To obtain the zeroorder problem we break the various quantities into unperturbed components (superscript ) and firstorder corrections (prefix ). In particular we have
From Equation (7) we can then obtain the governing equations for the zeroorder and perturbed problems.To lowest order Equation (7) becomes
The solution to this equation can be written in terms of the Green’s tensor (Section 3.3): where summation over the index is implied. From Equation (23) together with the definition of the observable in terms of and the definition of the Green’s functions for the observable we obtain the solution for the observable in terms of the wave source From Equation (24), supplemented with a model for the statistics of , we can obtain all of the information about the unperturbed problem, in particular the power spectrum, the timedistance crosscorrelation, and the ingressionegression correlation.Let us now consider the power spectrum of the observable . There are two reasons to want to know the power spectrum of any particular model: Comparison between the model power spectrum and observations can be used to verify that the wave excitation model is reasonable, and local power spectra are the zeroorder model for ringdiagram analysis (see Section 4.2). We define the power spectrum as
where is the area and the time duration that the observations are taken over. The factor is included in the definition for the sake of clarity in some of the following results. Here is the spatial and temporal Fourier transform of the observable. When the source model is translation invariant in the horizontal directions and stationary in time we can write the power spectrum as For the definition of the source covariance see Equation (10). For the definition of the Green’s function for the observable see Equation (16).Notice that we have not attempted to include various corrections to the power spectrum (although it could in principle be done). For example, we have neglected the effects of the temporal and spatial window functions. Having observations over a finite spatial area or a finite amount of time will lead to smearing and lack of resolution in the and domains. Also, we have ignored sphericity. Analyzing solar data as if the Sun were flat, which is common in localhelioseismology, leads to distortions in power spectra due to use of FFTs on data that have been projected onto a plane. Lineofsight projection effects have also been ignored (local helioseismology typically uses the Doppler velocity as input data). For example, waves moving toward disk center are more visible than waves moving in the perpendicular direction; this results in anisotropic power spectra.
The telescope also introduces artifacts into the power spectrum. Because of the finite spatial resolution of the instrument, given by the pointspread function, the power at high wavenumbers is reduced below what it would be for an instrument with perfect spatial resolution. In general the point spread function is not azimuthally symmetric, and so the power is reduced more in some directions than others. The effect of the instrument on the power spectrum is summarized in the modulation transfer function (MTF) which satisfies
where is the power spectrum seen by the instrument and is solar power spectrum.Figure 4 shows an example comparison of a model power spectrum with real data. The model power is computed from a model with spatially uncorrelated quadrupole sources located 100 km below the photosphere. In order to accurately model the linewidths seen in local power spectra it is crucial to take into account the distortion of the wavefield introduced by the remapping (Birch et al., 2004).

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