View source for Crosscorrelation and autocorrelation

{{SDA publication 
 | title   = Seismic Data Analysis
 | image   = [[File:Seismic-data-analysis.jpg|frameless|180px]]
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 | series  = Investigations in Geophysics
 | part    = 
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 | author  = [[Öz Yilmaz]]
 | doi     = 10.1190/1.9781560801580
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 | store   = http://shop.seg.org/OnlineStore/ProductDetail/tabid/177/Default.aspx?ProductId=847
 | isbn    = ISBN 978-1-56080-094-1
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[[processing of 3-D seismic data|Seismic data processing]] often requires measurement of the similarity or time alignment of two traces. ''Correlation'' is another time-domain operation that is used to make such measurements. Consider the following two wavelets:

:Wavelet 1: (2, 1, −1, 0, 0)
:Wavelet 2: (0, 0, 2, 1, −1)

Although these [[wavelet processing by shaping filters|wavelets]] are identical in shape, wavelet 2 is shifted by two samples with respect to wavelet 1. The time lag at which they are most similar can be determined. To do this, perform the operation on wavelet 1 as described in Table 1-5 without reversing wavelet 2 (omit Step 1). This is ''crosscorrelation'' and the result is shown in Table 1-7. Crosscorrelation measures how much two time series resemble each other.

{|class="wikitable" style="text-align:center; width:300px; height:200px; "border="1"
|+ '''Table 1-5.''' Mechanics of the convolutional process.
|-
|colspan="13"|Fixed Array:
|-
|colspan="13"|''a''<sub>0</sub>, ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>, ''a''<sub>4</sub>, ''a''<sub>5</sub>, ''a''<sub>6</sub>, ''a''<sub>7</sub>
|-
|colspan="13"|Moving Array:
|-
|colspan="13"|''b''<sub>0</sub>, ''b''<sub>1</sub>, ''b''<sub>2</sub>
|-
|colspan="13"|Given two arrays, ''a<sub>i</sub>'' and ''b<sub>j</sub>'':
|-
|colspan="13"|Step 1 : Reverse moving array ''b<sub>j</sub>''.
|-
|colspan="13"|Step 2 : Multiply in the vertical direction.
|-
|colspan="13"|Step 3 : Add the products and write as output ''c<sub>k</sub>''.
|-
|colspan="13"|Step 4 : Shift array ''b<sub>j</sub>'' one sample to the right and repeat Steps 2 and 3.
|-
|colspan="13"|Convolution Table:
|-
|  ||  ||''a''<sub>0</sub> ||''a''<sub>1</sub> ||''a''<sub>2</sub> ||''a''<sub>3</sub> ||''a''<sub>4</sub> ||''a''<sub>5</sub> ||''a''<sub>6</sub> ||''a''<sub>7</sub> ||  ||  ||Output
|-
|''b''<sub>2</sub> ||''b''<sub>1</sub> ||''b''<sub>0</sub> ||  ||  ||  ||  ||  ||  ||  ||  ||  ||''c''<sub>0</sub>
|-
|  ||''b''<sub>2</sub> ||''b''<sub>1</sub> ||''b''<sub>0</sub> ||  ||  ||  ||  ||  ||  ||  ||  ||''c''<sub>1</sub>
|-
|  ||  ||''b''<sub>2</sub> ||''b''<sub>1</sub> ||''b''<sub>0</sub> ||  ||  ||  ||  ||  ||  ||  ||''c''<sub>2</sub>
|-
|  ||  ||  ||''b''<sub>2</sub> ||''b''<sub>1</sub> ||''b''<sub>0</sub> ||  ||  ||  ||  ||  ||  ||''c''<sub>3</sub>
|-
|  ||  ||  ||  ||''b''<sub>2</sub> ||''b''<sub>1</sub> ||''b''<sub>0</sub> ||  ||  ||  ||  ||  ||''c''<sub>4</sub>
|-
|  ||  ||  ||  ||  ||''b''<sub>2</sub> ||''b''<sub>1</sub> ||''b''<sub>0</sub> ||  ||  ||  ||  ||''c''<sub>5</sub>
|-
|  ||  ||  ||  ||  ||  ||''b''<sub>2</sub> ||''b''<sub>1</sub> ||''b''<sub>0</sub> ||  ||  ||  ||''c''<sub>6</sub>
|-
|  ||  ||  ||  ||  ||  ||  ||''b''<sub>2</sub> ||''b''<sub>1</sub> ||''b''<sub>0</sub> ||  ||  ||''c''<sub>7</sub>
|-
|  ||  ||  ||  ||  ||  ||  ||  ||''b''<sub>2</sub> ||''b''<sub>1</sub> ||''b''<sub>0</sub> ||  ||''c''<sub>8</sub>
|-
|  ||  ||  ||  ||  ||  ||  ||  ||  ||''b''<sub>2</sub> ||''b''<sub>1</sub> ||''b''<sub>0</sub> ||''c''<sub>9</sub>
|-
|colspan="13"|where
|-
|colspan="13"|<math>{c_k} = \sum\limits_{j = 0}^n {{a_{k - j}}\;{b_j},} \quad k = 0,\;1,\;2,\; \cdots \;,m\; + \;n\; - \;1.</math>
|}

{|class="wikitable" style="text-align:center; width:300px; height:200px; "border="1"
|+ '''Table 1-7.''' Crosscorrelation of wavelet 1 with wavelet 2.
|-
|  ||  ||  ||  ||2 ||1 || -1 ||0 ||0 ||  ||  ||  ||  ||Output ||Lag
|-
|0 ||0 ||2 ||1 || -1 ||  ||  ||  ||  ||  ||  ||  ||  || -2 || -4
|-
|  ||0 ||0 ||2 ||1 || -1 ||  ||  ||  ||  ||  ||  ||  ||1 || -3
|-
|  ||  ||0 ||0 ||2 ||1 || -1 ||  ||  ||  ||  ||  ||  ||6 || -2
|-
|  ||  ||  ||0 ||0 ||2 ||1 || -1 ||  ||  ||  ||  ||  ||1 || -1
|-
|  ||  ||  ||  ||0 ||0 ||2 ||1 || -1 ||  ||  ||  ||  || -2 ||0
|-
|  ||  ||  ||  ||  ||0 ||0 ||2 ||1 || -1 ||  ||  ||  ||0 ||1
|-
|  ||  ||  ||  ||  ||  ||0 ||0 ||2 ||1 || -1 ||  ||  ||0 ||2
|-
|  ||  ||  ||  ||  ||  ||  ||0 ||0 ||2 ||1 || -1 ||  ||0 ||3
|-
|  ||  ||  ||  ||  ||  ||  ||  ||0 ||0 ||2 ||1 || -1 ||0 ||4
|}

From Table 1-7, note that maximum correlation occurs at lag −2. This suggests that if wavelet 2 were shifted two samples back in time, then these two wavelets would have maximum similarity.

Table 1-8 shows the crosscorrelation values that result when the arrays are interchanged. This time the maximum correlation occurs at lag 2. Thus, if wavelet 1 were shifted by two samples forward in time, these two wavelets would have maximum similarity.

{|class="wikitable" style="text-align:center; width:300px; height:200px; "border="1"
|+ '''Table 1-8.''' Crosscorrelation of wavelet 2 with wavelet 1.
|-
|  ||  ||  ||  ||0 ||0 ||2 ||1 || -1 ||  ||  ||  ||  ||Output ||Lag
|-
|2 ||1 || -1 ||0 ||0 ||  ||  ||  ||  ||  ||  ||  ||  || -0 || -4
|-
|  ||2 ||1 || -1 ||0 ||0 ||  ||  ||  ||  ||  ||  ||  ||0 || -3
|-
|  ||  ||2 ||1 || -1 ||0 ||0 ||  ||  ||  ||  ||  ||  ||0 || -2
|-
|  ||  ||  ||2 ||1 || -1 ||0 ||0 ||  ||  ||  ||  ||  ||0 || -1
|-
|  ||  ||  ||  ||2 ||1 || -1 ||0 ||0 ||  ||  ||  ||  || -2 ||0
|-
|  ||  ||  ||  ||  ||2 ||1 || -1 ||0 ||0 ||  ||  ||  ||1 ||1
|-
|  ||  ||  ||  ||  ||  ||2 ||1 || -1 ||0 ||0 ||  ||  ||6 ||2
|-
|  ||  ||  ||  ||  ||  ||  ||2 ||1 || -1 ||0 ||0 ||  ||1 ||3
|-
|  ||  ||  ||  ||  ||  ||  ||  ||2 ||1 || -1 ||0 ||0 || -2 ||4
|}

{|class="wikitable" style="text-align:center; width:300px; height:200px; "border="1"
|+ '''Table 1-9.''' Crosscorrelation of wavelet 1 with wavelet 2 and crosscorrelation of wavelet 2 with wavelet 1.
|-
|Wavelet 1 with Wavelet 2 ||Crosscorrelation Output Lag ||Wavelet 2 with Wavelet 1
|-
| -2 || -4 ||0
|-
|1 || -3 ||0
|-
|6 || -2 ||0
|-
|1 || -1 ||0
|-
| -2 ||0 || -2
|-
|0 ||1 ||1
|-
|0 ||2 ||6
|-
|0 ||3 ||1
|-
|0 ||4 || -2
|}

Unlike [[convolution]], ''crosscorrelation is not commutative — the output depends on which array is fixed and which is moved''. Table 1-9 shows a comparison of the crosscorrelation results listed in Tables 1-7 and 1-8.

Crosscorrelation of a time series with itself is known as ''autocorrelation''. Table 1-10 shows the autocorrelation lags of wavelet 1. Note that maximum correlation occurs at zero lag, an important property of autocorrelation. Moreover, ''the autocorrelation function is symmetric''. This is a property of a real time series. Therefore, only one side of the autocorrelation needs to be computed.

It is heuristically shown in the next section on [[random noise and frequency-wavenumber filtering|frequency filtering]] that convolution in the time domain is equivalent to multiplication in the frequency domain <ref name=ch01r1>Bracewell, R. N., 1965, The Fourier transform and its applications: McGraw-Hill Book Co.</ref>. Since correlation is a convolution without reversing the moving array (Table 1-5), a similar frequency-domain operation also applies to correlation.

{|class="wikitable" style="text-align:center; width:300px; height:200px; "border="1"
|+ '''Table 1-10.''' Autocorrelation of wavelet 1.
|-
|  ||  ||  ||  ||2 ||1 || -1 ||0 ||0 ||  ||  ||  ||  ||Output ||Lag
|-
|2 ||1 || -1 ||0 ||0 ||  ||  ||  ||  ||  ||  ||  ||  || -0 || -4
|-
|  ||2 ||1 || -1 ||0 ||0 ||  ||  ||  ||  ||  ||  ||  ||0 || -3
|-
|  ||  ||2 ||1 || -1 ||0 ||0 ||  ||  ||  ||  ||  ||  || -2 || -2
|-
|  ||  ||  ||2 ||1 || -1 ||0 ||0 ||  ||  ||  ||  ||  ||1 || -1
|-
|  ||  ||  ||  ||2 ||1 || -1 ||0 ||0 ||  ||  ||  ||  ||6 ||0
|-
|  ||  ||  ||  ||  ||2 ||1 || -1 ||0 ||0 ||  ||  ||  ||1 ||1
|-
|  ||  ||  ||  ||  ||  ||2 ||1 || -1 ||0 ||0 ||  ||  || -2 ||2
|-
|  ||  ||  ||  ||  ||  ||  ||2 ||1 || -1 ||0 ||0 ||  ||0 ||3
|-
|  ||  ||  ||  ||  ||  ||  ||  ||2 ||1 || -1 ||0 ||0 ||0 ||4
|}

[[file:ch01_fig1-19.png|thumb|{{figure number|1.1-19}} The frequency-domain description of convolution and correlation.]]

Figure 1.1-19 is a summary of frequency-domain descriptions of convolution and correlation. Note the following properties for convolution and correlation:

# Both convolution and correlation produce an output with a spectral bandwidth that is common to both of the input series. The immediate example is the band-pass filtering process.
# Phases are additive in case of convolution and subtractive in case of correlation <ref name=ch01r1/>.
# For autocorrelation, this implies that the output series is zero-phase. This fact already was verified by the example in Table 1-10 where it was shown that the autocorrelation is symmetric with respect to zero lag.

As a measure of similarity, crosscorrelation is used widely at various stages of data processing. For instance, traces in a CMP gather are crosscorrelated with a pilot trace to compute [[Residual statics corrections|residual statics shifts. The fundamental basis for computing velocity spectra is crosscorrelation ([[velocity analysis]]). The constituent elements of the [[Optimum wiener filters|Wiener filter]] are crosscorrelation of the desired output waveform with the input wavelet, and autocorrelation of the input wavelet.

==References==
<references/>

==See also==
*[[Analog versus digital signal]]
*[[Frequency aliasing]]
*[[Phase considerations]]
*[[Time-domain operations]]
*[[Convolution]]
*[[Vibroseis correlation]]
*[[Frequency filtering]]
*[[Practical aspects of frequency filtering]]
*[[Bandwidth and vertical resolution]]
*[[Time-variant filtering]]

==External links==
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[[Category:Fundamentals of Signal Processing]]
[[Category:The 1-D Fourier transform]]