Linear seismic inversion
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 (1983) 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. As such, this method of inversion to which linear inversion falls under is posed as an minimization problem and the accepted earth model is the set of model parameters that minimizes the objective function in producing a numerical seismogram which best compares with collected field seismic data. On the other hand, stochastic inversion methods are used to generate constrained models as used in reservoir flow simulation, using geostatistical tools like kriging. As opposed to deterministic inversion methods which produce a single set of model parameters, stochastic methods generate a suite of alternate earth model parameters which all obey the model constraint. However, the two methods are related as the results of deterministic models is the average of all the possible non-unique solutions of stochastic methods. Since seismic linear inversion is a deterministic inversion method, the stochastic method will not be discussed beyond this point. (Wikipedia).
