Correlation analysis

The purpose of the correlation analysis is to determine an “optimal” separation between indi-vidual hydrophones.

Autocorrelations

The plots in Figure4.6were produced by averaging autocorrelations from noise records ac-quired at5 knotsduring sea state 0.5 (calm) and sea state 4 (1.5 mwaves), respectively. Similar images were also obtained at other velocities. Figure4.6(a)shows that in calm weather, raw unfiltered noise records are correlated out to around120 ms, corresponding to30 cmby the frozen field hypothesis (Taylor,1938). When a low-cut filter is applied, the correlation length is shortened. For example, with a6 Hzlow-cut filter, it is about5 cm.

During sea state 4, seen in Figure4.6(b), the correlation lengths are similar. However, for the unfiltered data the correlation length is now almost300 msor70 cm. This can probably be explained by the presence of low-frequency swell or bulge wave noise produced by cross-flow over the streamer (Elboth et al.,2009b).

The integral length scaleΛ, is defined as Λ =

r_c 0

Cxxdx. (4.3)

Herercis where the autocorrelation coefficientCxxhas its first zero crossing.

Velocity 6.5m separation 175m separation

Table 4.1: The relative amount of coherent background noise at varying velocities during our experiment. The data was acquired at sea-state 0.5.

Λcan be interpreted as collecting spatially correlated energy from0torc, to find a physical length representing the fully correlated energy within this range. This length can be seen as a weighted midpoint of energy. It is reasonable to use2Λseparation to ensure uncorrelated en-ergy between neighboring hydrophones, ensuring that two adjacent measurements are (nearly) uncorrelated.

By doing the integration to the data displayed in Figure4.6we get2Λ ≈ 26 cmand64 cm, respectively. When a2 Hzlow-cut filter to remove influence of swell noise is applied to the data in Figure4.6(b),2Λis reduced to40 cm. Physically this autocorrelationΛrelates to the size of the largest noise producing features (eddies) in the TBL.

Ocean ambient noise

The relative amount of coherent background noise in a gather can also be estimated by the peak crosscorrelating coefficient of noise records from widely separated hydrophones. This is done in Table4.1for pairs of hydrophones separated by175 mand6.5 m. For the175 mseparation the results reflect the amount of coherent ambient noise from distant shipping, onshore industrial activity or possibly the towing vessel itself. The6.5 mseparation was computed by picking out the zero-lag correlation coefficient. The strong correlations found at this distance can probably be explained by a combination of various sources of noise, such as surface wave effects, local ambient turbulence, currents (even though none were observed), bulge waves on the streamer cable, and far-field coherent noise. The coherent noise at6.5 mseparation can be seen as a

“zero-level” for crosscorrelations of closely spaced hydrophones, and will be used in the next subsection.

Crosscorrelations

Figure4.7was produced by crosscorrelating traces with spatial separation≤7 min our special streamer cable. From each crosscorrelation we picked the correlation value found at zero-lag.

The curves were then produced by fitting a spline through a large number of such correlations to show the average spatial correlation length of the noise.

We have also indicated a probable zero level, taken from Table4.1. With this zero level flow-noise for typical seismic vessel velocities (4-5.5 knots) is spatially correlated up to between 0.4-0.6 m. The integral length-scaleΛis ≈ 0.25 m, and a hydrophone separation of2Λ ≈

0 100 200 300 400 500 600 700

Figure 4.7: Average spatial correlation distance of noise recorded along the streamer at varying velocities. The horizontal stapled black line indicates the probable zero level.

0 50 100 150 200 250 300

Figure 4.8: Averaged spatial correlation length of noise at different frequencies. The vessel velocity was5.5 knots.

50 cmseems reasonable. This number is somewhat larger than the separation suggested by the autocorrelation analysis for sea-state 0.5. However, in the autocorrelation computations we relied on theTaylor(1938) frozen field hypothesis to convert between the time and space domains. No such assumptions are made here. In the above computations, uncertainty exists concerning the exact placement of the zero level. If the zero level was to be placed at 0.5 or 0.4, this would correspond to2Λ≈60 cmand2Λ≈80 cm, respectively.

Finally, we investigated correlation length at different frequencies. Figure4.8was produced in the same way as Figure4.7. However, in Figure4.8we also applied low-cut filters. The high-frequency content of the noise(≥20 Hz)appears to be correlated over very long distances.

Based upon field experience with seismic acquisition, and aided by the Wenz-curves (Wenz, 1962), we hypothesize that the coherent high frequency noise originates from distant shipping, industrial onshore activity or the towing vessel itself.

Figure4.9shows the maximum spatial crosscorrelation for a large number of traces. The thick line shows how the average correlation develops with spatial separation while the crosses show individual correlations. Recently,Tutkun et al.(2009) reported two-point correlations in high Reynolds number TBL flows of up to seven times the boundary layer thickness. In our

Figure 4.9: The maximum spatial crosscorrelation for a large number of traces. The line in-dicates the average, while each cross inin-dicates the maximum correlation from two traces. The vessel speed was 5.5 knots.

Figure 4.10: Power spectral estimates of noise records acquired on the streamer. The spectral estimates were obtained by using a multi-taper method. The bottom image is a zoom of the data in the range from2to60 Hz. The steamer was at5 mwhen the data were acquired, and the vessel velocity was4.5 knots.

case, with a boundary layer of30 cm, the maximum expected correlation length is therefore as much as2.1 m. However, high correlations between hydrophones far apart might also be explained by far-field coherent noise or streamer bulge-wave noise.