# Differencing schemes

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

Finite-difference migration algorithms are based on differential solutions to the scalar wave equation that are used to downward continue the input wavefield recorded at the surface. A simple numerical example illustrates the finite-difference method of solving differential equations [1]. Assume that you have $100 today. Given an annual inflation rate of 10 percent, for the same buying power next year, you need$110. A computer algorithm can determine the face value of the present \$100 in future years. Table 4-3 shows the results of extrapolation from one year to the next.

Given the present value, 100, find future values in the data column. The following equation solves for the unknown x:

 ${\displaystyle (1.0)\times x+(-1.1)\times (100)=0,}$ (6a)

which yields x = 110. We used a two-point operator and aligned it with the data column as indicated in Table 4-3. Similarly, we have

 ${\displaystyle (1.0)\times x+(-1.1)\times (110)=0,}$ (6b)

which yields x = 121. By using the new value for x, we obtain

 ${\displaystyle (1.0)\times x+(-1.1)\times (121)=0,}$ (6c)

which yields x = 133, and so on. By moving the operator down in the time direction as shown in Table 4-3, we extrapolate the data column into the future.

 Operator Data Time Step -1.1 100 0 1.0 x 1 100 0 -1.1 110 1 1.0 x 2 100 0 110 1 -1.1 121 2 1.0 x 3

Equation (6a) is generalized as

 ${\displaystyle (1.0)\times P(t+1)+(-1.1)\times P(t)=0,}$ (7a)

which is rewritten in the form

 ${\displaystyle P(t+1)-P(t)=(0.1)\times P(t),}$ (7b)

where t is the time variable and P is the quantity being extrapolated. Instead of defining the time interval as one unit, we can define it as an arbitrary increment of time Δt. Also, assume that the inflation rate is a. Equation (7b) then takes the more general form

 ${\displaystyle P(t+\Delta t)-P(t)=a\,P(t).}$ (8a)

Alternatively, we could use the average of the present and future values on the right side of this equation:

 ${\displaystyle P(t+\Delta t)-P(t)={\frac {a}{2}}{\big [}P(t+\Delta t)+P(t){\big ]}.}$ (8b)

Equations (8a) and (8b) now can be put into the form of equation (6a) as

 ${\displaystyle P(t+\Delta t)+(-1-a)P(t)=0,}$ (9a)

and

 ${\displaystyle \left(1-{\frac {a}{2}}\right)P(t+\Delta t)+\left(-1-{\frac {a}{2}}\right)P(t)=0.}$ (9b)

By using either equation (9a) or equation (9b), we compute the future values of P(t) from a given initial value as shown in Table 4-4.

The operator in which the coefficient of the future value P(t + Δt) is unity is called the explicit operator. Stability of the finite-difference solution — the problem of wave amplitudes growing from one extrapolation step to another, can be an issue with this type of operator (Section D.6) An implicit operator produces stable results because of averaging on the right side of equation (9b), known as the Crank-Nicolson scheme. For the differential equations used in finite-difference migration algorithms, such as the parabolic equation described in Section D.3, scalar a becomes a matrix coefficient. Implicit schemes require inversion of this matrix. However, no inversion is needed with explicit schemes, since future values can be written explicitly in terms of only past values.

 Explicit Operator Implicit Operator Data Column −1 − a −1 − a/2 P(t) 1 1 − a/2 P(t + Δt)

Equation (9a) is rewritten by redefining scalar a as a Δt to obtain

 ${\displaystyle {\frac {P(t+\Delta t)-P(t)}{\Delta t}}=a\,P(t).}$ (10)

The left side of equation (10) is the discrete representation of the continuous derivative of P with respect to time, dP/dt. Therefore, equation (10) is the finite-difference equation that corresponds to the differential equation

 ${\displaystyle {\frac {dP}{dt}}=a\,P(t).}$ (11)

We have derived the differential equation that describes the inflation of money (equation 11). Now consider the analysis in reverse order. We start with the differential equation (11), and write the corresponding difference equation (10), which is the equation that is solved in the computer. This equation is written in either the explicit (equation 9a) or implicit (equation 9b) form to extrapolate the present value of P to the future.

This example illustrates how finite-difference schemes can solve differential equations in the computer. The scalar wave equation can be treated in a similar, but more complicated manner. Complications arise because it is a partial differential equation that contains the second derivatives of the wavefield with respect to depth, time, and spatial axes. Setting up the computer algorithm is more involved and is not discussed here. Claerbout [2], [1]. provides details of various aspects of the finite-difference migration methods.

## References

1. Claerbout, 1985, Claerbout, J.F., 1985, Imaging the earth’s interior: Blackwell Scientific Publications.
2. Claerbout (1976), Claerbout, J.F., 1976, Fundamentals of geophysical data processing: McGraw-Hill Book Co.