The helical coordinate

Blind deconvolution and the solar cube

An area of applications that leads directly to spectral factorization is ``blind deconvolution.'' Here, we begin with a signal. We form its spectrum and factor it. We could simply inspect the filter and interpret it, or we might deconvolve it out from the original data. This topic deserves a fuller exposition, say for example as defined in some of my earlier books. Here, we inspect a novel example that incorporates the helix.

Solar physicists have learned how to measure the seismic field of the sun surface. It is chaotic. If you created an impulsive explosion on the surface of the sun, what would the response be? James Rickett and I applied the helix idea along with Kolmogoroff spectral factorization to find the impulse response of the sun. Figure 9 shows a raw data cube and the derived impulse response. The sun is huge, so the distance scale is in megameters (Mm). The United States is 5-Mm wide. Vertical motion of the sun is measured with a videocamera-like device that measures vertical motion by an optical doppler shift. From an acoustic/seismic point of view, the surface of the sun is a very noisy place. The figure shows time in kiloseconds (Ks). We see roughly 15 cycles in 5 Ks which is 1 cycle in roughly 333 seconds. Thus, the sun seems to oscillate vertically with roughly a 5-minute period. The top plane of the raw data in Figure 9 (left panel) happens to have a sun spot in the center. The data analysis here is not affected by the sun spot, so please ignore it.

solar
Figure 9. Raw seismic data on the sun (left). Impulse response of the sun (right) derived by Helix-Kolmogoroff spectral factorization.

The first step of the data processing is to transform the raw data to its spectrum. With the helix assumption, computing the spectrum is virtually the same thing in 1-D space as in 3-D space. The resulting spectrum was passed to Kolmogoroff spectral factorization code, a 1-D code. The resulting impulse response is on the right side of Figure 9. The plane we see on the right top is not lag time $\tau=0$ ; it is lag time $\tau=3$ Ks. It shows circular rings, as ripples on a pond. Later lag times (not shown) would be the larger circles of expanding waves. The front and side planes show tent-like shapes.

The slope of the tent gives the (inverse) velocity of the wave (as seen on the surface of the sun). The horizontal velocity we see on the sun surface turns out (by Snell's law) to be the same as that at the bottom of the ray. On the front face at early times we see the low-velocity (steep) wavefronts and at later times we see the faster waves. Later arrivals reach more deeply into the sun except when they are late because they are ``multiple reflections,'' diving waves that bend back upward reaching the surface, then bouncing down again.

Multiple reflections from the sun surface are seen on the front face of the cube with the same slope, but double the time and distance. On the top face, the first multiple reflection is the inner ring with the shorter wavelengths.

Very close to $t=0$ see horizontal waveforms extending only a short distance from the origin. These are electromagnetic waves of essentially infinite velocity.

The helical coordinate