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.
Linear inversion The deterministic nature of linear inversion requires a functional relationship which models, in terms of the earth model parameters, the seismic variable to be inverted. This functional relationship is some mathematical model derived from the fundamental laws of physics and is more often called a forward model. The aim of the technique is to minimize a function which is dependent on the difference between the convolution of the forward model with a source wavelet and the field collected seismic trace. As in the field of optimization, this function to be minimized is called the objective function and in convectional inverse modeling, is simply the difference between the convolved forward model and the seismic trace. As earlier mentioned, different types of variables can be inverted for but for clarity, these variables will be referred to as the impedance series of the earth model. In the following subsections we will describe in more detail, in the context of linear inversion as a minimization problem, the different components that are necessary to invert seismic data.
Forward model The centerpiece of seismic linear inversion is the forward model which models the generation of the experimental data collected. According to Wiggins (1972), it provides a functional (computational) relationship between the model parameters and calculated values for the observed traces. Depending on the seismic data collected, this model may vary from the classical wave equations for predicting particle displacement or fluid pressure for sound wave propagation through rock or fluids, to some variants of these classical equations. For example, the forward model in Tarantola (1984) is the wave equation for pressure variation in a liquid media during seismic wave propagation while by assuming constant velocity layers with plane interfaces, Kanasewich and Chiu (1985) used the brachistotrone model of John Bernoulli for travel time of a ray along a path. In Cooke and Schneider (1983), the model is a synthetic trace generation algorithm expressed as in Eqn. 3, where R(t) is generated in the Z-domain by recursive formula. In whatever form the forward model appears, it is important that it not only predicts the collected field data, but also models how the data is generated. Thus, the forward model by Cooke and Schneider (1983) can only be used to invert CMP data since the model invariably assumes no spreading loss by mimicking the response of a laterally homogeneous earth to a plane-wave source
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