Time/space series modeling, data processing and inversion

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The interior of our Earth is, essentially, unavailable to direct observation. With the exception of information that is available from instrumentation that is placed in deep, exceedingly expensive wells, our information is always derived in indirect fashion, for example, by means of active surface seismic sources, passive sources, such as earthquakes, and non seismic techniques, electromagnetic, gravitational and isotopic measurements, to name but a few. The data that we record are, invariably, amplitude variations in time, space or both. The interpretation of the data in terms of Earth parameters entails, basically, three tasks:[1] Time and/or Space Series Modelling. Before the recorded data may be interpreted, we must envisage a process that can be assumed to have generated them. This step is called modeling. A model is desigened on the basis of experience, intuition and luck. The purpose of this model is to enable us to generate, synthetically, a data set that resembles the one recorded.[2] Data Processing. The recorded data are always contaminated by some type of noise. This noise may be random, partly deterministic, additive, convolutional, multiplicative and, as generally happens, an unknown mixture of all of the above. The task is to separate signal from noise, so that this signal may be compared with the signal modelled in step [1]. Many other sub-tasks are involved here, such as deconvolution, resolution enhancement etc., but these will be described in the proposal below.[3] Inversion. Step [1] models the signal in the data in terms of Earth parameters. This is called the "forward problem", i.e., assumed parameters >>> data. The desideratum is, however, data >>> parameters. the "inverse problem". Centrally important, fascinating and hugely difficult. I will not go into reasons why here. Sufficient be it to pose the following, "Given the number 12, find two numbers, a and b that, when added together, give 12. An infinity of answers. The question is, is one of these useful, and if so, which one?