Estimation of time shifts

Estimating time shifts is a two step procedure where local stepouts are first estimated and then integrated. The goal of dip estimation is to find a local stepout, p_h, that destroys the local plane wave such that,

$$0 \approx \frac{\partial u}{\partial h} + p_h \frac{\partial u}{\partial \tau},$$ (1)

where u is the wavefield at time $\tau$, midpoint x and offset h. For all gathers, we evaluate the slope p_h with a method based on high-order plane-wave destructor filters Fomel (2002). This technique has the advantage of being accurate for steep dips. The estimation of p_h is based on a non-linear algorithm (Gauss-Newton method) that includes a regularization term. This regularization smooths dips across offset and CMP location. One problem with the current technique is that the dip estimation algorithm cannot properly handle conflicting dips. This can be troublesome when, for instance, multiples are present in the data. To solve this problem, Fomel (2002) and Brown (2002) show how two dips can be estimated. Then if multiple reflections are present, the stepouts corresponding to the primaries are kept and those of the multiples are rejected.

Dips estimates lead to a vector of local stepouts ${\bf p}_h$that are integrated one CMP at a time to obtain time shifts. Dips are smoothed along both spatial directions but not in time $\tau$. To enforce smoothness in the $\tau$ direction Lomask and Guitton (2004) introduce a time component ${\bf p_\tau}$ to ${\bf p}$. The relationship between the local time shift vector, ${\bf t_s}={\bf t_s}( \tau,x,h)$ and the local dip vector ${\bf p}=({\bf p}_h, {\bf p_\tau})^T$((.)^T being the transpose) at a constant x is:  

$${\bf p} = \left ( \begin{array} {c} {\bf p}_h \\ {\bf p_\t... ...l_h \bf{t_s} \\ \partial_\tau \bf{t_s} \end{array} \right ),$$ (2)

where $\partial_\tau$ and $\partial_h$ are the partial derivative in $\tau$, h respectively. In practice, we choose ${\bf p_\tau=1}$ and control the amount of smoothness by introducing a trade-off parameter $\epsilon$ as follows Lomask and Guitton (2004):  

$$\left ( \begin{array} {c} \partial_h \bf{t_s} \\ \epsilon ... ...} {\bf p}_h \\ \epsilon {\bf p_\tau} \end{array} \right ).$$ (3)

Using equation (3) we wish to minimize the length of a vector ${\bf r_{t_s}}$ that measures the difference between ${\bf p_\epsilon}$ and $\nabla_\epsilon {\bf t_s}$ as follows:  

$${\bf 0} \approx {\bf r_{t_s}} = \nabla_\epsilon {\bf t_s}- {\bf p_\epsilon},$$ (4)

where $\nabla_\epsilon=(\partial_h, \epsilon \partial_\tau)^T$, and ${\bf p_\epsilon}=({\bf p}_h, \epsilon {\bf p_\tau})^T$.We then minimize the following objective function:  

$$f({\bf t_s})=\Vert{\bf r_{t_s}}\Vert^2,$$ (5)

where $\Vert.\Vert$ is the L₂ norm. By increasing $\epsilon$, the estimated time shifts ${\bf \hat t_s}$ become smoother in time. We solve equation (5) analytically in the Fourier domain Lomask (2003), which speeds up the estimation of ${\bf t_s}$:

 

$${\bf \hat t_s} \quad = \quad {\rm FFT_{\rm 2D}}^{-1} \left[{... ...epsilon Z_\tau^{-1} +2+2\epsilon -Z_h -\epsilon Z_\tau} \right]$$ (6)

where $Z_h = e^{i w \Delta h} \$and$\ Z_\tau = e^{i w \Delta \tau}$.The dip integration yields the desired time shifts plus a constant, i.e., a DC frequency component. The zero frequency component is removed by subtracting the near offset panel from the other offsets. Therefore, the time shifts are a measure of the moveout errors relative to the near-offset panel. At the end of the dip integration process, we end up with a map of time shifts, $t_s(\tau,x,h)$. These time shifts can be used to flatten the CMP gathers without any velocity analysis. Our goal, however, is to find an interval velocity function consistent with the estimated time shifts using a tomographic inversion procedure.