Digital prediction

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Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
DigitalImaging.png
Series Geophysical References Series
Title Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
Author Enders A. Robinson and Sven Treitel
Chapter 15
DOI http://dx.doi.org/10.1190/1.9781560801610
ISBN 9781560801481
Store SEG Online Store

The prototype minimum-delay digital signal is the causal damped geometric signal for k = 0, 1, 2, …, where (Table 3, first row below the column headings). We want to let this signal be the input to a causal, linear time-invariant system, which we call the extrapolation (or prediction) system E(Z). For the desired output, we choose the advanced signal (say, for the advance for k = ..., –3, –2, –1, 0, 1, 2, … (Table 3, second row). The front end , (Table 3, third row) of the advanced signal is anticausal, whereas the tailgate (Table 3, fourth row) is causal. Because the prediction system is causal, it cannot reach the anticausal front end; thus, it can predict only the causal tailgate. For the tailgate, we demand perfect prediction. We will allow no prediction error whatsoever.

Table 3. Minimum-delay signal, advanced signal, front end of advanced signal, and tailgate of advanced signal.
Time k ... –3 –2 –1 0 1 2 3 4 ...
Minimum-delay signal ... 0 0 0 1 a ...
Advanced signal ... 0 1 a ...
Front end of ... 0 1 a 0 0 0 0 0 ...
Tailgate of ... 0 0 0 ...

The problem is to find an expression for the causal prediction filter . A naive person would proceed in this way. Both the input and the desired output (i.e., the tailgate) are causal, so we can use the one-sided Z-transform. We recall that the one-sided Z-transform is denoted by Z (equation 27). The one-sided Z-transform of the input is


(99)

whereas the one-sided Z-transform of the tailgate of the desired output is


(100)

The prediction distance, or advance (), is always a positive integer. The required prediction filter has a transfer function given by the ratio of the output Z-transform over the input Z-transform; that is,


(101)

This formula is correct, because if we multiply the input signal by , we indeed get the advanced value . A little thought tells us that everywhere the geometric signal has the same shape, so by using a constant attenuation , we can change the present shape into the future shape . (Note: Two curves are said to have the same shape if one is a constant factor times the other.)

The formula certainly works in the case in which is a geometric signal. Why did we go to the bother of defining minimum delay? One reason is that this formula for works not only for a geometric signal but also for any minimum-delay signal. In fact, we can say this: Perfect prediction in the sense defined above is possible if and only if the input signal is minimum delay. For a minimum-delay signal, gives the causal prediction system for prediction distance . The prediction of a nonminimum-delay signal will be treated in the next section.

Let us look at the AR(2) system


(102)

We want to find the prediction system that predicts the impulse response . First of all, we know that the impulse response of an AR system is necessarily minimum delay. The impulse response satisfies the difference equation


(103)

The Z-transform of the impulse response is the transfer function


(104)

where the poles and lie outside the unit circle; that is, and . The impulse response can be found by expanding in partial fractions. We have


(105)

where


(106)

Thus, the impulse response is the causal minimum-delay signal


(107)

That is, the AR(2) impulse response is a weighted average of two geometric signals with the weights and given above. We want to feed into the causal prediction filter and obtain as output the advanced signal . As we stated above, E(Z) is the ratio of the one-sided Z-transform of the output over the one-sided Z-transform of the input. Of course, we know that the input Z-transform is F(Z). We can obtain symmetry in our expression for E(Z) if we write the input Z-transform F(Z) as


(108)

The output Z-transform is


(109)

Thus, the required prediction system is


(110)

which is


(111)

If we simplify this expression, we obtain


(112)

This is the general expression for the prediction system with prediction distance . Therefore, for unit prediction distance , the prediction system is


(113)

Because the relationship between the AR coefficients and and the roots and is


(114)

we have


(115)

Thus, the prediction system for is


(116)

If is the input (at time index k) to the prediction system (with prediction distance ), then we define as the output (at the same time index k). The notation should be read as “the predicted value (that is obtained at the present time k) of the future value (that will not be known until the future time ).” In other words, the positive integer is the prediction distance. In the symbol , the caret stands for predicted value, the subscript k stands for the time at which the predicted value occurs, and stands for the prediction distance. That is, is the prediction of the future value , the prediction being made at the present time k. Because here , we can write


(117)

which is


(118)

Because is causal, it is zero for negative k. Thus, this equation gives


(119)

With the initial condition , the above equations 119 are equivalent to the equations


(120)

which generate the impulse response . Therefore, it follows that


(121)

which shows that the prediction system perfectly predicts the tailgate , , ... of the impulse response. In a nutshell, we can find the prediction system with for the impulse response of an AR(2) system by inspection. If the AR(2) system is


(122)

then the prediction system is the MA(1) system given by


(123)

As an example, let us find the prediction system that (for ) predicts the impulse response of the AR(p) system given by


(124)

By inspection, the prediction system is


(125)

Let us look at the minimum-delay MA(1) system


(126)

By inspection, we see that the causal impulse response is


(127)

which in longhand is


(128)

Thus


(129)

and in general,


(130)

Thus, the prediction system for prediction distance is


(131)

So if the prediction distance is greater than one, only the trivial prediction can be obtained. The one-step prediction system is


(132)

Therefore, the prediction is


(133)

which is


(134)

Because f(k) is causal, we have the system of equations


(135)

Because the impulse response is minimum delay, perfect prediction is obtained. Thus,


(136)

so the above system of equations becomes


(137)

With the initial condition , we find


(138)

which indeed is the impulse response of the given MA(1) system.

Let us look next at the digital minimum-delay ARMA(1,1) system


(139)

so and . The transfer function is


(140)

The minimum-delay impulse response is


(141)

so


(142)

Thus, the prediction system is


(143)

which is an AR(1) system.

Next let us consider the minimum-delay ARMA(2,1) system


(144)

where , , and . The partial fraction expansion is


(145)

where


(146)

The expansion of the fractions gives


(147)

so the impulse response is


(148)

The advanced impulse response is


(149)

Thus, the prediction system is


(150)

which, simplified, is


(151)

We see that the prediction system is an ARMA(1,1) system.


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