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Inversion Theory:
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| 1.1. Field observations must include the measurements, as well as accurate details about sources, receivers, locations, and all data processing; that is, everything that affects the ability to reproduce the data. For specifics about how surveys work please see "Geophysical surveys" in the "Foundations" chapter. 1.2. Inversion requires a realistic estimate of the error of any datum. If details were not recorded in the field, then an estimate must be made. Examples of error sources include additive noise, inaccuracies in positioning, violations in assumptions, and many more. 1.3. Inversion cannot be performed unless the forward problem can be solved. Specifics about the physics for individual surveys, including forward modelling equations, are included in the "Geophysical surveys" in the "Foundations" chapter. 1.4. Results should be consistent with geological, structural, logical, and other information known prior to inversion. Often, such information can be included to constrain the inversion result. For example, the reference model can be determined, the type of structure might be characterized by weightings, or there may be upper and lower bounds on physical properties that could be defined Inversion algorithms incorporate prior information in several ways. These are discussed in the several sections that describe how inversion works. 2. The Earth must be divided into enough cells so that adequate structure can be modelled, yet not so many that the computing problem becomes too large. This is the problem known as "mesh" design. 3.1. "Misfit" describes how close synthetic data generated from the inversion result (called "predicted data") match the field survey measurements. Usually, the misfit criterion involves specifying standard deviations for survey measurements. 3.2. A "model objective function" is needed to measure the "size" of any model. This ultimately allows us to select a single model out of many that adequately fit the data. The manner in which this function is specified serves as one of the most important ways of incorporating prior information about the model into the inversion. 4. Numerically, the inverse problem is solved using methods from mathematical optimization theory. 5. The inversion result must be evaluated carefully to see that the algorithm progressed properly, and that the model is geologically sensible. 6. Successful inversion is rarely a once-only process. Upon the first completion of step 5, several more models must be generated by varying model objectives, or by adjusting misfit criteria. Therefore, completion of step 5 implies a return to step 1.3, or 1.4, or 2, or 3.1, or 3.2, etc. Adjustments will be made based upon your prior understanding of the geologic situation, and upon initial inversion results. A preferred model should be chosen only after there are several inversion results to work with, and after consultation with other members of the team who are working on the geological or geotechnical problem. See also "Notes on Iteration" below. close Before inversion can be carried out, the requirements of this flow chart must be addressed. Click each link on this chart to see a summary of that item.
Notes on iterationGeophysical inversion is an iterative process. Owing to the nonuniqueness of the problem, several equally valid solutions should be obtained. Generally a model obtained from a first successful inversion should be refined by exploring the importance of misfit and by adjusting the model objective function. These adjustments should be made within the context of as much understanding of the problem as possible. Then the preferred model of the earth can be chosen based upon the range of acceptable models, and what is already known about the problem and the geology. In other words, the person doing inversion must work as a member of a team with professionals who have geologic, geotechnical, geochemical, and/or other relevant expertise. The use of geophysics in general is also usually part of an iterative process. Geophysical information can build on geologic information already obtained, and it can help guide further investigations as the project proceeds from preliminary reconnaissance, through follow-up of anomalies in the field, delineation of subsurface details, and further project development. |
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