Moveout inversion by time warping

In general, we can formulate the problem of moveout inversion in the context of time-warping as follows (Burnett and Fomel, 2009):

$$t^2-t^2_0 = F(\mathbf{m},\mathbf{x})~,$$ (2)

where $t^2-t^2_0$ denotes the measured traveltime squared shifts from non-physical flattening, $F$ is a linear or non-linear operator describing the geometry of the reflection traveltime surface, i.e, the moveout approximation of choice, $\mathbf{m}$ denotes the vector of pertaining moveout parameters associated with that choice of $F$, and $\mathbf{x}$ denotes the vector of offsets ($x$ and $y$). Here, $t$ denotes the recorded reflection traveltime and $t_0$ is its value at zero offset. For each event at $t_0$ in a CMP gather of $N$ traces, we have $N$ equations with only several unknowns ( $\mathbf{m}$) depending on the choice of $F$. Therefore, this generally leads to a highly overdetermined system for estimating the associated moveout parameters $\mathbf{m}$. For example, in the case of 2D hyperbolic reflection traveltime, we have $F(\mathbf{m},\mathbf{x}) = x^2\;W(t_0)$ leading to

$$t^2-t^2_0 = x^2\;W(t_0)~.$$ (3)

Therefore, this highly overdetermined system can be solved using, for instance, a least-squares inversion (Burnett, 2011; Burnett and Fomel, 2009). The number of parameters in $\mathbf{m}$ varies depending on the choice of $F$ and the kind of considered data (2D or 3D). Nevertherless, in general, the number of traces in a CMP gather far exceeds the number of unknown parameters and the resulting system is almost always overdetermined. We note that in practice, the automatic traveltime picks from recorded data may include complications from various causes such as noise and errors from slope estimation, which may lead to a system of equations that is mathematically inconsistent — no set of values for $\mathbf{m}$ satisfies the system exactly. In this study, we propose to use the Bayesian inversion framework to find the posterior distributions of possible solutions instead of seeking only one solution associated with least-squares misfit minimization criterion.