# Kirchhoff summation

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

## Migration principles

The diffraction summation that incorporates the obliquity, spherical spreading and wavelet shaping factors is called the Kirchhoff summation, and the migration method based on this summation is called the Kirchhoff migration. To perform this method, multiply the input data by the obliquity and spherical spreading factors. Then apply the filter with the above specifications and sum along the hyperbolic path that is defined by equation (4). Place the result on the migrated section at time τ corresponding to the apex of the hyperbola. In practice, the order of the filter application, specified by factor (c), and summation can be interchanged without sacrificing accuracy because the summation is a linear process and the filter is independent of time and space.

The velocity used in equation (4) is taken as the rms velocity, which can be allowed to vary laterally. However, lateral variation in velocity distorts the hyperbolic nature of the diffraction pattern and somehow must be considered. The value for the rms velocity typically is that of the output time sample; that is, the apex time τ of the hyperbola.

 ${\displaystyle t^{2}=\tau ^{2}+{\frac {4x^{2}}{v_{rms}^{2}}}.}$ (4)

What was determined from a physical point of view in the preceding discussion can be rigorously described by the integral solution to the scalar wave equation. Schneider [1], Berryhill [2] and Berkhout [3] are excellent references for the mathematical treatment of the Kirchhoff migration method. The integral solution of the scalar wave equation yields three terms; the far-field term which is proportional to (1/r), and two other terms which are proportional to (1/r2). Hence, it is the far-field term that makes the most contribution to the summation that is used in practical implementation of Kirchhoff migration. The output image Pout(x0, z = /2, t = 0) at a subsurface location (x0, z) using only the far-field term is computed from the 2-D zero-offset wavefield Pin(x, z = 0, t), which is measured at the surface (z = 0), by the following summation over a spatial aperture

 ${\displaystyle P_{out}={\frac {\Delta x}{2\pi }}\sum \limits _{x}\left[{\frac {\cos \theta }{\sqrt {v_{rms}r}}}\rho (t)*P_{in}\right],}$ (5)

where vrms is the rms velocity at the output point (x0, z) and ${\displaystyle r={\sqrt {{{\left(x-{{x}_{0}}\right)}^{2}}+{{z}^{2}}}},}$ which is the distance between the input (x, z = 0) and the output (x0, z) points. The asterisk denotes convolution of the rho filter ρ(t) with the input wavefield Pin.

The rho filter ρ(t) corresponds to the time derivative of the measured wavefield, which yields the 90-degree phase shift and adjustment of the amplitude spectrum by the ramp function ω of frequency (Table A-1 of Appendix A). For 2-D migration, the half-derivative of the wavefield is used. This is equivalent to the 45-degree phase shift and adjustment of the amplitude spectrum by a function of frequency defined as ${\displaystyle {\sqrt {\omega }}.}$ Since the rho filter is independent of the spatial variables, it actually can be applied to the output of the summation in equation (5). Finally, the far-field term in equation (5) is proportional to the cosine of the angle of propagation (the directivity term or the obliquity factor) and is inversely proportional to vr (the spherical spreading term) in three dimensions. In two dimensions, the spherical spreading term is ${\displaystyle {\sqrt {vr}}.}$

 Operation Time Domain Frequency Domain (1) Shifting x(t − τ) exp(−iωτ)X(ω) (2) Scaling x(at) ${\displaystyle {{\left|a\right|}^{-1}}X\left({\omega }/{a}\;\right)}$ (3) Differentiation dx(t)/dt iωX(ω) (4) Addition f(t) + x(t) F(ω) + X(ω) (5) Multiplication f(t) x(t) F(ω)* X(ω) (6) Convolution f(t)* x(t) F(ω) X(ω) (7) Autocorrelation x(t)* x(−t) ${\displaystyle {{\left|X\left(\omega \right)\right|}^{2}}}$ (8) Parseval’s theorem ${\displaystyle \int {{\left|x\left(t\right)\right|}^{2}}\ dt}$ ${\displaystyle \int {{\left|X\left(\omega \right)\right|}^{2}}\ d\omega }$
 * denotes convolution.

Equation (5) can be used to compute the wavefield at any depth z. The output image Pout is computed at (x0, z = vτ/2, t = 0) using the input wavefield Pin at (x, z = 0, t − r/v). To obtain the migrated section at an output time τ, equation (5) must be evaluated at z = vτ/2 and the imaging principle must be invoked by mapping amplitudes of the resulting wavefield at t = 0 onto the migrated section at output time τ. The complete migrated section is obtained by performing the summation in equation (5) and setting t = 0 for each output location. The range of the summation is called the migration aperture.

## 3-D prestack depth migration

Kirchhoff’s integral solution to the scalar wave equation

 ${\displaystyle {\frac {\partial ^{2}P}{\partial x^{2}}}+{\frac {\partial ^{2}P}{\partial y^{2}}}+{\frac {\partial ^{2}P}{\partial z^{2}}}={\frac {1}{v^{2}(x,y,z)}}{\frac {\partial ^{2}P}{\partial t^{2}}}}$ (1)

gives the pressure wavefield P(x, y, z; t) propagating in a medium with velocity v(x, y, z) at a location (x, y, z) and at an instant of time t. As such, the Kirchhoff solution is a mathematical statement of Huygen’s principle which states that the pressure disturbance at time tt is the superposition of the spherical waves generated by point sources at time t [5].

Figure H-1  Geometry of a point diffractor to derive the Kirchhoff integral solution to the scalar wave equation.

The discrete form of the integral solution to equation (8-1) as used in practical implementation of Kirchhoff migration is given by (Section H.1)

 ${\displaystyle P_{out}={\frac {\Delta x\Delta y}{4\pi }}\sum _{A}{\frac {\cos \theta }{vr}}{\frac {\partial }{\partial t}}P_{in},}$ (2)

where Δx and Δy are inline and crossline trace intervals, Pout = p(xout, yout, z; τ = 2z/v) is the output of migration using the input wavefield Pin = P(xin, yin, z = 0; τ = t − r/v) within an areal aperture A. The geometry associated with equation (8-2) is given in Figure H-1.

The Kirchhoff summation method requires

1. computing nonzero-offset traveltimes through a 3-D, spatially varying velocity medium, and
2. scaling and summation of the amplitudes along the computed traveltime trajectory based on the Kirchhoff integral solution to the scalar wave equation.

The scaling of amplitudes before summation includes application of the obliquity factor cos θ, the spherical divergence factor 1/vr, and the amplitude and phase corrections, |ω| exp(/2), implied by the derivative operator ∂P/∂t (migration principles). Additionally, as for any migration, undersampling of the input data in x and y directions needs to be compensated for by a suitable antialiasing filter. Nevertheless, it is the traveltime computation that poses numerical accuracy and efficiency challenges when implementing the Kirchhoff summation method for 3-D prestack depth migration.

## References

1. Schneider, 1978, Schneider, W., 1978, Integral formulation for migration in two and three dimensions: Geophysics, 43, 49–76.
2. Berryhill (1979), Berryhill, J.R., 1979, Wave-equation datuming: Geophysics, 44, 1329–1333.
3. Berkhout (1980), Berkhout, A.J., 1980, Seismic migration – Imaging of acoustic energy by wavefield extrapolation: Elsevier Science Publ. Co., Inc.
4. Bracewell, 1965, Bracewell, R. N., 1965, The Fourier transform and its applications: McGraw-Hill Book Co.
5. Officer, 1958, Officer, C. B., 1958, Introduction to the theory of sound transmission: McGraw-Hill Book Co.