And now the variance images

Radar Video qualitative comparison

I think this summarizes it all

N-th iteration, 4th analysis:

Now I combined previous analysis in a single plot, that I think will represent what we want to show in the paper. The history of this graph is the following:

1. For each pair H, T the value of hc is computed.
2. With this value, I can compute the bar height Delta h by multiplying the value of hc times the nondimensional distance between the imaginary profile and the bar crest. All bars are going to be similar geometrically.
3. This gives me the bar height associated to that H,T pair.
4. Now, I scan a range of depths (h=[0-20] m and identify at which depth the bar can not be seen anymore.
5. Double check that the bar is not piercing the surface.
6. Plot using imagesc the resulting maximum depths.
7. On top, plot using contour the associated bar heights.

So this graph gives me the maximum depth any bar height can be seen, as a function fo both the period and the amplitude. The results are shown below.

At the end, it appears that there still is a dependency on both frequency and wave height. However, the analysis thus done enforce that a certain bar height can be associated to a very limited set of conditions (the one that generated it according to Hsu et al), which is not general enough. It is a good starting point though.

To read the graph above, you pick a bar height, say the 1 meter we used before. Follow the contour lines to see when this bar can be seen by a H,T pair. In this particular case, the 1 m contour line agrees with what was shown before. (Good, I am being consistent.)

Just a tiny bit of extrapolation can be done. For instance, if we are looking at which kind of bars a given condition see. Say we pick T=4 sec and H=1.5 m. That put us roughly with bar with a height Delta h=0.75 m, which can be seen in roughly h=3 m, according to the graph. Bars larger than that, would be seen at that depth (and smaller) too by that condition.

However, this plot does not show what is the minimum bar that can be seen at any given depth, because it depends on the wave conditions.

5th analysis
This a new approach. I created a matrix of water depths [0.1-20] m, every 20 cm, and of bar heights [0.5-2.0]m, every 25 cm.

From this matrix, I selected only those combinations where the bar is not surface piercing. Then, for the conditions surviving this filter, I scanned the H,T matrix used before (H=0.1 to 3 m, T=2.7 to 14 s) finding which pairs would enable observation of the bar and counting them.

At the moment I am not interested in the conditions that can see the bar, only if the bar could be seen. Hence, a value of 1, that is only one H,T pair is capable of observing the bar is enough. Obviously, the larger the number, the better.

Since it is a 4 dimensional analysis, it may take a while to compute, so keep on waiting. It finished!

So here is the result:

So this figure says that we can only see a Delta h=0.5 m bar in depths smaller than h=3.0 m, and more than 50% of the wave conditions used will be able to see it if the depth is less than h=2.0 m.

It also says that we can see 2 meter bars up 12 m in waters up to 12 m deep.

I really think this is something we want to show.

Tan tan.

How deep we can see

So here is a new approach. For a the same matrix of periods and wave heights, now I scanned at which depth the error Delta h becomes larger than the bar height, which I set to 1 m.



The result is shown in the image above and it can also be seen than longer waves with small nonlinearity are more likely to detect the bar at deeper absolute depths.
It is interesting that the depths are relatively small though, which may indicate that the technique is more suitable in relatively shallow waters. This can be seen if the same critical depth is used to calculate kh, as shown below.

Quick update

Below is the resulting graph for the same analysis as the DELILAH and DUCK94, but for the bar we used in the REU experiments. Again, it seems that the bar can be seen.

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)

Bars are not good for your health, steer clear from them

So here we are, using a linear profile and different wave heights and periods.



This first figure shows the relative error in depth (meters) at a certain true depth, for several combinations of wave heights and periods. It can be seen that the main factor is the wave period, which makes the error to vary significantly. It can be seen that at any given depth, the longer the period, the smaller the error (or uncertainity region). The area below each curve is the uncertainity area, that is, any wiggle returned by a possible DIA that falls below the curve we can not tell for sure whether is true or not. In that sense, it is more convenient to have longer periods to reduce this area.

Naturally, this begs the question if it is possible to express this in terms of kh. Well, here is the same plot but in terms of kh.



It is interesting to note that the curves do not collapse into one!. The reason for this is the nonlinear relation between k,A,T and h in the composite model, that makes impossible to have two different sets having the same properties, something that can happen with the linear dispersion relation. Again, this plots shows that longer periods (lower kh) are better. In other words, it is better to be in intermediate-to-shallow water.

Can we trust that bar?

Let's see.

I am struggling a little bit on how to bring down to earth this error bussiness, because it seems to me that I have too many variables hanging loose out there. So I tried this procedure:



1. With the true bathymetry, I compute the true wavenumber k, and a realistic wave height profile H(x).
2. With these, I can go to the analytical expression for the amplification factor associated with errors in wavenumber (F_2 in these equations).
3. Since we are assuming the composite model gives a 8% accuracy at all water depths, we can multiply the local water depth of a given bathymetric profile by the amplification factor and we would get the error bar in the bathymetric retrieval.

4. This is the next important assumption: The key thing is to what bathymetric profile we apply this error estimate. One approach is to apply it to the actual depth, but the result is somewhat hard to interpret, at least for me. Besides, the question is what kind of features can be safely identified by the DIA. Hence, I changed the approach. Let's say we have the right wavenumber, the right wave amplitude, and the right radian frequency. In other words, we measured everything perfectly!.

5. I don't have a step 5 (yet)! :(

Reflection, anyone?

Looks like a reflected wave finding its incident soul mate

Exciting!

WOW! After several pages, I finally derived the analytic expression of the percentage error in depth (Delta h /h) and its dependency on percentage errors in the radian frequency (Delta sigma / sigma), wavenumber (Delta k /k) and wave amplitude (Delta A /A). The surprising feature is that the error in h shows a varying dependency on the value of A.

That is, when A=0 (linear case), the error exhibits a monotonic dependency on kh, ever increasing. But as A increases, the curve shows a a shape that resembles a parabolic curve. If A is large enough, the trend is that for low kh, the error is larger.

Error in Delta h/h as a function of kh, when perturbations are induced in A, sigma, k. Note the strong influence of A on the functional response of the error. However, errors in A induce small errors in h

The figure above shows three amplitude conditions, when evaluated over the DELILAH bathymetry and wave conditions. The influence of A is evident, and suggest that perhaps a measure of nonlinearity needs to be included in the analysis. Ursell anyone?