### 3.3 Response to an impulsive source

Of central importance to the theory of local helioseismology is the concept of Green’s functions. The Green’s functions are the impulse responses of the solar model and solve
where is the Dirac delta function in one or two dimensions. For each , the is the unit vector in the direction at , and is the wave operator for the displacement (see Equation 7). For example, in the non-magnetic case as given by Equation (5). Because it describes a displacement vector, is a vector. Taken together, the three Green’s vectors form a Green’s tensor, . The function is the displacement in the direction at that results from a unit source acting in the direction at

There are three approaches to solving Equation (11). In the case of very simple problems it is sometimes possible to solve analytically for the Green’s functions in the Fourier domain (see, e.g., Gizon and Birch, 2002). Another, more general approach, is direct numerical solution (Section 3.3.1). An efficient approximate solution is normal mode summation (Section 3.3.2).

#### 3.3.1 Direct solution in plane-parallel models

For plane-parallel steady models with horizontal translation invariance, the Green’s functions will be of the form where now and are the decompositions of and into horizontal, and , and vertical, and , components. In this case we can write the Green’s functions as the inverse Fourier transforms of Fourier domain Green’s functions

where is the horizontal wavevector and is the temporal angular frequency. We can then obtain the equation satisfied by from Equation (11), for any particular choice of . The result will be an inhomogeneous set of ordinary differential equations (ODEs) in the variable for the components of . These ODEs can then be integrated numerically to obtain for given , , and . Care must be taken with the treatment of the delta function on the right hand side of Equation (11); this can treated by folding the computational domain so that the jump condition across the delta function becomes a non-local boundary condition (Birch et al., 2004).

#### 3.3.2 Normal-mode summation approximation

Dahlen and Tromp (1998) give an excellent discussion of the normal-mode summation approach to the computation of Green’s functions. For a detailed discussion in the context of helioseismology, see Birch et al. (2004). The basic notion of the normal-mode summation approximation is that the Green’s function can be represented as a sum over normal modes. Intuitively, the idea is to compute the amplitude to which a particular source excites each mode in the model, and then to let each mode evolve in time. For example, for the case of undamped modes, Dahlen and Tromp (1998) write

Here the index refers to the direction of the source, and the index to the component of the response that we are looking at. For the Green’s function is zero. The sum over is taken over all normal modes of the background model. By normal mode we mean a solution of Equation (4) of the form , with the normalization . In the case of spherical background models the modes are described by the radial order the angular degree and the azimuthal order . In this case we have . For the case of plane-parallel models, modes can be labeled by a radial order and a horizontal wavevector . In this case we have . The situation is somewhat more complicated for the case of non-adiabatic modes, although it can still be addressed in much the same way (see Chapter six of Dahlen and Tromp, 1998).

Figure 3 compares Green’s functions computed numerically and approximated via normal-mode summation. Notice that the exact numerical result shows a discontinuity at the source depth; this comes from the jump across the delta function on the right hand side of Equation (11). The normal mode approximation, being a finite sum of continuous functions, is everywhere continuous.

In the case of plane-parallel translation-invariant isotropic models where the only restoring forces are pressure and gravity, it can be shown that the Green’s functions, in the Fourier domain, can be decomposed as

This decomposition is useful as now for any source we need only to compute two components of the response rather than three. Also notice that and depend only on the wavenumber , defined by . A similar decomposition can be done for the dependence on the source direction (Birch et al., 2004).

#### 3.3.3 Green’s functions for the observable

For the remainder of this review, it will be convenient to have a Green’s function for the response of the observable to a wave source. We denote the observable wavefield by the scalar . For most current helioseismology work, the observable is the line-of-sight Doppler velocity. As a result, we define

where is the Eulerian velocity at horizontal location , at depth , and time . The line-of-sight unit vector may depend on . The operator describes the filter used in the data analysis, which includes the time window, instrumental effects, and other filtering.

As will become obvious in the following sections, it is convenient to introduce a Green’s function for the observable (see Equation 15), given by:

The function is the response of the observable to a unit source acting in the direction. Notice that for the special case when the steady background flow is constrained to be horizontal at the surface and the line of sight is vertical (), we simply have

For plane-parallel steady models with horizontal translation invariance, the Green’s functions for the observable are of the form where . In this case we can write the Fourier domain Green’s functions for the observable as , according to

In the short notation , the always refers to the vertical position of the source of excitation.