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ISRN Applied Mathematics
Volume 2012 (2012), Article ID 539359, 10 pages
Green's Theorem for Sign Data
The University of Louisiana at Lafayette, Lafayette, LA 70504-4210, USA
Received 14 March 2012; Accepted 19 April 2012
Academic Editors: A. Bellouquid and M.-H. Hsu
Copyright © 2012 Louis M. Houston. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Sign data are the signs of signal added to noise. It is well known that a constant signal can be recovered from sign data. In this paper, we show that an integral over variant signal can be recovered from an integral over sign data based on the variant signal. We refer to this as a generalized sign data average. We use this result to derive a Green's theorem for sign data. Green's theorem is important to various seismic processing methods, including seismic migration. Results in this paper generalize reported results for 2.5D data volumes in which Green's theorem applies to sign data based only on traditional sign data recovery.
In certain cases, data, consisting of coherent signal and random noise, is summed repeatedly in order to enhance the signal and reduce the noise. It has been found that when the signal-to-noise ratio is between 0.1 and 1, if only the signs of the data are retained prior to summation, then the signal can be recovered . We refer to this type of data as sign data. Sign data has advantages in significantly reducing the amount of space needed to record the data. The reduction in space can amount to a ratio of 20 to 1 bits of data storage .
In the seismic industry, it has been discovered that most of the data processing methods used on regular data are also effective on sign data . Many of these data processing methods are dependent on Green’s theorem. In particular, Kirchhoff migration is dependent on Green’s theorem. Migration is a seismic process that corrects data coordinates which are distorted by attributes of the seismic experiment. In accordance with Green’s theorem, we show that an integral of an operation on sign data over a volume is equal to an integral of an operation of the original data over a surface.
This result generalizes the work of Houston and Richard  in which, based only on traditional sign data recovery, it was shown that Green’s theorem is satisfied when the sign data encompasses a 2.5D volume. 2.5D data is data which is three-dimensional but only has variations along two dimensions. In this paper, we find that Green’s theorem is satisfied for sign data when the data volume is largely symmetric. For cases in which the data volume is of arbitrary shape we have derived a variant of Green’s theorem.
2. A Generalized Sign Data Average
It was shown by O’Brien et al.  that there is a recovery of amplitude from sign data in the presence of uniform random noise. If the random noise has amplitude , the signal is designated by the function , and the number of iterations is , then we can write where
Figure 1 illustrates the data recovery process including sign data.
Multiply both sides of (2.1) by the function : Now sum both sides of (2.3) over the index: If we allow , then (2.4) becomes or It is clear that if and , then (2.6) becomes which is essentially (2.1). A continuous version of (2.6) might be written as An argument for the consistency of (2.8) is as follows. Let . Then we have Now integrate over all values of : Let be a uniform probability density, .
3. A Generalized Average for Sign Data Derivatives
Consider the following th order forward finite difference : with .
If we make the variable, discrete by choosing a small real interval, and writing where is an integer index, then (3.1) becomes Because the finite difference is a linear operator, we can use (2.6) to derive Equation (3.4) takes into account the fact that where is an arbitrary index.
Since (3.4) suggests the following equation:
Let and .
4. The Application to Green’s Theorem
Equation (3.7) implies the special case: Employing three variables in (4.1) yields which can be simplified if , , and : Summation over the variables yields Consequently, (4.4) can be written as If we make the mapping then (2.8) becomes Summation over the variables yields which can be written as Subtracting (4.9) from (4.5) yields Green’s theorem is Consequently, we can write a variant of Green’s theorem for sign data as If the spatial intervals are uniform, that is, , , and , then (4.12) becomes In the case of (4.13), we see that Green’s theorem is satisfied by replacing the function with sign data and dividing the surface integral by the noise amplitude. Therefore, all processing of real data based on Green’s theorem will be effective for sign data with an associated variance. We also note that the effectiveness of Green’s theorem on sign data is enhanced by operating over a symmetric volume. When the volume is not symmetric, we can apply the variant of Green’s theorem given by (4.12).
5. The 2.5D Case
Now let us consider the special case for which the functions and have only two-dimensional variation. That is, Without using a generalized sign data average, this implies or Equation (5.3) is Green’s theorem for sign data when the data encompasses a 2.5D volume and is consistent with results reported in Houston and Richard .
6. Computational Tests
Equation (2.6) derives from a weighted average of the function . We can thus write the expectation value of as We find that The variance, , can be written as This can be reduced to a simpler form by using the fact that Thus, we have We can demonstrate (2.6), computationally. Let Consequently, we want to demonstrate that We can compute the average percentage error as We should see a correlation between and . The results of this comparison are shown in Table 1.
Using the results of sign data signal recovery leads to a derivation of a generalized sign data average. Extending these results to incorporate derivatives leads to a variant of Green’s theorem for sign data. We find that Green’s theorem directly applies to sign data when the data volume is symmetric and the surface integral is divided by the noise amplitude. A specific application of this result is that Green’s theorem applies to sign data when the data volume is 2.5D and a generalized sign data average is not required.
Discussions with Maxwell Lueckenhoff are appreciated.
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