A common objective of geophysics is to probe the properties of the Earth's interior based on data from observations. Geoscientists often use seismic data to build a model of the subsurface as a representation of various assessments of some simplified key aspects of the real world. The validity of each model depends on its consistency with observations. All observable datasets constitute a data space, and all possible models constitute a model space. Data fitting and model inversion are two complementary approaches in geophysics to relate the data space to the model space. Data fitting uses forward modeling to search for models that fit well with the observed data and satisfy our scientific intuition. Model inversion uses our scientific intuition to set up rules about how the models should behave and then determines the model variations that fit best with the available data. The usefulness of data fitting and model inversion is evident in many applications illustrated in this chapter.

The basic theories of seismic modeling and inverse theory are reviewed here. Data fitting is introduced in the first two sections via several seismic forward modeling methods and a simple example of regression. The basic theories on inverting a system of linear equations are given in the next three sections, in conjunction with the tomographic velocity analysis in Section 8.4. The least squares method as a classic linear inversion is widely applicable in geophysical data analysis and beyond. Some mathematic insights on inversion of linear equations are illustrated via several common ways of matrix decomposition. The common causes of non-uniqueness in geophysical inversion include insufficient constraining power of the data, the non-linear relationship between data and model, and dependency of that relationship on the solution. Several practical inverse solutions discussed here include the Backus–Gilbert method and the LSQR algorithm. Practically, seismic inversion is synonymous with the inverse imaging in Section 8.4. The inverse approach has the advantage of subjectively determining the values of model properties based on the given model parameterization and data. For some applications such as inverse filtering and tomographic velocity analysis, inversion is the preferred method because of its objectiveness in obtaining the solutions.