3.6 Tests of Born approximation for sound speed and flow perturbations

3.6 Tests of Born approximation for sound speed and flow perturbations

The range of validity of the Born approximation is an important question for local helioseismology, as the linear forward problem proceeds naturally in the Born approximation.

Birch et al. (2001 ) simulated the scattering of sound waves by a spherical region of perturbed sound speed. The geometry and the essential result are shown in Figure 5 . The main result, which was also found by Hung et al. (2001 ) in the context of geophysics, is that in the limit of weak perturbations the Born approximation is valid, while the ray approximation can fail badly. The Born approximation becomes less accurate as the strength, or spatial extent, of the perturbation is increased. Notice that there have not yet been any studies, in the context of helioseismology, on the accuracy of waveforms, rather than travel times, computed in the Born approximation.

Figure 5:

A numerical experiment of wave scattering by a spherical region of perturbed sound speed. The left panel shows the geometry and a single frame from a numerical simulation. The circular contours are contours of the sound-speed perturbation, which is a raised cosine with radius $R$ and with a maximum fractional sound speed perturbation of $A$ . The curved heavy lines show example ray paths leaving the source, at $x = 0$ , $y = - 45 Mm$ , and going through the receiver location, at $x = 0$ , $y = 45 Mm$ . The middle and right-hand panels show the numerical travel times perturbations (solid lines), the Born approximation travel time perturbations (dashed lines), and the first-order ray approximation for the perturbed travel times (dotted lines), for positive and negative five percent changes in the sound speed ( $A = ±0.01$ , middle panel) and positive and negative ten percent changes in the sound speed ( $A = ± 0.1$ , right panel). In both cases, the different travel time perturbations $dt$ are shown as functions of the sphere radius $R$ . Also in both cases, the travel times for negative sound-speed perturbations have $dt > 0$ and for positive sound-speed perturbations have $dt < 0$ . From Birch et al. (2001

).

Birch et al. (2004

) studied the validity of the Born approximation for the effect of flows on time-distance travel times. Figure 6

shows the geometry for the numerical experiment, and Figure 7

shows the basic results. For weak flows, the Born approximation is valid as long as the acoustic waves were not traveling nearly upstream or downstream. They also found that strong flows can reflect incoming waves, which leads to a failure of the Born approximation.

Figure 6:

Geometry for a numerical test of the Born approximation. The wave source is located at the origin of the coordinate system. The open circles show the locations where the wavefield is observed. There is a background flow in the $+^x$ direction confined between the solid horizontal lines. The flow strength varies as a raised cosine, centered between the two horizontal lines, on the $y$ coordinate and has a maximum of value of $1/6$ of the background sound speed, which is spatially uniform and equal to $10 km s-1$ . The lines emanating from the source show example ray traces. The flow reflects waves that hit it at $x > 50$ . The travel times are shown in Figure 7

. From Birch and Felder (2004

).

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Figure 7:

Travel times in the Born approximation (heavy line), ray approximation (thin line), and computed numerically (open circles) for the geometry and jet-flow configuration described in Figure 6

. The top panel is for waves that cross the jet-like flow and the bottom panel is for the waves that do not cross the jet. The horizontal axis is the distance in the upstream or downstream direction traveled by the wave before it is observed, i.e., $x = 0$ is for waves traveling perpendicular to the direction of the flow. For the waves that cross almost perpendicularly to the flow direction the Born and ray approximations are good. For waves that hit the jet at a glancing angle, a strong reflected wave is seen and the Born approximation fails. From Birch and Felder (2004 ).