The ingression and egression are better understood of we answer the following question: Given the signal observed at the solar surface, what is the best estimate of the wavefield at some point in the solar interior assuming that the observed wavefield resulted entirely from waves generated from that point (for the egression) or waves converging towards that point (for the ingression)?
Holography has better analogy with optics and as a result the holography literature often employs optical terminology. The target point in the solar interior at which we attempt to estimate the wavefield is termed the “focus point”. In practice, only data in a restricted area on the surface above the focus point is used to compute the egression and ingression. This region is called the “pupil”.Helioseismic Holography from:
(Lindsey and Braun,THE ASTROPHYSICAL JOURNAL, 485:895–903, 1997 August 20)
As we have already mentioned, shallow subsurface holography can be done very rapidly with modest computational resources. Again, as in conventional holography, it is not necessary to run the computation over the entire surface, rather only over the surface substantially covering the area of interest. In this case, the plane-parallel approximation becomes accurate. This allows us to express normal modes in terms of simple plane waves now multiplied by functions, R, of depth, z:
In this case, the computations essentially reduce to fast Fourier transforms. The coefficients a(, ) are computed simply by Fourier transforming the emergent noise, (, z, t) and equating the result to a(, )R(, k, z = 0). The extrapolation backward to a depth z is then accomplished simply by computing the integral stated by equation (3), also a fast Fourier transform, with R(, k, z) computed at the desired depth, z.
These simplifications then make it fairly straightforward to do relatively shallow holography of selected regions on a workstation. For example, an 8 hr time series of SOHO images is approximately 512 frames. The time needed for a DEC 3000 to Fourier transform a 256 × 256 pixel region (0.5 solar radii × 0.5 solar radii) of a SOHO image along any axis is timed at 0.4 s. At this rate, the time needed to fully transform an 8 hr data cube (256 × 256 × 512) data cube both spatially (in two dimensions) and temporally (one dimension) is just over 10 minutes. After extrapolating the radial function, R(, k, z), to the desired depth, z, the computation to reverse the transformin the spatial dimensions onlyrequires approximately 7 minutes. Once the initial transform is accomplished, the time needed to extrapolate the surface acoustic power to any selected depth is the time needed to determine the radial functions at the appropriate depth plus 7 minutes to reverse the Fourier transform spatially.
This is the procedure that we actually used in the numerical simulation that is shown in Figure 2, with R the radial function for a uniform medium (constant sound speed), the equivalent of electromagnetic holography in a vacuum. In this case,
where , the z-component of the wave vector, which is unobservable, is determined by
where c is the sound speed.
As we have already mentioned, holography can be accomplished from the spatial-temporal (time-distance) perspective just as from the spectral perspective described above. We will begin by considering simple subsurface acoustic sources that manifest a diffuse acoustic signature at the surface of the acoustic medium. We will then discuss how these principles apply to scatterers, which do not manifest the acoustic excess that makes sources or absorbers directly visible.
Consider a plane parallel medium in which waves propagate freely without absorption with a speed, c, that is dependent only on depth, z. Let T(, z) be the time required for sound to travel from a point acoustic source, P (see Fig. 3), a distance z below the surface to any point on the circle, C(), a distance from Q, the vertical projection of P to the surface. By this we mean it to be implicit that T(, z) is the travel time
along the path, (, z), along which the travel time is a minimum. Standard techniques in variational calculus reduce the determination of (, z) to Snell's law: the horizontal phase speed of the wavefront perpendicular to must be constant. So T(, z) is easily determined. Let the position of Q, the surface projection of P, be indicated by the two-dimensional vector . Now consider a single omnidirectional pulse emanating from P at time t and propagating to the surface. Such a pulse produces a surface disturbance in the form of a simple expanding ring. We can express this Green's function in the form
The disturbance expressed by equation (7) sweeps out a surface in -t space (see Fig. 4), which we will call the surface geometrical wavefront of the pulse, or simply the wavefront, after its optical analogy in electromagnetic radiation (see Born & Wolf 1975b). We will refer to the point P from which the disturbance emanates as the focal point, or simply the focus, of the wavefront. The temporal profile of the pulse is expressed as a simple Dirac delta function, and the function T(, z), then, simply expresses the time delay between the emission of the pulse, at time t, and the arrival of the disturbance a distance from the overlying point, Q, on the surface. The function f(, z) expresses the amplitude of the pulse at a radius from Q, integrated over the duration of the pulse.Fig. 4
If some acoustic disturbance, , at the surface of the medium contains G+ as a significant component, we can make a statistical assessment of just this component by integrating weighted by the amplitude of G+ over the surface occupied by the wavefront. This is automatically accomplished by the integral
since G+ is nonzero only on the wavefront. In practical application, however, one saves a great deal of computation if the integral expressed in equation (8) is in fact confined to the wavefront itself:
The integral over t in equation (8) is then accomplished simply by evaluating at t satisfying equation (9). In practice this integral must be limited to a finite domain around Q, which we will suppose to be the interior of a circle of radius a:
We will call H+(z, , t) the acoustic egression in from the point located at (z, ) at time t. We will regard this as a coherent assessment of the contribution to of a pulse emanating from a source at horizontal location , depth z at time t. We will call the domain of integration defined by a in equation (10) the acoustic pupil of the computation.
The integral over the surface domain expressed in equation (10) can also be expressed as a sum of line integrals around the circles, C() concentric about Q, along each of which the wave travel time, T, from the source is constant:
While this exudes a certain appeal in terms of axial symmetry about Q, we suspect that this equation has less computational utility than equation (10), since in practice the sum is most conveniently performed over individual pixels and accurately computing for each, rather than selecting a set of values of to examine and searching for those pixels with centers that lie satisfactorily close to C() for each.
The function H+ is basically sensitive only to wave sources or sinks. For example in the case of a simple point source, i.e., a compact locality that generates its own acoustic power so as to manifest a clear acoustic excess at the surface, we can assess its depth by examining H+ at focal points that may coincide with such a source. We can, for example, make acoustic power maps, P(z, ), by integrating H+(z, , t)2 over a considerable time interval, t, for each point, (, z), on a specified surface of constant depth, z:
It should be clear that if a source happens to lie on a point, (z, ), on this surface, then its wavefront will coincide with that of the integral expressed by equation (10), and P(z, ) will manifest a strong response. If we shift the source to some other point, P, the wavefront of the source will be horizontally shifted with respect to that of the integral, so as to intersect it only along a thin locus passing directly between Q and Q, the point overlying P. The coherence of the integral is then quickly lost resulting in a greatly reduced response. Because of this, P(z, ) is highly sensitive to the location of the source, and this is what gives us the spatial discrimination we desire with respect to the horizontal location of the source. At greater or shallower depths the surface over which the integral in equation (10) is specified will warp to a different curvature, so as to be unable to satisfactorily encompass the wavefront of the disturbance at any horizontal location; so, the coherence of the integral will be only partial for all locations. In this case, the signature manifested by P(z, ) will be simply diffuse, literally out of focus.
So far, we have assumed that no practical limit is imposed on either spatial or temporal resolution and that the pulse is an infinitely sharp Dirac delta function. In practice, an acoustic pulse is limited to a finitesimal duration, t, and correspondingly in spatial extension. Further limits may be imposed by the instruments that observe the wave motion. When this is the case, then the wavefront of the disturbance can deviate from the surface defined by T(, z) by roughly as much as t before coherence is seriously impaired. The result is that the response of P(z, ) to any source, no matter how localized the source itself, is smeared to a spatial extent characteristic of the deviation in required to destroy coherence. This is simply diffraction. As in electromagnetic optics, the spatial extent of this smearing is simply proportional to t. It should also be clear, by optical analogy, that the greater the pupil, a, of the computation, the smaller the smearing of the image due to diffraction will be. However, once the pupil radius, a, becomes comparable to the depth, z, of the source, the spatial resolution permitted by diffraction approaches a limit, roughly the surface wavelength of the highest frequency waves resolved. Efforts to extend the pupil past this limit therefore begin to yield rapidly diminishing returns.
Simple acoustic power maps computing P(z, ) at various depths will show absorbing regions just as they will emission, given that there exists a sufficient background of acoustic waves to be absorbed. The basic quality of the signature is the same as that for sources, except that one sees silhouettes, deficits in acoustic power in place of excesses. The silhouettes will be sharp when the surface on which P(z, ) is computed passes through a well-defined absorber. When the surface on which we determine P(z, ) lies substantially above or below the absorber, then the silhouette will become diffuse.
As we briefly indicated in § 1, when the features we are interested in are neither emitters nor absorbers, then we expect them to become invisible in acoustic power, P(z, ).
If a scatterer is illuminated by an isotropic distribution of waves, the acoustic power emanating from it will be the same as that from the background, rendering it invisible by lack of contrast, however fine the resolution.
As in the spectral perspective, phase-sensitive techniques are needed to detect acoustical perturbations that are purely refractive. In the spatial perspective, these are based on the effect of the wave perturbations on the correlation between ingoing and outgoing waves.
The time-distance helioseismology described and exercised by Duvall et al. (1996) examines the acoustical correlation between a point on the solar surface and a surrounding concentric ring. Refractive perturbations in the optical paths connecting the reference point with points on the surrounding ring introduce a time delay into the waveform that characterizes the correlation between the ingoing and outgoing waves. These time delays are sensitive not only to local refractive perturbations but also to flows.
The basic conceptual extension needed to incorporate holographic techniques into time-distance statistics lies in recognition of the ingressing analog,
to the acoustic egression defined in equations (8), (9), and (10). In the context of simple time-distance statistics, then, holography proposes to examine the temporal correlation between the ingressing and egressing waves:
While a localized refractive perturbation, n, most anywhere in the medium will generally have a minimal effect on this correlation, it will manifest a strong, focused, coherent perturbation when it is centered at a mutual focal point, (z, ), of H+ and H-. A perturbation in the shape of a small sphere with radius b centered at (z, ), for example, should introduce a time delay of order
to a wave focused on (z, ), shifting the peak of the correlation function, C(z, , ), from = 0 to t. Refractive perturbations, then, should be clearly rendered and localized in the holographic perspective by maps of the time delay, , characterizing the maximum of C as a function of (z, ).