# Green's Theorem for Sign Data

Authors: Louis M. Houston
Journal: ISRN Applied Mathematics
 (2012)

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## Abstract

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.


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## Body

## 1. Introduction

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 [1]. 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 [1].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 [2]. 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 [3] 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. [1] that there is a recovery of amplitude from sign data in the presence of uniform random noise. If the random noise Xj has amplitude a, the signal is designated by the function fk, and the number of iterations is M=∑j, then we can write
(2.1)fk=aM∑jsgn⁡(fk+Xj),
where
(2.2)sgn⁡(y)={+1,y>00,y=0-1,y<0.Figure1 illustrates the data recovery process including sign data.Figure 1
A synthetic signal (sinc(x/3)e-(x/15)2) and the computer-generated uniform random noise used to examine sign-bit amplitude recovery.  This test has noise of unit magnitude (a=1) and a signal-to-noise ratio of one. Shown from bottom to top is the signal, the signal plus noise, the sign of the signal plus noise, the average over 200 iterations of signal plus noise, and the average over 200 iterations of the sign of signal plus noise.Multiply both sides of (2.1) by the function gk:
(2.3)gkfk=aM∑jgksgn⁡(fk+Xj).
Now sum both sides of (2.3) over the k index:
(2.4)∑kgkfk=aM∑k∑jgksgn⁡(fk+Xj).
If we allow k→j, then (2.4) becomes
(2.5)∑jgjfj=aM∑j∑jgjsgn⁡(fj+Xj)
or
(2.6)∑jgjfj=a∑jgjsgn⁡(fj+Xj).
It is clear that if fj→f and gj→g, then (2.6) becomes
(2.7)f=aM∑jsgn⁡(f+Xj),
which is essentially (2.1).  A continuous version of (2.6) might be written as
(2.8)∫g(v)f(v)dv=a∫g(v)sgn⁡(f(v)+v)dv.
An argument for the consistency of (2.8) is as follows. Let f(v)→f. Then we have
(2.9)f=a∫g(v)sgn⁡(f+v)dv∫g(v)dv.
Now integrate over all values of v:
(2.10)f=a∫-∞∞g(v)sgn⁡(f+v)dv∫-∞∞g(v)dv.
Let g(v) be a uniform probability density, ρ(v).That implies(2.11)∫-∞∞g(v)dv=1,
and (2.10) becomes
(2.12)f=a∫-∞∞ρ(v)sgn⁡(f+v)dv,
which was shown by Houston et al. [4].


## 3. A Generalized Average for Sign Data Derivatives

Consider the followingnth order forward finite difference [5]:
(3.1)Δhnf(v)=∑i=0n(-1)i(ni)f(v+(n-i)h),
with (ni)=n!/i!(n-i)!.If we make the variable,v discrete by choosing a small real interval, q and writing
(3.2)fj=f(jq),
where j is an integer index, then (3.1) becomes
(3.3)Δhnfj=∑i=0n(-1)i(ni)f(jq+(n-i)h).
Because the finite difference is a linear operator, we can use (2.6) to derive
(3.4)∑jgjΔhnfj=a∑jgjΔhnsgn⁡(fj+Xj).
Equation (3.4) takes into account the fact that
(3.5)∑jg(jq)f(jq+lh)=a∑jg(jq)sgn⁡(f(jq+lh)+Xj),
where l is an arbitrary index.Since(3.6)dndvn=lim⁡h→0Δhnhn,
(3.4) suggests the following equation:
(3.7)∫g(v)dndvnf(v)dv=a∫g(v)dndvnsgn⁡(f(v)+v)dv.Letdn/dvn→dn/dun and f(v)→f(u).Then from (3.7) we have
(3.8)dndunf(u)=a∫g(v)(dn/dun)sgn⁡(f(u)+v)dv∫g(v)dv.
Once again, integrate over all v, let g(v)=ρ(v), and (3.8) becomes
(3.9)dndunf(u)=a∫-∞∞ρ(v)sgn⁡(f(u)+v)dv,
which was shown by Houston et al. [4].


## 4. The Application to Green’s Theorem

Equation (3.7) implies the special case:
(4.1)∫g(v)d2dv2f(v)dv=a∫g(v)d2dv2sgn(f(v)+v)dv.
Employing three variables in (4.1) yields
(4.2)∫g(v1,v2,v3)∂2∂vi2f(v1,v2,v3)dv1dv2dv3=a∫g(v1,v2,v3)∂2∂vi2sgn⁡(f(v1,v2,v3)+vi)dv1dv2dv3,
which can be simplified if g=g(v1,v2,v3), f=f(v1,v2,v3), and dV=dv1dv2dv3:
(4.3)∫g∂2∂vi2fdV=a∫g∂2∂vi2sgn⁡(f+vi)dV.
Summation over the variables yields
(4.4)∑i‍∫g∂2∂vi2fdV=a∑i‍∫g∂2∂vi2sgn⁡(f+vi)dV.
Consequently, (4.4) can be written as
(4.5)∫g∇2fdV=a∫g∑i∂2∂vi2sgn⁡(f+vi)dV.
If we make the mapping
(4.6)g(v)⟶∂2∂vi2g,
then (2.8) becomes
(4.7)∫f∂2∂vi2gdV=a∫sgn⁡(f+vi)∂2∂vi2gdV.
Summation over the variables yields
(4.8)∑i‍∫f∂2∂vi2gdV=a∑i‍∫sgn⁡(f+vi)∂2∂vi2gdV,
which can be written as
(4.9)∫f∇2gdV=a∫∑i‍sgn⁡(f+vi)∂2∂vi2gdV.
Subtracting (4.9) from (4.5) yields
(4.10)∫(g∇2f-f∇2g)dV=a∫(g∑i∂2∂vi2sgn⁡(f+vi)-∑i‍sgn⁡(f+vi)∂2∂vi2gdV).
Green’s theorem is
(4.11)∮S(U1∂U2∂n-U2∂U1∂n)dS=∫V(U1∇2U2-U2∇2U1)dV.
Consequently, we can write a variant of Green’s theorem for sign data as
(4.12)∮S(g∂f∂n-f∂g∂n)dS=a∫(g∑i∂2∂vi2sgn⁡(f+vi)-∑i‍sgn⁡(f+vi)∂2∂vi2g)dV.
If the spatial intervals are uniform, that is, v1∈[a1,b1], v2∈[a2,b2], v3∈[a3,b3] and [a1,b1]=[a2,b2]=[a3,b3], then (4.12) becomes
(4.13)∮S(g∂f∂n-f∂g∂n)dS=a∫(g∇2sgn⁡(f+vi)-sgn⁡(f+vi)∇2g)dV.
In the case of (4.13), we see that Green’s theorem is satisfied by replacing the function f 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 functionsf and g have only two-dimensional variation. That is,
(5.1)f⟶f(v1,v2)g⟶g(v1,v2).
Without using a generalized sign data average, this implies
(5.2)a∫(g∇2sgn⁡(f+vi)-sgn⁡(f+vi)∇2g)dV=(b3-a3)∫(g∇2f-f∇2g)dv1dv2
or
(5.3)∮S(g∂f∂n-f∂g∂n)dS=(b3-a3)∫(g∇2f-f∇2g)dv1dv2.
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 [3].


## 6. Computational Tests

Equation (2.6) derives from a weighted average of the function yj=sgn(fj+Xj). We can thus write the expectation value of yi as
(6.1)E(Y)=∑jgjyj∑jgj.
We find that
(6.2)E(Y)=∑jgjfja∑jgj.
The variance, Var(Y), can be written as
(6.3)Var⁡(Y)=E(Y-E(Y))2.
This can be reduced to a simpler form by using the fact that
(6.4)E(Y2)=1.
Thus, we have
(6.5)Var⁡(Y)=1-(E(Y))2.
We can demonstrate (2.6),
(6.6)∑jgjfj=a∑jgjsgn⁡(fj+Xj),
computationally. Let
(6.7)fj=sin⁡(jq),gj=cos(jq).
Consequently, we want to demonstrate that
(6.8)∑jcos⁡(jq)sin⁡(jq)=a∑jcos⁡(jq)sgn⁡(sin⁡(jq)+Xj).
We can compute the average percentage error β as
(6.9)β=100〈(∑jcos⁡⁡(jq)sin⁡⁡(jq)-a∑jcos⁡(jq)sgn⁡(sin⁡(jq)+Xj))2〉(∑jcos⁡(jq)sin⁡(jq))2.
We should see a correlation between β and Var(Y). The results of this comparison are shown in Table 1.Table 1
Twenty trials which compare the values of the indicated expressions forM iterations per trial. The trials are divided into ten trials for two different noise amplitudes. Recall that M=∑j. Let M=10000 and q=0.0001.
∑
j
=
1
M
cos
(
j
q
)
sin
(
j
q
)
a
∑
j
=
1
M
cos
(
j
q
)
sgn
(
sin
(
j
q
)
+
X
j
)
a
=
1
3540.594404
3529.425407
(1)
↓
3569.469208
(2)
3460.5978
(3)
3540.394914
(4)
3542.955134
(5)
3644.993387
(6)
3441.978126
(7)
3510.495952
(8)
3544.098984
(9)
3645.64803
(10)
β
=
0.56447%,Var(Y)=0.822949
a
=
2
3540.594404
3578.540737
(1)
↓
3468.32837
(2)
3063.26568
(3)
3629.793227
(4)
3562.692845
(5)
3605.376332
(6)
3284.005063
(7)
3375.67384
(8)
3356.467161
(9)
3500.608956
(10)
β
=
1.731154%,Var(Y)=0.955737


## 7. Conclusions

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.



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*Source: 539359-2012-06-21.xml*