Improvement of signal/noise ratio by stacking

Problems in Exploration Seismology and their Solutions

Series	Geophysical References Series
Title	Problems in Exploration Seismology and their Solutions
Author	Lloyd P. Geldart and Robert E. Sheriff
Chapter	6
Pages	181 - 220
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	SEG Online Store

Problem 6.22

Select random numbers between $\pm 9$ to represent noise $N_{j}$ and add to each a signal $S=2$. Sum 4 values of $S+N_{j}$ and determine the mean, standard deviation $\sigma$ of $(S+N_{j})$, and values of the ratio $S/(S+N)$. Repeat for 8, 16, and 32 values. Note how the mean converges toward $S$ as the number of values increases, how $\sigma$ approaches a limiting value (which depends on the statistical properties of the noise), and how the ratio $S/(S+N)$ converges toward $+1$.

Background

The standard deviation $\sigma$ is a measure of the scatter of measured values of a quantity, large values corresponding to large variations. It is given by the equation

${\begin{aligned}\sigma =[(1/n)\Sigma (x-{\bar {x}})^{2}]^{1/2},\end{aligned}}$

(6.22a)

where $n$ is the number of values and ${\bar {x}}$ is the mean value.

Table 6.22a. Random numbers.

20897	13007	95217	19221	15433	94882	23741	86571	20504	22169
20737	19305	71148	04035	03180	79506	12771	34806	37279	62739
31552	59282	16856	38655	31802	84283	08694	06945	19286	16924
60605	97685	26147	51379	39553	04893	25469	96469	57436	97888
42094	17446	27775	99466	63704	60957	55029	02764	91845	76174
54774	15832	04324	73597	42328	74303	58231	85798	16725	27836
89730	31886	34683	07814	57000	63721	43798	12003	04676	08367
12049	18538	96266	62439	81839	13093	22659	75018	31494	89519
22364	15913	51674	94189	10336	97801	21025	58966	40663	26197
80102	39977	78674	29634	38652	85289	47962	16594	50834	93484

To obtain a sequence of random numbers within a given range, adopt a rule for selecting digits from Table 6.22a; use this rule to get the units digit, then get the tens digit, etc. To determine the algebraic sign of each number, adopt another rule for fixing the sign, e.g., letting even/odd digits signify plus/minus, respectively.

Solution

Using Table 6.22a, we get 32 values of $N_{i}$ and sums $S+N$ in Table 6.22b.

Writing $({\overline {S+N}})$ for the mean of ($S+N_{i}$), we get the results in Table 6.22c.

Table 6.22b. Values of $N_{i}$ and $S+N_{i}$.

$N_{i}$	$S+N_{i}$	$N_{i}$	$S+N_{i}$	$N_{i}$	$S+N_{i}$	$N_{i}$	$S+N_{i}$
$-2$	0	$-2$	0	$-3$	$-1$	$+5$	$+7$
$-1$	$+1$	$+1$	$+3$	$-3$	$-1$	$+8$	$+10$
$+9$	$+11$	$+7$	$+9$	$-8$	$-6$	$-8$	$-6$
0	$+2$	$+2$	$+4$	0	$+2$	$-3$	$-1$
$-1$	$+1$	$-6$	$-4$	0	$+2$	$+5$	$+7$
$+9$	$+11$	$+3$	$+5$	$-4$	$-2$	$-4$	$-2$
$+2$	$+4$	$-5$	$-3$	$+4$	$+6$	$-6$	$-4$
$-8$	$-6$	$-1$	$+1$	$+7$	$+9$	$+1$	$+3$

Table 6.22c. Calculated values of $({\overline {S+N}})$, $\sigma$, and $S({\overline {S+N}})$ for different numbers of samples.

	$\Sigma (S+N)$	$({\overline {S+N}})$	$\sigma$	$S/({\overline {S+N}})$
1st 4	14	3.5	4.4	0.57
1st 8	24	3.0	5.3	0.67
2nd 8	15	1.9	4.0	1.05
3rd 8	9	1.1	4.4	1.82
4th 8	14	1.8	5.5	1.11
1st 16	39	2.4	4.8	0.83
2nd 16	23	1.4	5.0	1.43
32	62	1.9	4.9	1.05

We see that as the number of samples increases, $\sigma$ approaches 4.9, $({\overline {S+N}})$ approaches 2, and $S/({\overline {S+N}})$ approaches $+1$.

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Table of Contents (book)

Also in this chapter

• Characteristics of different types of events and noise
• Horizontal resolution
• Reflection and refraction laws and Fermat’s principle
• Effect of reflector curvature on a plane wave
• Diffraction traveltime curves
• Amplitude variation with offset for seafloor multiples
• Ghost amplitude and energy
• Directivity of a source plus its ghost
• Directivity of a harmonic source plus ghost
• Differential moveout between primary and multiple
• Suppressing multiples by NMO differences
• Distinguishing horizontal/vertical discontinuities
• Identification of events
• Traveltime curves for various events
• Reflections/diffractions from refractor terminations
• Refractions and refraction multiples
• Destructive and constructive interference for a wedge
• Dependence of resolvable limit on frequency
• Vertical resolution
• Causes of high-frequency losses
• Ricker wavelet relations
• Improvement of signal/noise ratio by stacking

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