THE FORWARD OPERATOR AND ITS CONJUGATE

A migration operator is conjugate to the corresponding forward modeling operator (Claerbout, 1992). In order to find the conjugate operator, the linear forward modeling operator must be found first. For the sake of simplicity, we formulate the modeling algorithm from the datuming viewpoint, which requires only an extrapolation operator. The migration operator can be obtained by changing the input from the wavefield to the subsurface image and adding the imaging and transformation operator into the extrapolation operator.

To start, we formulate a simple modeling algorithm using phase-shift extrapolation when the recording surface is not flat. For simplicity, the number of geophone groups on an irregular surface is eight as illustrated in Figure .

 

syngeometry Figure 2 Synthetic surface recording geometry. Solid squares represent geophone location on an undulating surface.

The wavefield recorded at each geophone is the wavefield propagated up to the depth level where the geophones are located. Therefore, the modeling can be formulated by propagating the wavefield upward without regard for the irregular surface boundary. Then at each altitude, various portions (shaded in Figure ) are extracted from the wavefield. The sum of these extracted portions makes up the wavefield along the irregular surface. Figure  explains this modeling algorithm in the datuming sense.

 

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Figure 3 Modeling scheme 1: The schematic diagram for exploding reflector modeling when the surface is not flat. W_j represents the upward propagating operator at the j-th depth level shown in equation (2), FT^* represents the inverse Fourier transform. A, B and C are spatial filter shown in equation (3).

In order to find the conjugate operator of the algorithm shown in Figure , the algorithm must be written down in algebraic form. The algebraic representation for the forward modeling scheme in Figure  is given by

$$\left[ \begin{array} {ccc} A&B&C\end{array}\right] \left[ \b... ...t] = \left[ \begin{array} {c} d_{\rm surface}\end{array}\right]$$ (1)

where $D_{\rm datum}$ and $d_{\rm surface}$ represent the wavefield in the frequency-wavenumber domain at the datum just below the deepest geophone location and the wavefield in the frequency-space domain at the nonflat surface, respectively, and where A,B, and C are masking operators defined in Figure  (and later in equation (2)). Equation (1) shows that the modeling can be decomposed into three steps. The first step is the extrapolation step. The first matrix multiplied to the input has three diagonal matrices, U₁, U₂ and U₃, along its column for the given model and the number of matrices will change as the topography changes. The matrix U_j consists of j operators, from W_j to W₁, which do extrapolation by phase shift from j-th depth level to the surface. The extrapolation operator at each depth level W_j is a diagonal matrix such as:

$$\left[ \begin{array} {cccccccc} \exp^{ik_{z_j}\Delta z}&.&.&... ...&.\\ .&.&.&.&.&.&.&\exp^{ik_{z_j}\Delta z}\\ \end{array}\right]$$

where k_z is a function of frequency($\omega$) and horizontal wavenumber(k_x), defined by the dispersion relation of the acoustic wave equation. Some of these elements will be set to zero instead of $\exp^{ik_z\Delta z}$if k_z falls into an evanescent branch of the dispersion relation. The second step is to inverse Fourier transform along the horizontal axis x in preparation for the third step in which the wavefield is picked where a geophone is located at each depth level. The operator applied in the third step in equation (1) consists of three diagonal matrices on each of which is designed to pick out the wavefield at the geophone location on each depth level. For the synthetic model given in Figure , A, B and C have the following form:

$$A= \left[ \begin{array} {cccccccc} 1& & & & & & & \\ &1& & ... ...&1& & \\ & & & & & &.& \\ & & & & & & &.\\ \end{array}\right]$$

$$\rm and \quad C= \left[ \begin{array} {cccccccc} .& & & & & ... ...&.& & \\ & & & & & &.& \\ & & & & & & &.\\ \end{array}\right]$$ (2)

where dots represent zeros.

Now, the datuming operator can be easily found by transposing and taking the complex conjugate of the forward operator shown in equation (1). Therefore, we get the datuming operator for the data gathered on an irregular surface as

$$\left[ \begin{array} {ccc} U_1^T&U_2^T&U_3^T\end{array}\righ... ...}\right] = \left[ \begin{array} {c} D_{datum}\end{array}\right]$$ (3)

where FT is the Fourier transform along the horizontal axis and W_j^T represents downward phase-shift operator at each depth level. Equation (3) says that the conjugate operator of wavefield picking at several depth levels is the summation of the wavefields which are separately extrapolated from those depth levels at the point where the wavefields were picked. Figure  shows a schematic of this datuming operator.

 

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Figure 4 Datuming scheme 1 : Schematic diagram for datuming when the surfaces are irregular as the conjugate operator to modeling scheme 1. W₁^T represents the downward propagation operator at each depth level.