Kirchhoff migration

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Problem 9.27a

Prove that the equation of a diffraction curve (see Figure 9.27a) is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} z^{2} -x^{2} =z_{0}^{2}, \end{align}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z_{0} is the depth of the diffracting point (assuming constant velocity).

Background

Migration is the process of repositioning reflection data so that reflection events are located where the reflector in the subsurface is, as shown in Figure 9.27a. $ S_{0} $, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S_{1} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S_{2} , ..., are a series of coincident sources and receivers and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): P is a diffracting point on a plane reflector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): PQ . The data recorded at $ S_{0} $ are ordinarily plotted directly below Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S_{0} before migration is performed. If the velocity is constant, the reflection from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): P that is observed at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S_{0} appears on the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S_{0} trace at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): M , and the reflection point must lie on a circle through Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): M centered at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S_{0} . The circles such as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): PM are wavefronts and radii such as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S_{0}P are raypaths which are orthogonal to the respective wavefronts.

Solution

The locus of point M as the source and coincident receiver are moved is a diffraction curve (see Figure 6.5a). If O is the origin and the coordinates of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): M are (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x,z ), then, because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S_{0} P=S_{0} M .


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} z^{2} =x^{2} +z_{0}^{2},\quad \mathrm{so}\quad z^{2} -x^{2} =z_{0}^{2}. \end{align} (9.27a)

Problem 9.27b

Show that an unmigrated reflection is tangent to the diffraction curve.

Solution

We find the equation of an unmigrated reflection Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): MN (Figure 9.27a) and determine its slope, then find the slope of the diffraction curve and show that the point where the slope of the diffraction curve is the same as that of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): MN is also on $ MN $. To get the equation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): MN , we need the equation of the reflector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): PQ because it meets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): MN at the outcrop on the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x -axis.

The line Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): PQ passes through Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): P(0,\; z_{0}) with slope Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\rm \; tan\;}\xi , where $ \xi $ is the dip, so its equation is


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} z=x{\rm \; tan\;}\xi +z_{0}. \end{align} (9.27b)
Figure 9.27a.  Intersecting wavefronts.

This line outcrops (that is, at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z=0 ) at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=-z_{0} {\rm \; cot\;}\xi , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): MN passes through the point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): (-z_{0} {\rm \; cot\;}\xi ,\; 0) as well as the point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): M , whose coordinates are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=z_{0} {\rm \; tan\;}\xi , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z=S_{0} M= S_{0} P=z_{0} /{\rm \; cos\;}\xi . The equation of $ MN $ now becomes


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \frac{z-z_{0}/\cos\xi}{x-z_{0}{\tan\xi}} =\frac{z_{0} /\cos\xi}{z_{0} \tan\xi +z_{0} \cot\xi} =\frac{1}{\sin\xi +\cos^{2} \xi /\sin\xi} =\sin\xi .\\ \mbox {Thus} \qquad \qquad z&=z_{0} /\cos\xi +x\sin\xi -z_{0} \tan\xi \sin\xi \\ &=x\sin\xi +z_{0} \cos\xi, \end{align} (9.27c)

and the slope of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): MN is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\hbox{d}}z/{\hbox{d}}x={\rm \; sin\;}\xi .

The slope of the diffraction curve in equation (9.27a) is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\hbox{d}}z/{\hbox{d}}x=x/z , so we need to find a point on the curve in equation (9.27a) where this slope equals that of $ MN $, that is, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x/z=\sin\xi . Substituting this value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x in equation (9.27a) gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} z^{2} -(z\sin\xi)^{2} =z_{0}^{2},\quad \mathrm{so}\quad z=z_{0} /\cos\xi, x=z_{0} \tan\xi. \end{align}

We now substitute these coordinates in equation (9.27c) to show that they satisfy the equation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): MN . Thus,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} z_{0} /\cos\xi =(z_{0} \tan\xi)\sin\xi +z_{0} \cos\xi \; ,\; 1=\sin^{2} \xi +\cos^{2} \xi ; \end{align}

therefore, the diffraction curve and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): MN both pass through this point and, since they have only one common point, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): MN must be a tangent.

Problem 9.27c

Show that the coordinates of $ P $ and the slope of the wavefront at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): P (hence also the dip Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \xi ) can be obtained from the recorded data.

Solution

Assuming that the seismic line is normal to strike, we obtain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z_{0} from the minimum value of the traveltime Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{0} , which occurs when the source is directly over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): P ; this fixes the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x -coordinate of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): P . If the velocity is constant, by swinging an arc with center at any source and radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \frac{1}{2} Vt_{0} and then drawing the tangent from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): P to the arc, we get the dip $ \xi $.

Problem 9.27d

How will parts (a), (b), and (c) change if the velocity changes in the vertical direction (e.g., if it increases with depth)?

Solution

In this case the raypath will curve, a wavefront may not be the arc of a circle, the diffraction curve will no longer be a hyperbola, an unmigrated reflection will still be tangent to the diffraction curve, and locating the reflecting point will be more complicated. However, assuming that a diffraction curve is hyperbolic often results in solutions that are close enough to be useful and most data processing makes this assumption.

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Kirchhoff migration – book
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