Cross-correlation technique

A visual inspection of the seven-year data set reveals that there is significant spatial variation in sand-wave migration. It appears as if the sand waves migrate in directions that are not perpendicular to the crests. Simply tracking the crests of sand waves along transects perpendicular to the crests would provide a limited picture. Following Duffy and Hughes-Clarke (2005), a simple but efficient cross-correlation technique was used to determine the spatial migration of the sand waves. Duffy and Hughes-Clarke (2005) used the ‘Weighted centroid’ technique to calculate migration vectors, rather than the maximum correlation. Moreover, they used spatial gradient maps rather than bathymetric maps. For the purpose of this research, the use of the ‘maximum correlation’ technique and bathymetric maps appeared to give robust results.

In the cross-correlation technique, spatial patterns on subsequent time steps t1 and t2

on DTM1 and DTM2 are compared (Fig. 5.3). A square box, i.e. ‘fit matrix’, centred

∆x ∆y

search area fit matrix

DTM₁ DTM₂

Figure 5.3. Illustration of the cross-correlation technique. To obtain the migration vector of the sand wave (shaded area) the following procedure is used. For every grid point, the fit matrix on DTM1 on time t1 is shifted in x and y directions within the search area on DTM2

on time t2. For every displacement a correlation between the height in the fit matrices on

DTM1 and DTM2 is calculated. The highest correlation within the search area yields the

displacements ∆x and ∆y of the fit matrix.

tion (h or h0) of t1 is moved around with prescribed displacement increments ∆x and ∆y

within a ‘search area’ on DTM2. When doing so, it is important to choose the maximum

displacement larger than the expected pattern migration. The size of the displacement increments, equal to a multiple of the grid size, determines the accuracy of the migra- tion vectors. To obtain a higher accuracy, the DTMs were regridded on 3-m grids by linear interpolation. Regridding is beneficial for a higher accuracy because the correlation technique tracks three-dimensional shapes. The rows of the fit matrices were placed in succession to yield vectors a and b at t1 and t2, respectively. Finally, for every displace-

ment, a correlation r between vectors a and b was calculated. If the patterns are exactly the same, r = 1, whereas if they are in antiphase, r = −1. The correlation coefficient is defined as:

r = Sab SaSb

, (5.1)

Sa (and Sb) the variance and Sab the covariance:

Sa2 = 1 n − 1 n X j=1 (aj − {a})2, Sab = 1 n − 1 n X j=1 (aj − {a})(bj − {b}),

where {a} and {b} are the mean elevations, and n the number of data points within each vector. After determining r for every displacement, its maximum value within the search area yielded the displacements ∆x and ∆y of the pattern within the fit matrix. To save computation time, migration vectors were only calculated for each 15-m grid point.

An example of the cross-correlation technique is presented in Fig. 5.4 for the period from December 18, 2001 to April 2, 2002 (105 days). In this example, DTMs based on h0 were used. The fit matrix has a size of 210 m. The displacement vectors in the left panel of Fig. 5.4 have directions ranging from east to north. The vector field is smooth and changes gradually. Only near the boundaries occasional outliers occur. The large migration vectors north of y = 556.5 are due to the absence of bedforms. Here the technique does not find

30 m displacement x [km] y [k m ] 114 114.5 115 554 554.5 555 555.5 556 556.5 0.7 0 .7 0 .7 0.95 0 .9 5 0 .9₅ r x [km] 114 114.5 115

Figure 5.4. Sand-wave migration (left) and correlation coefficient r (right) calculated with the cross-correlation technique for the period from December 18, 2001 to April 2, 2002 (105 days). The size of the fit matrix equals 210 m. In the left panel, the background is a shaded contour map of h0 with a median date of December 18, 2001 and the migration vectors are plotted every 90 m. In the right panel, dark (light) shades of grey indicate high (low) r values.

good matching patterns. These outliers correspond to lower correlation coefficients. In general however, the matches are good and correlation coefficients are higher than 0.9.

The sensitivity of the correlation technique was studied for fit matrix sizes of 60, 120, 180, 210, 240, and 270 m. The sensitivity is indicated by standard deviations σu and σv

of migration rates u and v along the x and y directions, respectively. Migration rates were calculated for time steps of three months, from March 1998 to March 2005. They are presented in the Results section. For every time step, σu and σv were calculated for

all u and v within areas I-IV (Fig. 5.8). Mean standard deviations were then simply calculated over all time steps and areas and plotted as a function of the fit matrix size in Fig. 5.5. Also plotted is the mean percentage of migration vectors with a correlation smaller than 0.9. Within the different areas there is some variation in migration speed (compare Fig. 5.4 and Fig. 5.8), and as a result, the standard deviations are never zero. Therefore, the results of the sensitivity analysis should be considered relative to each other. Fig. 5.5 shows a decrease in the standard deviations and the percentage of ‘bad fits’ with larger fit matrices. On a typical DTM, wave lengths range from 125 m to 250 m. Apparently, the fit matrix with the size of 60 m is not able to track the large bedform patterns correctly. There are too many incorrect pattern matches, resulting in a chaotic migration-velocity field and high standard deviations. The results pertaining to fit matrixes with sizes larger than 120 m are much better. On the other hand, it is more

size fit matrix [m] 60 120 180 210 240 270 0 5 10 15 σ u[m y −1_] σ v[m y −1_] % r < 0.9

Figure 5.5. Standard deviations σu and σv and the percentage of r < 0.9 as a function of

the size of the fit matrix.

difficult to track smaller bedforms with fit matrices that are too large. On the basis of this analysis and visual inspection of progressive vector diagrams, such as Fig. 5.7, a fit matrix size of 210 m was selected for the cross-correlation technique. The dots in Fig. 5.7 represent integrated migration vectors. With a size of 210 m, the clustering of the dots is minimal and the migration pathways of the dots are smoothest. The cross-correlation technique was applied to both h and h0. The differences between the results are small, but progressive vector diagrams show that the migration pathways are a little smoother for h0. Therefore h0 was used for the cross-correlation calculations. Fig. 5.7 will be further discussed in the Results section.

5.4

Results