic material through an aquatic microcosm. IntroductionThis paper is primarily concerned with an exploration of the relationships between the state model of a continuous time system and the data set obtained by periodically sampling the system state. The results obtained are based upon the properties of Prony's (1795) method, a scheme for fitting exponential functions to equally spaced data, as applied to state model indentification. Similar data versus model relationships using general regression analysis are invoked throughout the paper for purposes of comparison.The motivation for this work rests on a desire to determine the effect of the data sample period on the method for identifying the parameters of a state model for a continuous time system. Apart from noisy measurements, the accuracy of an identification procedure is determined by the discrete nature of the measured data, both in time and amplitude. As a practical matter, algorithms developed for model identification depend upon the time scale of action for the system response. For periodic sampling schemes, the sampling interval for collecting equally spaced discrete time data of a prescribed arithmetic precision must be carefully chosen in order to adequately capture the dynamics of continuous time systems. This problem has been solved for a bandlimited system response where relationships between the sample period and amplitude quantization have been discovered; however, these results do not generalize.
Techniques for exploring in structurally complex areas were derived from recording and interpreting a single seismic line across the thrust‐faulted west flank of the Casper Arch. Line 306 was recorded to delineate a suspected subthrust prospect. The line was oriented along true dip based on the attitude of outcropping beds. We used short arrays and high‐fold stack, resulting in good data quality, especially in imaging of steep dips from the hanging wall. A two‐dimensional depth model of the structure was constructed. Outcrop dips constrained the interpretation and helped model the hangingwall. A correct model was confirmed when its synthetic time response (time model) matched the real reflections of a near‐offset section. Key to this match was accurate delineation of the hangingwall, which was confirmed when the model’s time response from the subthrust beds matched the static shifts of these same reflections on the real data. A perfect depth model was not achieved, however, due to constraints of the modeling program and the complexity of the thrust‐fault zone. Not all objectives of the project were met: no prospect was found, and proper migration of the data was never achieved. However, the field techniques used did provide good record quality, and the static problems were adequately handled. The final depth model was a reasonably accurate structural picture, although greater confirmation of its accuracy might have been achieved if a crossline had been available. In addition, the model might have been constructed more accurately and in less time using a real‐time, interactive modeling program.
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