Constant velocity gradient

The velocity model I now consider has a constant gradient, with both the vertical and horizontal velocity gradients equal to 1.0 km/s/km. The dimensions are the same as the previous constant velocity model. In this case, the velocity ranges linearly from 1.0 km/s in the upper left corner to 13 km/s in the lower right corner, which is an extremely strong gradient compared to what may be naturally observed.

I presume that we have two previously calculated traveltime fields at surface source positions of x = 2.75 km and 3.25 km, i.e., a sparse traveltime table sampling every 500 m along the line (my other rule of thumb goal in addition to beating 1% relative error). Figure shows the contours of traveltime due to the source at 2.75 km (upper panel), and due to a source at 3.25 km (lower panel). Since I interpolate traveltime gradients, and not the traveltimes themselves, Figures shows the horizontal and vertical traveltime gradients $\tau_{1x}$ and $\tau_{1z}$ for the first source position, and Figure shows the $\tau_{3x}$ and $\tau_{3z}$ for the source given at 3.25 km.

I then solve system (13) using a combination of Cramer's Rule where nonsingular, and Singular Value Decomposition otherwise. I also use the cosine estimate and correction discussed previously from (8) and (17). The interpolated traveltime field is shown in the upper panel of Figure for a desired surface source position at 3.0 km. The lower panel of Figure shows the relative error between the interpolated traveltime field, and the true traveltime field obtained by an analytic raytracing algorithm for constant gradient velocity media (Zhang, 1992, pers. comm.). The error contours are at 1% farthest from the source region, and increase to 10% in the near vicinity of the source. The interpolation is highly accurate, except for some small error directly beneath the source position. This is caused by the previously mentioned instability at the source position in the vertical traveltime gradient, and it is propagated somewhat into all of the traveltimes by the subsequent spatial integration. To help stabilize the singularity, I calculate the $\nabla\tau$ values within a few mesh points of the source location by assuming straight rays (constant velocity) from the source. Figure shows the interpolated horizontal and vertical traveltime gradients respectively.

 

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Figure 7 Linear gradient velocity model. Contours of traveltime field for the two given surface source positions at 2.75 km (top) and 3.25 km (bottom).

 

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Figure 8 Linear gradient velocity model. Horizontal (top) and vertical (bottom) traveltime gradient fields for the surface source positioned at 2.75 km.

 

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Figure 9 Linear gradient velocity model. Horizontal (top) and vertical (bottom) traveltime gradient fields for the surface source positioned at 3.25 km.

 

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Figure 10 Linear gradient velocity model. Interpolated traveltime field. The top panel is the interpolated traveltime field for a surface source positioned at 3.00 km. The lower panel is the relative error between the interpolated traveltime field, and the correct traveltimes modeled by an analytical solution to the eikonal. The contour farthest from the source region is at 1% relative error, and the contour values increase to 10% error right at the source location, in 1% contour increments.

 

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Figure 11 Linear gradient velocity model. Horizontal (top) and vertical (bottom) traveltime gradient fields for the interpolated surface source positioned at 3.00 km.