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.