The bar of Hsu at al 2006

This is somewhat different to the previous analysis, but not that much. In this case, I am interested to see when the input wave condition is insufficient to identify its own bar, if ever.

To do so, we need a functional form for the profile and we use that proposed by Hsu et al(2006). Sadly, on their paper they used data from Duck on the analysis but no values are reported for the 5 parameters needed to create the bar shape. Hence, I had to manually calibrate them (I know, it would have been better to use a least squares approach, but when I figured that out, I was almost ready).


The selected parameter space is this [A1 A2 B1 B2 B3]=[2.2 1.3 3.0 2.8 -0.5], which is very different to the values they present for the beach at Japan, but for some reason they did not present values at Duck, right? Anyway, I decide to focus the calibration on the bar, so I decided to stop the profile at X=1.5. The non-dimensional profile shows very good agreement with the non-dimensional versions of DELILAH and DUCK94, as shown in the figure to the right.

With this non-dimensional profile, it is possible to measure the nondimensional distance between the bar crest and the underlying semi-parabolic profile defined by the first term on the equation of Hsu et al (2006). However, typically this
term deviates significantly from the actual profile in deeper water (due to the parameter space used), so I decided to use a simpler linear interpolation between the bar trough and the offshore end of the profile, and measure the bar height accordingly. Both bar height values are shown in the figure below(in nondimensional values). It can be seen that for the linearly interpolated case, the value is Hbar_non=0.51, which is slightly less than the value measured on the real profiles (0.54 and 0.68, respectively). Furthermore, a smaller bar height is a more conservative approach, and hence is selected.

The next step is to transform this into dimensional space. For each T, Hs pair it is possible to determine the bar crest depth and cross-shore location, with the use of other two parameters. For the beach slope I used beta=0.035, following Stockdon & Holman (2000); and I used a sediment fall velocity of w=1 cm/s. The bars thus predicted are slightly deeper and offshore than the real bars, but this can be attributed to a poor model to estimate these parameters.

Anyway, with this information is possible to calculate the error at each wave condition and compare it with its own bar height, and calculate the difference error in the same way as before.

The main result is that each wave condition can see its own bar, but it can also be noticed that the trend is that larger waves and longer periods would enable better identification, mainly because the bar itself would be larger.

Analysis of the DELILAH and DUCK 94 bars

Here we study under which kind of wave conditions the bars observed at Duck, NC during the DELILAH and DUCK94 experiments.

DELILAH
In this case the peak period is about Tp=7.5 s and the significant wave height Hs=1.8 m along the pressure array at y=984. The nearest surveyed profile showed a bar with its crest at a depth of hc=1.56 m, and it was measured that this bar represented a perturbation of about Hbar=1.06 m respect to an imaginary profile without the bar.

DUCK94
In this case the peak period is about Tp=8 s and the significant wave height Hs=2 m along the pressure array at y=984. The nearest surveyed profile showed a bar with its crest at a depth of hc=1.88 m, and it was measured that this bar represented a perturbation of about Hbar=1.01 m respect to an imaginary profile without the bar.

Results

With these data, is possible to estimate the relative error in depth retrieval Delta h. If the value of the error is less than the bar height, it means that is possible to be characterize the bar as real feature and not an error artifact.

The figures above show the difference error

D=Delta h - Hbar

which basically says if the value of D is negative, the bar can be identified. The white dot is the input wave condition associated with the real bar. As can be seen, for all the wave periods and wave heights considered, the bars can be identified. It can be noticed also that our previous results seem to be confirmed, in the sense that longer periods can identify the bar with more certainity (i.e., they have smaller Delta h)