Sign Data Derivative Recovery

Abstract

Given only the signs of signal plus noise added repetitively or sign data, signal amplitudes can be recovered with minimal variance. However, discrete derivatives of the signal are recovered from sign data with a variance which approaches infinity with decreasing step size and increasing order. For industries such as the seismic industry, which exploits amplitude recovery from sign data, these results place constraints on processing, which includes differentiation of the data. While methods for smoothing noisy data for finite difference calculations are known, sign data requires noisy data. In this paper, we derive the expectation values of continuous and discrete sign data derivatives 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 geophone and ask only whether its output is positive or negative, whether it is going up or coming down. 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, the final result of sign-bit recording, after stacking, correlating, and other processing, looks no less good, to the eye, than the result from full-fidelity recording. This is considered to be as intriguing as it is surprising [1]. Alternatively, what we present in this paper is evidence that the processing of sign-bit data (i.e., sign data) can be limited for certain cases relative to the processing of the full-bandwidth data.

Model signal appears as a one-dimensional function, 𝑓(𝑣), and noise as a random variable, 𝑋. In industries like the seismic industry, measurements of signal, 𝑓(𝑣)∶ℝ→ℝ and noise, 𝑋∶Ω→ℝ, 𝑓(𝑣)+𝑋 are recorded for multiple iterations of the noise. The average of the measurement (i.e., the expectation 𝐸) recovers the signal𝐸(𝑓(𝑣)+𝑋)=𝑓(𝑣).(1.1) If the noise is chosen to be uniform, where 𝜌(𝑥) is the density function such that𝜌(𝑥)=021,𝑎,−el𝑎se≤,𝑥≤𝑎(1.2) then the variance, 𝐸(𝑓(𝑣)+𝑋)²−(𝐸(𝑓(𝑣)+𝑋))², reduces toVar(𝑓(𝑣)+𝑋)=13𝑎².(1.3) As reported by O’Brien et al. [2], it was empirically discovered that the average of the signs of 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−𝑓𝐸(sgn(𝑓(𝑣)+𝑋))𝜌=(𝑥𝑓)𝑑(𝑣𝑥)(1.5)𝑎,𝑓∈[−𝑎,𝑎].(1.6) The variance is 𝐸(sgn(𝑓(𝑣)+𝑋))²−(𝐸(sgn(𝑓(𝑣)+𝑋)))², reducing toVar(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 record full amplitude data [2].

The goal of this paper is to examine the recovery of derivatives from sign data in uniform noise. The issue is that recovery of signal from sign data can be extended to recovery of derivatives of the signal through the use of finite differences and that recovery is constrained by the size of the variance. In this paper, we first examine sign data derivatives for both the discrete and continuous case. We follow with a derivation of variance. We conclude our analysis with a computational test, which lists the true variance versus the variance estimate derived 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 sign data, it can be shown that derivatives of the signal are also recoverable. Using the linearity of the expectation value,𝐸Δ^𝑛𝑣(Δ𝑣)^𝑛sgn(𝑓(𝑣)+𝑋)=Δ^𝑛𝑣(Δ𝑣)^𝑛𝐸(sgn(𝑓(𝑣)+𝑋)),(2.1) where Δ^𝑛𝑣 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 writeΔ^𝑛𝑣(Δ𝑣)^𝑛sgn(𝑓(𝑣)+𝑋)=1(Δ𝑣)^𝑛𝑛_𝑖=0(−1)^𝑖⎛⎜⎜⎝𝑛𝑖⎞⎟⎟⎠sgn𝑓𝑣+(𝑛−𝑖)Δ𝑣+𝑋_𝑖,(2.2) where the notation (^𝑛𝑖) represents the binomial coefficient 𝑛!/𝑖!(𝑛−𝑖)! and where 𝑋_𝑖=𝑋₀,𝑋₁,… are independent representations of the random variable, 𝑋.

Substituting from (1.6) into (2.1) yields𝐸Δ^𝑛𝑣(Δ𝑣)^𝑛sgn(𝑓(𝑣)+𝑋)=1𝑎Δ^𝑛𝑣𝑓(𝑣)(Δ𝑣)^𝑛.(2.3) In the limit of infinitesimal step size, this becomes a continuous derivativelim_Δ𝑣→0𝐸Δ^𝑛𝑣(Δ𝑣)^𝑛sgn(𝑓(𝑣)+𝑋)=1𝑎𝑑^𝑛𝑓(𝑣)𝑑𝑣^𝑛(2.4) or𝐸𝑑^𝑛𝑑𝑣^𝑛sgn(𝑓(𝑣)+𝑋)=1𝑎𝑑^𝑛𝑓(𝑣)𝑑𝑣^𝑛.(2.5) Equation (2.4) presents an alternative solution to direct integration. For example, using the rule, ∫𝑓(𝑥)𝛿^(𝑛)(𝑥)𝑑𝑥=−∫(𝜕𝑓/𝜕𝑥)𝛿^(𝑛−1)(𝑥)𝑑𝑥, [7], the integral𝐸𝑑³𝑑𝑣³sgn(𝑓(𝑣)+𝑋)=^∞−∞2𝑑²𝛿𝑑𝑢²𝑑𝑑𝑓𝑣³+6𝑑𝛿𝑑𝑢𝑑𝑑𝑓𝑣𝑑²𝑓𝑑𝑣²+2𝛿𝑑³𝑓𝑑𝑣³𝜌(𝑥)𝑑𝑥(2.6) loses all terms with derivatives of the delta functional, reducing to=2𝜌(−𝑓)𝑑³𝑓𝑑𝑣³||||_𝑓=−𝑥.(2.7) In general,𝐸𝑑^𝑛𝑑𝑣^𝑛sgn(𝑓(𝑣)+𝑋)=2𝜌(−𝑓)𝑑^𝑛𝑓𝑑𝑣^𝑛||||_𝑓=−𝑥=1𝑎𝑑^𝑛𝑓𝑑𝑣^𝑛.(2.8) It follows that the noise is restricted such that 𝑎≥|𝑓|.

3. The Variance of Sign Data Derivatives

Letting 𝑆_𝑛≡(Δ^𝑛𝑣/(Δ𝑣)^𝑛)sgn(𝑓(𝑣)+𝑋), compute the variance, 𝐸(𝑆^2𝑛)−(𝐸(𝑆_𝑛))². From (2.3), it follows that (𝐸(𝑆_𝑛))²=(Δ^𝑛𝑣𝑓/𝑎(Δ𝑣)^𝑛)². 𝐸(𝑆^2𝑛) can be found by inductively generalizing from 𝑛=2:𝐸𝑆²²=1Δ𝑣⁴𝑏₀sgn𝑓₀+𝑋₀+𝑏₁sgn𝑓₁+𝑋₁+𝑏₂sgn𝑓₂+𝑋₂²=1Δ𝑣⁴𝑏²⁰+𝑏²¹+𝑏²²+2𝑏₀𝑏₁𝑓⁰𝑓₁𝑎²+2𝑏⁰𝑏₂𝑓⁰𝑓₂𝑎²+2𝑏¹𝑏₂𝑓¹𝑓₂𝑎²,(3.1) where 𝑓_𝑖=𝑓(𝑣+(𝑛−𝑖)Δ𝑣), 𝑓_𝑘=𝑓(𝑣+(𝑛−𝑘)Δ𝑣), and 𝑏_𝑖=(−1)^𝑖(^𝑛𝑖).

These results generalize toVar𝑆_𝑛=1(Δ𝑣)^𝑛2𝑛_𝑖=0⎛⎜⎜⎝𝑛𝑖⎞⎟⎟⎠²+2(Δ𝑣)^𝑛2𝑛_𝑖≠𝑘(−1)^𝑖+𝑘⎛⎜⎜⎝𝑛𝑖⎞⎟⎟⎠⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠𝑓_𝑖𝑓_𝑘𝑎²−Δ^𝑛𝑣𝑓𝑎(Δ𝑣)^𝑛².(3.2) Since 𝑓 is differentiable, |(Δ^𝑛𝑣𝑓/(Δ𝑣)^𝑛)−(𝑑^𝑛𝑓/𝑑𝑣^𝑛)|<𝜀 and, thus, Δ^𝑛𝑣𝑓/(Δ𝑣)^𝑛 is finite. Based on definition, Var(𝑆_𝑛)>0.

Consequently, lim_Δ𝑣→0Var(𝑆_𝑛)=+∞. Similarly, lim_𝑛→∞Var(𝑆_𝑛)=+∞,0<Δ𝑣<1. The variance of a discrete sign derivative approaches infinity with decreasing step size and increasing order. In addition, since lim_Δ𝑣→0(𝑆_𝑛)=(𝑑^𝑛/𝑑𝑣^𝑛)sgn(𝑓(𝑣)+𝑋), Var((𝑑^𝑛/𝑑𝑣^𝑛)sgn(𝑓(𝑣)+𝑋))=+∞, so in the case of the continuous derivatives (2.5) the variance 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+𝑓²⁰𝑎².(3.3) The variance of the second discrete sign derivative (𝑛=2) is similarly computed asVar𝑆₂=1(Δ𝑣)⁴6−𝑎1₂𝑓²⁰+4𝑓²¹+𝑓²².(3.4)

4. Computational Tests

These results can be tested computationally. Variance can be estimated for 𝑁 iterations withVar_𝑁𝑆_𝑛=1𝑁^𝑁_𝑚=1𝑆_𝑛(𝑚)−𝐸𝑆_𝑛,(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 Var_𝑁(𝑆₁), and using the second-order sign data derivative (𝑛=2), compare Var(𝑆₂) to Var_𝑁(𝑆₂) for 𝑁=1000, 𝑎=1, and 𝑣=3. The results are shown in Tables 1 and 2.

Table 1 

True variance, Var(𝑆₁), versus the variance estimate, Var_𝑁(𝑆₁), for the function 𝑓=sin(𝑣), with the number of iterations 𝑁=1000, 𝑎=1, and 𝑣=3.

Table 2 

True variance, Var(𝑆₂), versus the variance estimate, Var_𝑁(𝑆₂), for the function 𝑓=sin(𝑣), with the number of iterations 𝑁=1000, 𝑎=1, and 𝑣=3.

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

Figure 1 

The expectation value curves for 𝑆𝑛=(1/(Δ𝑣)𝑛)∑𝑛𝑖=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 is 𝑁=1000. 𝐸(𝑆0) corresponds to the blue curve and approximates 𝑓, 𝐸(𝑆1) corresponds to the green curve and 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 depends on the noise amplitude, while recovery of discrete derivatives from the signs of signal plus noise (i.e., sign data) incurs a variance which grows infinite for infinitesimal step size and infinite order.

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

Appendix

Clarification of 𝐸(𝑆²²)

𝐸𝑆²²=1Δ𝑣⁴𝑏₀sgn𝑓₀+𝑋₀+𝑏₁sgn𝑓₁+𝑋₁+𝑏₂sgn𝑓₂+𝑋₂²=1Δ𝑣⁴+𝑏²⁰2s𝑏g₂n²𝑓₀+𝑋₀+2𝑏₀sgn𝑓₀+𝑋₀𝑏₁sgn𝑓₁+𝑋₁+𝑏²¹sgn²𝑓₁+𝑋₁+𝑏₂₂sgn𝑓₂+𝑋₂𝑏₀sgn𝑓₀+𝑋₀+2𝑏₂sgn𝑓₂+𝑋₂𝑏₁sgn𝑓₁+𝑋₁sgn²𝑓₂+𝑋₂.(A.1) This simply reduces to𝐸𝑆²²=1Δ𝑣⁴𝑏²⁰+2𝑏₀𝑏₁𝐸sgn𝑓₀+𝑋₀sgn𝑓₁+𝑋₁+𝑏²¹++22𝑏𝑏₂₂𝑏₀𝐸sgn𝑓₂+𝑋₂sgn𝑓₀+𝑋₀𝑏₁𝐸sgn𝑓₂+𝑋₂sgn𝑓₁+𝑋₁+𝑏²².(A.2) In order to compute (A.2), we must compute an integral of the form𝐸sgn𝑓_𝑖+𝑋sgn𝑓_𝑘+𝑋=^∞−∞sgn𝑓_𝑖+𝑥_𝑖sgn𝑓_𝑘+𝑥_𝑘𝜌𝑥_𝑖𝜌𝑥_𝑘𝑑𝑥_𝑖𝑑𝑥_𝑘.(A.3) The probability densities are both uniform:𝜌𝑥_𝑖=𝜌𝑥_𝑘=021,𝑎,e−l𝑎se≤𝑥≤𝑎,(A.4) and using the results of (1.6),𝐸sgn𝑓_𝑖+𝑋sgn𝑓_𝑘+𝑋=𝑓^𝑖𝑓_𝑘𝑎².(A.5) Consequently, (A.2) reduces to𝐸𝑆²²=1Δ𝑣⁴𝑏²⁰+𝑏²¹+𝑏²²+2𝑏₀𝑏₁𝑓⁰𝑓₁𝑎²+2𝑏⁰𝑏₂𝑓⁰𝑓₂𝑎²+2𝑏¹𝑏₂𝑓¹𝑓₂𝑎².(A.6)

Acknowledgment

Thanks are due to Gwendolyn Houston for advice and proofreading.