Kirchhoff integral formula

Let us consider a limited volume domain D, bounded by a surface S. A scalar wave-field P (r, t) is assigned within the bounds of that domain and satisfies the following equation  

Let us assume that in formula (13.101) the function P satisfies equation (13.99), while Q is taken to be the Green s function of the wave equation  

After integrating both sides of relation (13.105) over t with infinite limits, we have 

Integrating by parts and taking into account the finite character of function G, we find that the integral on the right-hand side of equation (13.106) is equal to zero. Therefore, introducing a function F(r,t ), 

according to Gauss s theorem (13.100), we obtain the following integral 

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The validity of the Kirchhoff integration formula (3.111) can be verified graphically by consideration of the thermochemical cycle shown in Fig. 3.14. As shown in the figure, the enthalpy change AH(1 ) for the direct reaction path at T must match the total enthalpy change for the alternative path of... 

Expression (13.109) is called the Kirchhoff integral formula. This formula shows that the wavefield inside domain D can be determined by the values of this field (and its normal derivative as well) on the domain boundary S. 

As one can see from relation (13.112), the wavefield at the point r D may be viewed at the moment of time t as the sum of elementary fields of point and dipole sources distributed over the surface S with densities dP r,t)/dn and P r,t) respectively. The interference of these fields beyond the domain D results in complete suppression of the total wavefield. Thus, the Kirchhoff integral formula can be treated as the mathematical formulation of the classical physical Huygens-Fresnel principle. 

Generalized Kirchhoff integral formulae for the Lame equation and the vector wave equation... 

In particular, assuming that the domain D with the external forces is located outside the volume V, we arrive at the generalized Kirchhoff integral formula for the Lame equation (Zhdanov, 1988) ... 

Assuming that all external forces are located outside the homogeneous domain V c = const), we arrive at the Kirchhoff integral formula for the vector wave equation... 

The generalized Kirchhoff integral formulae enable the values of the elastic displacement field (equation (13.121)) or vector wavefield (equation (13.125)) to be reconstructed everywhere inside the domain V from the known values of these fields and their normal derivatives at the domain boundary S. 

Based on the last expression, we can write the Kirchhoff integral formula in frequency domain as... 

Figure 13-2 Application of the Kirchhoff integral formula to a domain Vr bounded by a sphere Or of a radius r with the center at the origin of coordinates. The direction of the normal to the boundary of the domain at a point M e Or coincides with the direction of the radius-vector at that point.

A similar result holds true for the case shown in Figure 13-4b. Applying the inverse Fourier transform to both sides of the last expression, we cirrive at the Kirchhoff integral formula for an unbounded domain ... 

The Kirchhoff approximation is based on expressing the scattered field p (r, oj) at some point r in the upper half-space, using its values at the reflecting boundary (see Figure 14-2). To solve this problem we apply the Kirchhoff integral formula (13.197) to the scattered field in the upper half-space ... 

Expression (15.224) represents Stolt s (1978) Fourier-based migration formula. 15.4-5 Equivalence of the spectral and integral migration algorithms In the previous sections we analyzed independently two different approaches to wave-field migration 1) based on the Kirchhoff integral formula, and 2) Stolt s Fourier-based migration. It is important to understand that these two approaches are equivalent. 

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