Accumulator geometries - eoplus/apsfs GitHub Wiki

The backward Monte Carlo code can accumulate photons that reach the surface in three different geometries that will be appropriate for different conditions being simulated and different intended uses of those simulations.

Regardless of the accumulator, all simulations also return the direct transmission (contribution from photons that were not scattered between the surface and the sensor). The default is to simulate with an infinitesimal FOV (directional sensor) such that direct transmission occurs only at {0,0} and in absence of scattering events. Note that there is no relation between the Instantaneous Field of View (IFOV, the surface area projection of the FOV) and the resolution and geometry of the accumulator. For finite FOVs, direct transmittance occur from surface elements within the IFOV for photons that were not scattered between the surface and the sensor. That is, the diffuse transmission of photons is followed even over surface elements within the IFOV.

Annular geometry

For symmetric problems (e.g., nadir view over a flat Lambertian surface) the contribution of an area element to the diffusely transmitted photons reaching the sensor within the solid angle viewing the target surface depends only on its radial distance to the center of that surface. The result of the simulation is the contribution of each annulus (area integral) to the diffusely transmitted photons reaching the sensor:

$g_i = 2\pi \int_{r-\Delta r}^{r} f(r)rdr = 2\pi \overline{f(r)r}\Delta r,$

where is the contribution of annulus , $\Delta r$ is the requested resolution and $\overline{f(r)r}$ is the average radial product with the PSF within the annulus. The image below illustrate the annular accumulator geometry, the calculated radial weighted average density PSF and the calculated radial cumulative PSF. Note that for those plots the PSF was normalized to integrate to unity: $2\pi\int_0^\infty f(r)rdr= \sum_{i = 1}^n g_i = 1$ .

The break points indicating the limits of each annulus and the mid point of each annulus are also returned. The following convention is used for this geometry: the lower limit is always zero and the first radial distance is half of the requested annular resolution. The next break points are spaced by the requested resolution up to the nearest value lower than the requested radial extent. To this, an upper infinity limit is added, creating an interval that accumulates all photons from surface elements at distances larger than the upper resolved limit (extent).

Sectorial geometry

When the conditions of the simulation are not symmetrical, but include a directional component (e.g., non-nadir viewing angles over Lambertian surfaces) the annuli can be subdivided in constant sectors. The simplest configuration is when the arcs formed by the sector boundaries subtend the same angles over consecutive annuli and with constant angular spacing. The result of the simulation is the contribution of each annulus sector (area integral) to the diffusely transmitted photons reaching the sensor:

$g_{i,j} = \int_{\theta-\Delta \theta}^{\theta} \int_{r-\Delta r}^{r} f(r,\theta)rdrd\theta = \overline{f(r, \theta)r}\Delta r \Delta \theta,$

where $g_{i,j}$ is the contribution of sector of annulus , $\Delta r$ and $\Delta \theta$ are the requested radial and azimuthal resolutions, respectively, and $\overline{f(r,\theta)r}$ is the average radial product with the PSF within the annulus sector. The image below illustrate the sectorial accumulator geometry, the calculated weighted average density PSF and the calculated bidimensional cumulative PSF. Note that for those plots the PSF was normalized to integrate to unity: $\int_0^{2 \pi}\int_0^\infty f(r,\theta)r dr d\theta = \sum_{i = 1}^n\sum_{j = 1}^m g_{i,j} = 1$ .

An alternative representation can be made if the sectors are projected (disk to plane) such that the abscissa is the azimuth and the ordinate is the radius. This is the actual internal geometry of the accumulator and the output format for sectorial geometry. It is more convenient, since the bidimensional cumulative distribution on the disk is a surface between an helicoid and a tangent developable.

Note that the axis of the planar projection of the disk were chosen such as to make indexing of the data simpler. The first element of the first column is at the first mid point for the average density PSF and at the first break point for the cumulative PSF.

The break points indicating the limits of each annulus and the mid point of each annulus are also returned, but the break points and mid points of the azimuths are not returned since they are fixed: the sectorial geometry subdivide the annuli in constant spacing of 1°, from 0° to 360°. The other conventions used for this geometry are the same as for the annular geometry.

Grid geometry

A more general geometry for simulation is a regular grid in metric distances. The simplest configuration is when the the spacing in and are equal and constant. The result of the simulation is the contribution of each grid cell (area integral) to the diffusely transmitted photons reaching the sensor:

$g_{i,j} = \int_{y-\Delta y}^{y} \int_{x-\Delta x}^{x} f(x,y)dxdy = \overline{f(x, y)} \Delta x \Delta y,$

where $g_{i,j}$ is the contribution of grid cell $\{i,j\}$ , $\Delta x$ and $\Delta y$ are the requested spatial resolution, and $\overline{f(x,y)}$ is the average PSF within the grid cell. The image below illustrate the grid accumulator geometry, the simulated grid cell integral PSF and the calculated bidimensional cumulative PSF. Note that for those plots the PSF was normalized to integrate to unity: $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} f(x,y) dx dy = \sum_{i = 1}^n\sum_{j = 1}^m g_{i,j} = 1$ .

Here as well, that the axis were chosen such as to make indexing of the data simpler. The first element of the first column is at the first mid point for the average density PSF and at the first break point for the cumulative PSF.

The break points indicating the limits of each grid cell and the mid point of each cell are also returned, both as single vectors as they are the same for x and y. Other conventions used in this geometry are: the center cell has limits $\{-\Delta x/2, \Delta x/2\}$ such that its center is at {0,0}. From those limits of the center cell, the next limits are spaced by the requested resolution up to the nearest value to the $\pm$ extent. A lower and an upper limit at -infinity and infinity are added at the extremes in each dimension, to capture photons in the area beyond the $\pm$ extent.

Extent and resolution

The extent defines the spatial limits where the photons will be accumulated in bins defined by the requested resolution. The extent in the radial and sectorial geometries is the radius, while in the grid geometry is the positive limit, and is mirrored in the negative direction. An additional bin is added to all geometries to capture photons arising from surface areas beyond the extent.

The choice of extent depends on the intended application, but 15 km is a reasonable compromise. It will be sufficient to spatially resolve most (> 95 %) of the diffuse contribution from the majority of aerosol models. At further distances, the photon path probabilities become too small, increasing statistical fluctuation (noise) from insufficient sampling, requiring a large increase in the number of simulated photon paths. This problem will be more important at higher spatial resolutions, and in particular for the grid geometry. For the annular and sectorial geometries, constant sectors and annuli radius cause an increase of bin area with distance from the origin, partially compensating for the reduction of per area photon path probabilities. The author has successfully used an extent of 15 km with a spatial resolution of 0.003 km (3 m) for annular geometry and 0.03 km (30 m) for sectorial geometry, following 10^8 photons (6 min in a Pentium i5 machine).

Due to the more isotropic characteristic of the Rayleigh phase function, the spatial scale of the adjacency effect is much larger than for aerosols. When simulating the Rayleigh contribution, 15 km extent will spatially resolve about 70 % of the signal. When combining aerosol and Rayleigh, the fraction of the signal will be between 70 % and 95 %, depending on the relative magnitudes of aerosol diffuse transmittance and Rayleigh and Rayleigh+Aerosol interaction transmittance.

Depending on the desired extent and resolution combination, it may not be feasible to have sufficient random photon paths sampled for regions far from the origin. For those circumstances, it may be better to run annular or sectorial geometries to fit a model that can be reconstructed an in grid geometry, as will be discussed in the last chapter of this manual.

Concluding remarks

All example simulations shown here are for the symmetrical condition of nadir view, with the same input parameters. Therefore, the radial cumulative distribution of the annular geometry equals the radial profile at a constant $\theta =360^{\circ}$ on the cumulative distribution of the sectorial geometry.

Finally, while the grid geometry is great for visualization and ready to use if simulated at sensor resolution, inspection of the cummulative PSF show that that this surface is much harder to model since the PSF are highly peaked at {0,0}. Fitting of the CDF is necessary to calculate the derivative (univariate) or hessian (bidimensional) and generate grids at any sensors resolution. That said, the CDF in grid geometry could potentially be modeled as a bidimensional generalized logistic function with a large parameter to model the sharp rise after {0,0}. Such model however is not implemented. The recommendation is to use the annular and sectorial geometries if the intent is to have fitted coefficients that can be used to reconstruct the APSF at any sensor resolution and to use the grid geometry if the intent is to create a Look Up Table (LUT) for direct application to a given sensor.

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