Dear Almaz Torgoev,

Thank you very much for your interest in our paper.

For step 2 you simply need to smooth the curvature map at the required length. In our paper, we propose a simple way to do that in Equation 4: we use a double convolution of the curvature matrix with an n × n matrix of ones, n being equal to the baseline length Ls in meter divided by twice the DEM resolution (n = Ls/40 if pixel size is 20 m). You can easily perform such a convolution with Matlab. If you use ArcGis, you can perform the smoothing with the "focal statistics" function and give the value Ls in the "neighborhood settings" option.

Please be aware that n must be an odd number. With a 20 m resolution DEM, the FSC can only be computed for Ls=120 m, 200 m, 280 m... You'll need a higher-resolution DEM if you want to compute the curvature at smaller lengths.

Please let me know if you need anything else.

Emeline Maufroy

Dear Emeline!

Thank you for the answer.

I attach a figure from our planned paper - we deal with 2D dynamic studies. Did the presented calculation of 2D frequency-scaled curvature reccurred your 3D approach?

Dear Almaz,

I'm not sure that you can simply take the average of the curvature values in the baseline length. I never tried that, so I don't know if it is equivalent or not.

The procedure that we describe in our paper is totally equivalent to the ArcGis procedure, so I advice you to closely follow it as this procedure is commonly used by people in geomorphology and hydrology.

Best regards,

Emeline Maufroy

Dear Emeline!

I have performed a test in Spatial Analyst/Neighbourhood/Focal Statistics with 3 by 3 cells neighbourhood. I manually checked returned mean values and got the same results as automatic raster....so it means that Focal Statistics simply averages all values around certain neighborhood - I did the same for 2D curvature calculation (see figure above). An additional confirmation comes from http://help.arcgis.com/en/arcgisdesktop/10.0/help/index.html#/How_Focal_Statistics_works/009z000000r7000000/

You can see "Example" Chapter with the sum of pixel in 3 by 3 cells neighborhood - they simply sum all pixel values, so I guess they do the same with the mean value

Dear Almaz,

I've checked on my side. The averaging procedure you describe corresponds to one convolution in my procedure (it is indeed totally equivalent). That's interesting.

HOWEVER: two convolutions (or two smoothings, or two averagings) must be performed when computing the curvature in 3D. It means that the smoothing must be performed twice on a DEM (and therefore to take only half-Ls in each smoothing to reach the total required Ls). One smoothing is indeed required per derivative of the elevation map (the curvature being the second derivative; please see page 5 of our paper).

Yet I don't know if two smoothings are required for a 2D topographic profile. You may want to check that if you have the data available to perform such a test. I'll be glad to know your results.

Cheers,

Emeline

Dear Emeline,

I am not sure, if got good understanding of your explanation. Do you mean, that first I should smooth DEM, then calculate curvature for 40 m (DEM pixel size is 20 m) and then smooth curvature over specific baseline length? 

What is the physical meaning of double-smoothing? I can understand a smoothing of Curv40 values, but what is the second smoothing for?

Dear Almaz,

The smoothing (and the determination of lengths to compute the curvature) is required if you intend to correlate seismic wavelengths with the topography. The correlation that we show in our paper between wavelengths and topographic amplification was only reachable after two smoothings of the curvature (or of the DEM, it has no importance if the smoothing is performed before or after the curvature computation). Our linear correlation wasn't as good with only one smoothing. That's why we recommend two smoothings. We think one smoothing is required per derivative of the DEM. The slope being the first derivative, it requires one smoothing. The curvature being the second derivative, it requires two smoothings.

Anyway, you can adapt the smoothing to the aims of your study. As I don't know what you intend to do with the curvature computation, I cannot help you further on that point.

Emeline

Dear Emeline,

thank you for all valuable comments!!!

I got the answers for all of my questions