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3.4 The zero-order problem

The zero-order problem is to solve the driven equations of motion, Equation (7View Equation), when there are no perturbations to the background model. By the background model we mean a description of the background state together with specifications for wave damping and the statistics of wave excitation. As discussed in Section 3.1 we assume that the background state is a steady, wave-free, non-magnetic background solar model. The statistics of wave excitation which are needed to compute the wavefield covariance are described by the source covariance matrix M ij (see Section 3.2). Wave damping can be described either physically (e.g., Balmforth, 1992) or phenomenologically (e.g., Gizon and Birch, 2002Jump To The Next Citation Point).

To obtain the zero-order problem we break the various quantities into unperturbed components (superscript 0) and first-order corrections (prefix δ). In particular we have

0 ā„’ = ā„’ + δā„’, (19 ) ξ = ξ0 + δξ, (20 ) 0 S = S + δS. (21 )
From Equation (7View Equation) we can then obtain the governing equations for the zero-order and perturbed problems.

To lowest order Equation (7View Equation) becomes

ā„’0 ξ0 = S0. (22 )
The solution to this equation can be written in terms of the Green’s tensor G (Section 3.3):
∫ ξ0 (r, t) = d3r dt Gi (r,t;r ,t )S0(r ,t ), (23 ) s s s s i s s
where summation over the index i is implied. From Equation (23View Equation) together with the definition of the observable in terms of ξ and the definition of the Green’s functions for the observable we obtain the solution for the observable in terms of the wave source
∫ Φ (x, t) = d3rsdts š’¢i(x,t;rs,ts)S0i(rs,ts). (24 )
From Equation (24View Equation), supplemented with a model for the statistics of S0, we can obtain all of the information about the unperturbed problem, in particular the power spectrum, the time-distance cross-correlation, and the ingression-egression correlation.

Let us now consider the power spectrum of the observable Φ. 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 zero-order model for ring-diagram analysis (see Section 4.2). We define the power spectrum as

3 [ ] P(k, ω) = (2π-)-E |Φ (k,ω )|2 , (25 ) AT
where A is the area and T the time duration that the observations are taken over. The factor (2π )3 is included in the definition for the sake of clarity in some of the following results. Here Φ (k,ω ) 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
∫ ∫ 6 ′ i ∗j ′ ′ P (k,ω ) = (2π ) dz dz š’¢ (k, ω,z) š’¢ (k,ω,z )Mij (k,ω, z,z ). (26 )
For the definition of the source covariance Mij see Equation (10View Equation). For the definition of the Green’s function for the observable see Equation (16View Equation).

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 k and ω domains. Also, we have ignored sphericity. Analyzing solar data as if the Sun were flat, which is common in local-helioseismology, leads to distortions in power spectra due to use of FFTs on data that have been projected onto a plane. Line-of-sight 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 point-spread 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

P (k,ω) = MTF (k )PāŠ™(k, ω), (27 )
where P is the power spectrum seen by the instrument and P āŠ™ is solar power spectrum.

Figure 4View Image 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., 2004Jump To The Next Citation Point).

View Image

Figure 4: Comparison of a model filtered power spectrum with the filtered power spectrum of MDI data. The filter is such that most of the power is in the p 1 and p 2 ridges. The model power spectrum is computed for a model with spatially uncorrelated quadrupole sources located at 100 km below the photosphere. The model includes the effects of the spatial and temporal window functions and the broadening of the spectrum by the remapping. Without these corrections the model power spectrum would have linewidths much smaller than those of the data. Upper left panel: Observed filtered power spectrum. Upper right panel: Model filtered power spectrum. The solid lines in the upper panels show the location of the f-mode ridge, for reference. Lower left panel: Average over wavenumbers of the observed (dashed line) and model (solid line) filtered power spectra. Lower right panel: Average over frequencies of the observed (dashed line) and model (solid line) filtered power spectra. From Birch et al. (2004Jump To The Next Citation Point).

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