The Scientists

The actual code wasn't yet developed as the team energy was spent in develop the mathematical model and searching for applicable literature. Although , the code can be easy created by applying the equations in SAR images, as described below.

The demonstration of the equations can be seen below and the code doesn't have a demo yet.

Detailed description of the project.

As known, tropospheric effects can cause errors in the interferogram processing up to 20 centimeters. (Bekaert,Walters, et al., 2015; Fournier et al., 2011; Heleno et al., 2010; Hooper et al., 2012), so the development of mathematical methods can improve the accuracy of the data in processing interferograms. Various approaches have been applied in order to reduce the magnitude of the tropospheric errors. However, we still have an inaccuracy due to non-dependency of weather properties variability along height.

In order correctly represent the tropospheric effect it was developed a mathematical model which guarantees a better approximation of the error. The initial equation is shown below (HANSSEN, 2001):

As you can see, the weather parameters are not dependent of height. Considering only the weather conditions to be studied.

We can study each of them separately.

As the variation of the temperature in height depends on the pressure and the partial pressure of water vapor depends on the temperature, we can solve the Ordinary Differential Equation of the hydrostatics by applying the separation of variables method.

p is the density of dry air (the consideration of water particles in air molecules is considered by calculating the component of the partial pressure of water vapor); g is the acceleration of gravity, that varies with height as well, and p is the pressure in a given point in space. Integrating on both sides of the equation:

Of course, the integration is in height, but it was suppressed by simplicity. As mentioned, gravity varies on height so the above expression must consider this variation (LOPES, 2008). As p is constant we may not include it from inside of the integral, as the minus sing:

The left hand side of equation can be interpreted as the pressure at a given point in space minus the pressure on the ground, which we can consider for now P. So, the above equation can be rewritten as:

g0 is the gravity in sea level, at the Equator line, w is a latitude in degrees of the consider area, B is a coefficient, z is the altitude of radar satellite and R is the Earth radius, which can be approximated as 6.371 x 10exp6 meters.

As the pressure equation is already solved, we can make some algebraic manipulation to obtain the variation of temperature in height.

Being Ri the gas constant.

The last parameter necessary to complete the equation is the partial pressure of water vapor:

All parameters above can be accoupled to the first equation and then, calculate the error by the expression bellow (HASSAN, 2001; SHEN et. al. 2019):

The code will change the information of the SAR file that would be downloaded from ASF database. It is certainly a contribution that the code can be implemented in an website and instead of download the data from ASF for the later correction. The user will paste the link of the file from ASF on our website, the algorithm will calculate the errors, correct the SAR image, and give the correct file to be downloaded. This way, the user download the files just one time and already corrected.

The algorithm will operate inside the files as they already have the information of elevation of each pixel. This information already is related to the color of the pixel, so a simpler way to correct the error is changing the value association and not the “color” for each pixel. The method can be better explained with an example:

Imagine that you have a yellow square, a blue square and a red square.

The yellow square is taller than the blue, which is taller than the red.

So, the red square is the smaller one.

Each of them has a characteristic value: 4.5, 3.5 and 2.5 respectively.

However, there is a big square that holds thousands of little squares colored in yellow, blue and red.

Instead of change the height of each little square in the big square, you can just change the value that each color represents. Applying the algorithm and finding a value error of 0.5, for example, the value associated with yellow, blue and red will no longer be 4.5, 3.5 and 2.5. But, 4; 3; and 2.

Note that the colors remain the same and the effort applied to change the height of all little square is much lower than correcting square by square of the big square.

How to Create an Interferogram Using ESA’s Sentinel-1 Toolbox – ASF (alaska.edu)

ASF Data Search (alaska.edu)

UZH-S1-GC-AD-v112 (copernicus.eu)

Sentinel-1 Toolbox - Sentinel Online (esa.int)

http://step.esa.int/main/download/

This experience was unlike anything we ever lived before. We got the chance to have a small clue of what kind of problems NASA has to solve - totally out of our reality, completely new. We chose this challenge because of the Maths and technology that surrounds it, but also for it's applicability since it's all about changes in our own ground. The whole thing was tough, from the basic bibliography to the final interferogram. We used differents papers and books to help us solve the problem (or at least try to). We had to deal with a few setbacks, like informations that were no longer available in the dadabase specifyed by the authors that we based our work on, but we were able to find a way with our own equations.

Bekaert, D., Walters, R., Wright, T., Hooper, A., & Parker, D. (2015). Statistical comparison of InSAR tropospheric correction techniques.

Remote Sensing of Environment, 170, 40–47;

Berrada Baby, H., P. Golé, and J. Lavergnat (1988), A model for the tropospheric excess path length of radio waves from surface meteorological measurements, Radio Sci., 23, 1023–1038;

Elliott, J. R., J. Biggs, B. Parsons, and T. J. Wright (2008), InSAR slip rate determination on the Altyn Tagh Fault, northern Tibet, in the presence of topographically correlated atmospheric delays, Geophys. Res. Lett., 35, L12309, doi:10.1029/2008GL033659;

Fournier, T., Pritchard, M. E., & Finnegan, N. (2011). Accounting for atmospheric delays in InSAR data in a search for long-wavelength deformation in South America. IEEE Transactions on Geoscience and Remote Sensing, 49(10), 3856–3867;

Hanssen, R. F. (2001), Radar Interferometry, Data Interpretation and Error Analysis, Remote Sens. Digital Image Process., vol. 2, Kluwer Acad., Dordrecht, Netherlands;

Heleno, S. I., Frischknecht, C., d'Oreye, N., Lima, J., Faria, B., Wall, R., & Kervyn, F. (2010). Seasonal tropospheric influence on SAR interferograms near the ITCZ—The case of Fogo Volcano and Mount Cameroon. Journal of African Earth Sciences, 58(5), 833–856;

Hooper, A., Bekaert, D., Spaans, K., & Arıkan,M. (2012). Recent advances in SAR interferometry time series analysis for measuring crustal deformation. Tectonophysics, 514, 1–13;

LOPES, Wilson Variação da Aceleração da Gravidde com a Latitude e Altitude. Caderno Catarinense de Ensino de Física, v. 24, p. 561-568, 2008;

Shen, L., Hooper, A., & Elliott, J. (2019). A spatially varying scaling method for InSAR tropospheric corrections using a high-resolution weather model. Journal of Geophysical Research: Solid Earth, 124, 4051–4068. https://doi.org/10.1029/2018JB016189.

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