Simulating APSFs - eoplus/apsfs GitHub Wiki

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The adjacency effect arises from scattering in the atmosphere and therefore to simulate the Atmospheric Point Spread Function (APSF) it is necessary to resolve those scattering events.

The scattering of energy in different directions is described by the scattering phase function. More specifically, the scattering phase function describes the fraction of the incident (non-absorbed) energy that is scattered per solid angle in a given direction, for all directions in the unit sphere. It is intuitive that this function is a probability distribution, and by definition it integrates to unity. Each scatter in the atmosphere, with the exception of gases, treated collectively under Rayleigh scattering, has its own scattering phase function.

This probability distribution describes a single scattering event, but except when the atmosphere is optically thin (wavelength ranges of low molecular absorption and Rayleigh scattering, and under low aerosol load), a significant fraction of the photons reflected by the surface will suffer multiple scattering events before reaching the sensor. Therefore the probability of a given photon path between the surface and the sensor is given by the multiplication of the probabilities of each scattering angle, with the APSF being the sum of probabilities of all possible photon paths between a given area element and the sensor. For any realistic set of conditions, numeric solutions have to be used for the calculation.

The approach used in the code is to resort to Monte Carlo methods. The Monte Carlo method is based on drawing millions of samples from the probability distributions that describe the processes being simulated. That is, millions of photons paths are simulated, considering the probabilities of free path, absorption, interaction with different scatters, and the resulting scattering angles. Since random photon paths are simulated, is a measure of economy that the problem be reversed to simulate photon paths from the sensor to the surface. This reversal is justified by a symmetry similar to the one used to describe the PSF: the scattering angles for a given photon path are the same, regardless if considered from the surface to sensor or sensor to surface, therefore their probabilities are equal.

ADD FIGURE HERE: PHOTON PATH ANGLES DIRECT AND REVERSE

There is one important caveat on this approach. Think about a continuous probability distribution, like the Gaussian model.

A simple use of this probability distribution is to describe the transfer of energy when the atmosphere is optically thin (low molecular absorption, small aerosol load). Under those conditions, single scattering theory can be used, integrate the angular probability to a given position on the surface from different positions along the line of sight of the sensor.

When the atmosphere is optically thin (low molecular absorption, small aerosol load), single scattering theory can be used to calculate Atmospheric Point Spread Function (APSF). In this case, the average atmospheric scattering phase function due to Rayleigh and aerosol scattering is used as a probability weight to calculate the contribution of each desired position on the surface, integrating along the vertical path.

The photon paths from the surface area elements to the sensor through the atmosphere is

The adjacency effect described by the APSF arises due to multiple scattering in teh Since the Atmospheric Point Spread Function (APSF) describe the (relative) magnitude of the contribution of each surface area element to the signal received by the sensor, it is necessary to resort to Radiative Transfer theory for its simulation. Radiative transfer deals with the propagation of electromagnetic energy in a scattering medium.

ADD HERE: theory on the MC integration and environmental signal as in Tanre et al. 1981

But it is important to keep in mind that Monte Carlo methods are based on integration and cannot provide directly the PSF related to the area elements, but the PSF integrated over finite areas. With the area know, it is possible to calculate

The scheme of the right most figure illustrate how this code simulates the Atmospheric Point Spread Function with Backward Monte Carlo Radiative Transfer simulations, by using yet another symmetry: the photon paths from surface to sensor have the same probability as the reversed paths from sensor to surface.

To simulate an APSF the primary step is to define the atmospheric properties. The following information are required:

• Aerosol properties:
  • Scattering phase function;
  • Attenuation coefficient;
  • Single scattering albedo;

The function atm_prfl helps to build such a matrix. It requires as inputs a atmospheric profile of pressure at specified heights, the aerosol optical thickness at surface, the aerosol scale height in Km, the aerosol single scattering albedo and the Rayleigh optical thickness at surface

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Next: Aerosol properties