Traveltime extrapolation

Suppose I know the Green's function traveltime field $\tau_1({\bf x};{\bf x}_1)$for all subsurface coordinates ${\bf x}$ due to a source at a (surface) position ${\bf x}_1$. Estimating the traveltime field $\tau_2({\bf x};{\bf x}_2)$ due to an adjacent surface source position ${\bf x}_2$, from the given field $\tau_1$, is defined as traveltime extrapolation.

I assume that $\tau_1$ has been previously calculated by solving the ray-theoretic eikonal equation (Cervený et al., 1977):

 

$$\vert \nabla\tau_1({\bf x};{\bf x}_1) \vert^2 = s^2({\bf x})$$ (1)

where $\nabla$ is the vector gradient operator, and $s({\bf x})$ is the earth slowness (inverse velocity) field. The unknown field $\tau_2$ can be decomposed without loss of generality as follows:

 

$$\tau_2({\bf x};{\bf x}_2) = \tau_1({\bf x};{\bf x}_1) + \phi({\bf x};{\bf x}_1,{\bf x}_2)$$ (2)

where $\phi$ is the traveltime difference between the known field $\tau_1$ and the unknown field $\tau_2$, and is a function of subsurface location ${\bf x}$and the two specified source positions ${\bf x}_1$ and ${\bf x}_2$. Naturally, one requires that the extrapolated field $\tau_2$ also satisfies the eikonal:

 

$$\vert \nabla\tau_2\vert^2 = \vert \nabla\tau_1+ \nabla\phi \vert^2 = s^2({\bf x})\;.$$ (3)

Expanding the squared term in (3) and substituting (1) results in an equation for $\phi$ in terms of the known field $\tau_1$:

 

$$2 \nabla\tau_1{\bf \cdot}\nabla\phi + \vert \nabla\phi \vert^2 = 0\;.$$ (4)

In this form, solving for $\phi$ looks harder than solving for $\tau_2$ directly from the eikonal equation! However, given that $\phi=\tau_2-\tau_1$, the term $\vert\nabla\phi\vert^2$ can be expanded as follows:

 

$$\vert\nabla\phi\vert^2 = 2 \left( s^2 - \nabla\tau_1{\bf \cdot}\nabla\tau_2\right) = 2 s^2 \left( 1 - \cos\theta_{12} \right)$$ (5)

where $\theta_{12}({\bf x};{\bf x}_1,{\bf x}_2)$ is the angle at any subsurface point ${\bf x}$between the ray connecting ${\bf x}_1$ to ${\bf x}$, and the ray connecting ${\bf x}_2$ to ${\bf x}$. Substituting (5) into (4), the equation for $\phi$ in terms of $\tau_1$ becomes:

 

$$\nabla\tau_1{\bf \cdot}\nabla\phi + s^2 \left( 1 - \cos\theta_{12} \right) = 0\;,$$ (6)

which can also be expressed as an equation for $\tau_2$ directly in terms of $\tau_1$:

 

$$\nabla\tau_1{\bf \cdot}\nabla\tau_2- s^2 \cos\theta_{12} = 0 \;.$$ (7)

This last equation is simply a statement that the dot product of the two gradient fields is the cosine of the angle between the local ray directions, scaled by the local value of the slowness squared.

If $\cos\theta_{12}$ can be approximated in some physically reasonable and robust manner, then (6) or (7) represent a single first-order linear PDE to extrapolate the unknown field $\tau_2$. I believe such an approximation is attainable, since only the angle between the two rays is required, and not each ray angle individually. In fact, if the surface source location ${\bf x}_2$ is on the order of a few tens of meters away from ${\bf x}_1$, and we are interested in extrapolating traveltimes at imaging points a few kilometers distant from the source region, then $\cos\theta$can not stray too far from a value of unity, based on arguments of traveltime field continuity. This is a mathematical reinforcement of our intuition that, if we perturb our source location a bit, then the resulting perturbed traveltime field should not be tremendously different from the unperturbed field. Many of us have noted this effect while tracing traveltime maps along a line for prestack depth migration, and have been frustrated that we were forced to completely recalculate slightly different traveltime fields from CMP to adjacent CMP.

As a first approximation, the function $\cos\theta_{12}$ could be approximated by the Law of Cosines, as:

 

$$\cos\theta_{12} \;\sim \; \frac{ r_1^2 + r_2^2 - \vert{\bf x}_2-{\bf x}_1\vert^2 }{ 2 r_1 r_2 }$$ (8)

where r₁ is the (straight) ray distance from ${\bf x}_1$ to ${\bf x}$, and r₂ is the (straight) ray distance from ${\bf x}_2$ to ${\bf x}$. Although this constant velocity approximation would seem to be inappropriate for estimating ray angles in general $s({\bf x})$ media, it may not be so bad for estimating the angle between the two rays, as discussed above. In this sense, the constant velocity assumption means more like: the velocity field ``seen'' along ray 1 is about the same as the velocity field ``seen'' along ray 2. Note that this approximation is expected to deteriorate as the source separation $\vert{\bf x}_2-{\bf x}_1\vert^2$ increases or the ray lengths r₁, r₂ decrease. Conversely, the approximation (8) is expected to be good for cases in which $\vert{\bf x}_2-{\bf x}_1\vert <\!< \min \{r_1,r_2\}$.

Both (6) and (7) can be cast into the general 3-D form:

 

$$a \psi_x + b \psi_y + c \psi_z + d = 0 \;,$$ (9)

or the 2-D form:  

$$a \psi_x + c \psi_z + d = 0 \;,$$ (10)

where $\{a,b,c\}$ are the components of the gradient vector of the known field
$\{\tau_{1x},\tau_{1y},\tau_{1z}\}$, d is a function of squared slowness s² and $\cos\theta_{12}$, and $\{\psi_x,\psi_y,\psi_z\}$ are the components of the unknown traveltime gradient field, $\{\tau_{2x},\tau_{2y},\tau_{2z}\}$ or $\{\phi_x,\phi_y,\phi_z\}$, which are to be solved for. The extrapolation equation (9) can be solved by a finite difference method, for example, and represents a first-order linear PDE which should require less effort to solve than the nonlinear eikonal PDE for $\tau_2$.The solution will be in terms of the gradient of the unknown field, $\nabla\tau_2$, which can then be integrated to the traveltimes $\tau_2({\bf x})$ directly (refer to Equation (18)). This completes the extrapolation process of an unknown traveltime field $\tau_2({\bf x};{\bf x}_2)$ from a known traveltime field $\tau_1({\bf x};{\bf x}_1)$. This extrapolation equation might be useful for computing Frechet derivatives of the type

 

$$\frac{\partial \tau({\bf x};{\bf x}_s)}{\partial {\bf x}_s} \sim \phi({\bf x}; {\bf x}_s+ \delta{\bf x}_s)$$ (11)

for applications in data acquisition survey design, or shot gather seismic data continuation, for example.