page_title: Linear seismic inversion
section_title: Parameterization of the earth model space
hierarchical_section_title: Linear seismic inversion / Parameterization of the earth model space
caption_attribution_description: English: Impedance Well Log
context_page_description: Inverse modeling is a mathematical technique where the objective is to determine the physical properties of the subsurface of an earth region that has produced a given seismogram. Cooke and Schneider defined it as calculation of the earth's structure and physical parameters from some set of observed seismic data. The underlying assumption in this method is that the collected seismic data are from an earth structure that matches the cross-section computed from the inversion algorithm. Some common earth properties that are inverted for include acoustic velocity, formation and fluid densities, acoustic impedance, Poisson's ratio, formation compressibility, shear rigidity, porosity, and fluid saturation.
  The method has long been useful for geophysicists and can be categorized into two broad types: Deterministic and stochastic inversion. Deterministic inversion methods are based on comparison of the output from an earth model with the observed field data and continuously updating the earth model parameters to minimize a function, which is usually some form of difference between model output and field observation.
context_section_description: Irrespective of the variable to be inverted for, the earth's impedance is a continuous function of depth (or time in seismic data) and for numerical linear inversion technique to be applicable for this continuous physical model, the continuous properties have to be discretized and/or sampled at discrete intervals along the depth of the earth model. Thus, the total depth over which model properties are to be determined is a necessary starting point for the discretization. Commonly, as shown in Fig. 3, this properties are sampled at close discrete intervals over this depth to ensure high resolution of impedance variation along the earth's depth. The impedance values inverted from the algorithm represents the average value in the discrete interval.
  Considering that inverse modeling problem is only theoretically solvable when the number of discrete intervals for sampling the properties is equal to the number of observation in the trace to be inverted, a high-resolution sampling will lead to a large matrix which will be very expensive to invert. Furthermore, the matrix may be singular for dependent equations, the inversion can be unstable in the presence of noise and the system may be under-constrained if parameters other than the primary variables inverted for, are desired. In relation to parameters desired, other than impedance, Cooke and Schneider (1983) gives them to include source wavelet and scale factor.
  Finally, by treating constraints as known impedance values in some layers or discrete intervals, the number of unknown impedance values to be solved for are reduced, leading to greater accuracy in the results of the inversion algorithm.
caption: Figure 9b: Impedance Well Log
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