Sign Data Derivative Recoverydownloads.hindawi.com/journals/isrn/2012/630702.pdf · In industries like the seismic industry, measurements of signal, f v: R → R. 2 ISRN Applied Mathematics

International Scholarly Research NetworkISRN Applied MathematicsVolume 2012, Article ID 630702, 7 pagesdoi:10.5402/2012/630702

Research ArticleSign Data Derivative Recovery

L. M. Houston,1 G. A. Glass,2 and A. D. Dymnikov1

1 Louisiana Accelerator Center, The University of Louisiana at Lafayette, Lafayette, LA 70504-4210, USA2 Ion Beam Modification and analysis Laboratory, Department of Physics, University of North Texas,Denton, TX 76203, USA

Correspondence should be addressed to L. M. Houston, [email protected]

Received 2 November 2011; Accepted 29 November 2011

Academic Editors: J. Shen and F. Zirilli

Copyright q 2012 L. M. Houston et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

Given only the signs of signal plus noise added repetitively or sign data, signal amplitudes canbe recovered with minimal variance. However, discrete derivatives of the signal are recoveredfrom sign data with a variance which approaches infinity with decreasing step size and increasingorder. For industries such as the seismic industry, which exploits amplitude recovery from signdata, these results place constraints on processing, which includes differentiation of the data.Whilemethods for smoothing noisy data for finite difference calculations are known, sign data requiresnoisy data. In this paper, we derive the expectation values of continuous and discrete sign dataderivatives and we explicitly characterize the variance of discrete sign data derivatives.

1. Introduction

Sign-bit recording systems discard all information on the detailed motion of the geophoneand ask only whether its output is positive or negative, whether it is going up or comingdown. In a sign-bit system, therefore, the signal waveform is converted into a square wave.All amplitude information is lost [1].

It is well known that, for a range of signal-to-noise ratios between about 0.1 and 1, thefinal result of sign-bit recording, after stacking, correlating, and other processing, looks noless good, to the eye, than the result from full-fidelity recording. This is considered to be asintriguing as it is surprising [1]. Alternatively, what we present in this paper is evidence thatthe processing of sign-bit data (i.e., sign data) can be limited for certain cases relative to theprocessing of the full-bandwidth data.

Model signal appears as a one-dimensional function, f(v), and noise as a randomvariable, X. In industries like the seismic industry, measurements of signal, f(v) : R → R

2 ISRN Applied Mathematics

and noise, X : Ω → R, f(v) +X are recorded for multiple iterations of the noise. The averageof the measurement (i.e., the expectation E) recovers the signal

E(f(v) +X

)= f(v). (1.1)

If the noise is chosen to be uniform, where ρ(x) is the density function such that

ρ(x) =

⎧⎨

⎩

12a

, −a ≤ x ≤ a

0, else,(1.2)

then the variance, E(f(v) +X)2 − (E(f(v) +X))2, reduces to

Var(f(v) +X

)=

13a2. (1.3)

As reported by O’Brien et al. [2], it was empirically discovered that the average of the signsof signal plus noise recovers the signal if the signal-to-noise ratio is less than or equal to one.This can be shown mathematically [3] using the signum function [4], sgn(x) = +1, x > 0,sgn(x) = −1, x < 0, sgn(0) = 0,

E(sgn(f(v) +X

))=∫∞

−∞sgn(f(v) + x

)ρ(x)dx =

∫∞

−fρ(x)dx −

∫−f

−∞ρ(x)dx. (1.4)

Because ρ(x) is even and equals

∫f

−fρ(x)dx (1.5)

E(sgn(f(v) +X

))=

f(v)a

, f ∈ [−a, a]. (1.6)

The variance is E(sgn(f(v) +X))2 − (E(sgn(f(v) +X)))2, reducing to

Var(sgn(f(v) +X

))= 1 −

(f(v)a

)2

. (1.7)

Consequently, the error is minimal when the signal-to-noise ratio is near unity.The advantage of retaining only the signs of signal plus noise is the requirement of

approximately 1 bit to record the information as opposed to requiring 16 to 20 bits to recordfull amplitude data [2].

The goal of this paper is to examine the recovery of derivatives from sign data in uni-form noise. The issue is that recovery of signal from sign data can be extended to recovery ofderivatives of the signal through the use of finite differences and that recovery is constrainedby the size of the variance. In this paper, we first examine sign data derivatives for both

ISRN Applied Mathematics 3

the discrete and continuous case. We follow with a derivation of variance. We conclude ouranalysis with a computational test, which lists the true variance versus the variance estimatederived statistically for a test function for selected step sizes.

2. Sign Data Derivatives

Let the signal f(v) be an nth order differentiable function. Based on signal recovery from signdata, it can be shown that derivatives of the signal are also recoverable. Using the linearity ofthe expectation value,

E

(Δn

v

(Δv)nsgn(f(v) +X

))

=Δn

v

(Δv)nE(sgn(f(v) +X

)), (2.1)

where Δnv is the nth order finite difference operator with respect to the variable v [5]. In this

case, a nonunit step size, Δv, is used (e.g., [6]).In detail, we can write

Δnv

(Δv)nsgn(f(v) +X

)=

1(Δv)n

n∑

i=0(−1)i

(n

i

)

sgn(f(v + (n − i)Δv +Xi)

), (2.2)

where the notation ( ni ) represents the binomial coefficient n!/i!(n − i)! and where Xi = X0,

X1, . . . are independent representations of the random variable, X.Substituting from (1.6) into (2.1) yields

E

(Δn

v

(Δv)nsgn(f(v) +X

))

=1a

Δnvf(v)(Δv)n

. (2.3)

In the limit of infinitesimal step size, this becomes a continuous derivative

limΔv→ 0

E

(Δn

v

(Δv)nsgn(f(v) +X

))

=1a

dnf(v)dvn

(2.4)

or

E

(dn

dvnsgn(f(v) +X

))

=1a

dnf(v)dvn

. (2.5)

Equation (2.4) presents an alternative solution to direct integration. For example, using therule,

∫f(x)δ(n)(x)dx = − ∫(∂f/∂x)δ(n−1)(x)dx, [7], the integral

E

(d3

dv3sgn(f(v) +X

))

=∫∞

−∞

(

2d2δ

du2

(df

dv

)3

+ 6dδ

du

df

dv

d2f

dv2+ 2δ

d3f

dv3

)

ρ(x)dx (2.6)


loses all terms with derivatives of the delta functional, reducing to

= 2ρ(−f)d

3f

dv3

∣∣∣∣∣f=−x

. (2.7)

In general,

E

(dn

dvnsgn(f(v) +X

))

= 2ρ(−f)d

nf

dvn

∣∣∣∣f=−x

=1a

dnf

dvn . (2.8)

It follows that the noise is restricted such that a ≥ |f |.

3. The Variance of Sign Data Derivatives

Letting Sn ≡ (Δnv/(Δv)n) sgn(f(v)+X), compute the variance, E(S2

n)−(E(Sn))2. From (2.3), it

follows that (E(Sn))2 = (Δn

vf/a(Δv)n)2. E(S2n) can be found by inductively generalizing from

n = 2:

E(S22

)=

1Δv4

(b0 sgn

(f0 +X0

)+ b1 sgn

(f1 +X1

)+ b2 sgn

(f2 +X2

))2

=1

Δv4

(b20 + b21 + b22 + 2b0b1

(f0f1

a2

)+ 2b0b2

(f0f2

a2

)+ 2b1b2

(f1f2

a2

)),

(3.1)

where fi = f(v + (n − i)Δv), fk = f(v + (n − k)Δv), and bi = (−1)i( ni ).

These results generalize to

Var(Sn) =1

(Δv)2n

n∑

i=0

(n

i

)2

+2

(Δv)2n

n∑

i /= k

(−1)i+k(n

i

)(n

k

)(fifk

a2

)−(

Δnvf

a(Δv)n

)2

.

(3.2)

Since f is differentiable, |(Δnvf/(Δv)n)− (dnf/dvn)| < ε and, thus,Δn

vf/(Δv)n is finite. Basedon definition, Var(Sn) > 0.

Consequently, limΔv→ 0 Var(Sn) = +∞. Similarly, limn→∞ Var(Sn) = +∞, 0 < Δv < 1.The variance of a discrete sign derivative approaches infinity with decreasing step sizeand increasing order. In addition, since limΔv→ 0(Sn) = (dn/dvn) sgn(f(v) + X),Var((dn/dvn) sgn(f(v) + X)) = +∞, so in the case of the continuous derivatives (2.5) thevariance is infinite.

Use (3.2) to find the variance of the first discrete sign derivative by letting n = 1:

Var(S1) =1

(Δv)2

(

2 −(f21 + f2

0

)

a2

)

. (3.3)


Table 1: True variance, Var(S1), versus the variance estimate, VarN(S1), for the function f = sin(v), withthe number of iterationsN = 1000, a = 1, and v = 3.

Δv Var(S1) VarN(S1)

1.0 1.4073 1.45130.5 7.4281 7.35620.2 49.4169 49.32020.1 197.8356 199.51950.04 1231.1 1283.20.02 4913.4 4739.80.01 19629 20401

Table 2: True variance, Var(S2), versus the variance estimate, VarN(S2), for the function f = sin(v), with thenumber of iterationsN = 1000, a = 1, and v = 3.

Δv Var(S2) VarN(S2)

1.0 2.7695 2.71920.5 78.6422 80.22760.2 3688.2 3752.30.1 59698 590000.04 2.3 × 106 2.3 × 106

0.02 3.69 × 107 3.685 × 107

0.01 5.89 × 108 5.988 × 108

The variance of the second discrete sign derivative (n = 2) is similarly computed as

Var(S2) =1

(Δv)4

(6 − 1

a2

(f20 + 4f2

1 + f22

)). (3.4)

4. Computational Tests

These results can be tested computationally. Variance can be estimated forN iterations with

VarN(Sn) =1N

N∑

m=1

(Sn(m) − E(Sn)), (4.1)

where the index m designates the sample number.Consider the test function f = sin(v). Using the first-order sign data derivative (n = 1),

compare Var(S1) to VarN(S1), and using the second-order sign data derivative (n = 2),compare Var(S2) to VarN(S2) for N = 1000, a = 1, and v = 3. The results are shown inTables 1 and 2.

We illustrate the change in variance in Figure 1, which shows three curves, eachconsisting of N = 1000 iterations. The first curve in blue shows the sign data recovery ofthe function f = sin(v) or E(S0) for a = 1 and Δv = 0.5. The second curve in green shows thesign data recovery E(S1), which approximates f ′ for a = 1 and Δv = 0.5. The third curve inred shows the sign data recovery E(S2), which approximates f ′′ for a = 1 and Δv = 0.5.


−1.5

−1

−0.5

0

0.5

1

1.5

0 1 2 3 4 5 6

E(S0)

E(S1)E(S2)

Figure 1: The expectation value curves for Sn = (1/(Δv)n)∑n

i=0 (−1)i(ni ) sgn(f(v+(n− i)Δv+Xi)) or E(Sn)for n = 0, 1, 2, f(v) = sin(v), a = 1, and Δv = 0.5. The number of iterations in the expectation values isN = 1000. E(S0) corresponds to the blue curve and approximates f , E(S1) corresponds to the green curveand approximates f ′, and E(S2) corresponds to the red curve and approximates f ′′.

5. Conclusions

Recovery of signal from the signs of signal plus noise incurs a variance, which only dependson the noise amplitude, while recovery of discrete derivatives from the signs of signal plusnoise (i.e., sign data) incurs a variance which grows infinite for infinitesimal step size andinfinite order.

The application problem is that sign data can be used in the seismic industry in proc-esses which may differentiate the data. In such cases, if the step size or order of the finitedifference is not constrained, the process will incur large variance and convergence of theprocess will be minimized. While methods for smoothing noisy data for finite differencecalculations are known, sign data requires noisy data. In this paper, we have characterizedthe problem by explicitly evaluating the variance of discrete sign data derivatives.

Appendix

Clarification of E(S22)

E(S22

)=

1Δv4

(b0 sgn

(f0 +X0

)+ b1 sgn

(f1 +X1

)+ b2 sgn

(f2 +X2

))2

=1

Δv4

(b20sgn

2(f0 +X0)+ 2b0 sgn

(f0 +X0

)b1 sgn

(f1 +X1

)+ b21sgn

2(f1 +X1)

+ 2b2 sgn(f2 +X2

)b0 sgn

(f0 +X0

)+ 2b2 sgn

(f2 +X2

)b1 sgn

(f1 +X1

)

+b22sgn2(f2 +X2

)).

(A.1)


This simply reduces to

E(S22

)=

1Δv4

(b20 + 2b0b1E

(sgn(f0 +X0

)sgn(f1 +X1

))+ b21

+ 2b2b0E(sgn(f2 +X2

)sgn(f0 +X0

))

+2b2b1E(sgn(f2 +X2

)sgn(f1 +X1

))+ b22

).

(A.2)

In order to compute (A.2), we must compute an integral of the form

E(sgn(fi +X

)sgn(fk +X

))=∫∫∞

−∞sgn(fi + xi

)sgn(fk + xk

)ρ(xi)ρ(xk)dxi dxk. (A.3)

The probability densities are both uniform:

ρ(xi) = ρ(xk) =

⎧⎪⎨

⎪⎩

12a

, −a ≤ x ≤ a,

0, else(A.4)

and using the results of (1.6),

E(sgn(fi +X

)sgn(fk +X

))=

fifk

a2. (A.5)

Consequently, (A.2) reduces to

E(S22

)=

1Δv4

(b20 + b21 + b22 + 2b0b1

(f0f1

a2

)+ 2b0b2

(f0f2

a2

)+ 2b1b2

(f1f2

a2

)). (A.6)

Acknowledgment

Thanks are due to Gwendolyn Houston for advice and proofreading.

References

[1] N. A. Anstey, Seismic Prospecting Instruments, Gebruder Borntraeger, Berlin, Germany, 2nd edition,1981.

[2] J. T. O’Brien, W. P. Kamp, and G. M. Hoover, “Sign-bit amplitude recovery with applications to seismicdata,” Geophysics, vol. 47, no. 11, pp. 1527–1539, 1982.

[3] L. M. Houston and B. A. Richard, “The Helmholtz-Kirchoff 2.5D integral theorem for sign-bit data,”Journal of Geophysics and Engineering, vol. 1, no. 1, pp. 84–87, 2004.

[4] R. A. Gabel and R. A. Roberts, Signals and Linear Systems, Wiley, New York, NY, USA, 3rd edition, 1987.[5] W. G. Kelley and A. C. Peterson, Difference Equations, Academic Press, Boston, Mass, USA, 1991.[6] D. M. Dubois, “Computing anticipatory systems with incursion and hyperincursion, computing

anticipatory systems,” in Proceedings of the 1st International Conference on Computing Anticipatory Systems(CASYS ’98), vol. 437 of AIP Conference Proceedings, pp. 3–29, The American Institute of Physics, 1998.

[7] G. Arfken,Mathematical Methods for Physicists, Academic Press, New York, NY, USA, 1966.

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