Uncertainties in Finite-Fault Slip Inversions: To What Extent to Believe? (A Critical Review)

The matrix inversion of seismic data for slip distribution on finite faults is based on the formulation of the representation theorem as a linear inverse problem. The way the problem is posed and parameterized involves substantial, and often subjective, decision making. This introduces several levels of uncertainty, some of them recognized and some not adequately addressed. First, the inverse problem must be numerically stabilized and geologically constrained to obtain meaningful solutions. It is known that geologically irrelevant solutions also exist that may even better fit the data. It is also known that obtaining a stabilized, constrained solution compatible with the data does not guarantee that it is close to the true slip. Second, once the scheme has been set up, there still remains significant uncertainty in seismological parameterization. Synthetic tests have consistently shown that incorrect assumptions about the parameters fixed in the inversions, such as the rupture speed, fault geometry, or crustal structure, generate geologic artifacts, which are also dependent on array geometry. Third, solving the inverse problem involves numerical approximation of a continuous integral, with the generally grid-dependent result. Fourth, maintaining a linear inverse problem requires that the final slip on each subfault be the only variable to solve for. The slip functions used in the inversions are typically integrals of triangles or boxcars; they all involve a second parameter, slip duration, which has to be fixed. The effect of chosen duration cannot be disregarded, especially when frequencies higher than 0.1-0.5 Hz in the data are modeled. Fifth, the spectra of triangles and boxcars are sinc functions, whose relevance to realistically observed spectra is problematic. How close then could an inverted slip image be to the true one? There are reasons to believe that the fine structure resolved is often an artifact, dependent on the choice of a particular inversion scheme, variant of seismological parameterization, geometry of the array, or grid spacing. This point is well illustrated by examining the inversions independently obtained for large recent events, for example, the 1999 Izmit, Turkey, earthquake. There is no basis currently available for distinguishing between artificial and real features. One should be cautioned against any dogmatic interpretation of inhomogeneous features on inverted slips, except their very gross characteristics.