This page contains valuable information regarding the generation of the various scene representations, as well as on the way the user simulated results are evaluated:

Terminology | Information on the various terms and definitions used by the RT (Radiation Transfer) modelling community. |

Angular sign conventions | Information on the (view and illumination, zenith and azimuth) angular sign conventions used. |

Leaf angle distribution functions | Information on recommended leaf normal distributions. |

RPV model | Information on the parametric RPV (Rahman-Pinty-Verstraete) model and its parameters. |

RT model technicalities | Information regarding how to define reflectance measurements within RT models. |

Statistical measures | Information on selected metrics that are used to visualise the performance of individual models. |

#### Terminology:

All remote sensing measurements acquired from space or airborne imaging sensors in the solar domain turn out to be strongly dependent on the particular geometry of illumination and observation at the time these measurements were made. A similar situation occurs in the spectral domain: different measurement values are obtained when a target is observed in different spectral bands. The reflectance of a geophysical medium (at a particular wavelength) is thus dependent on both the orientation of the Sun and the orientation of the observer with respect to the target. Such a medium is called *anisotropic*, and the reflectance is characterized as *bi-directional*. By contrast, a perfectly reflecting, isotropic surface, i.e., a system that would reflect all incoming light equally in all directions, is called *Lambertian*.

The fundamental mathematical concept describing this anisotropic reflectance is the so-called Bidirectional Reflectance Distribution Function (BRDF). For practical reasons, the measured reflectance of a target is often normalized by the reflectance of a reference panel that is (ideally) a Lambertian surface, illuminated and observed under identical geometric conditions. The result of this normalisation is then called the Bi-directional Reflectance Factor (BRF). For a polar plot of BRF values observed over a particular target click here.

Observations that lie in the same plane as the local vertical and the incoming direct solar radiation are referred to as BRFs in the *principal plane*. Observations along a plane whose azimuth differs by ±90 degrees to that of the principal plane are referred to as BRFs in the *orthogonal (or cross) plane*. If the direction of observation coincides with that of the direct solar illumination, no shadows are observed within the target and a BRF maximum known as the *hot spot* effect is observed.

Since the field of radiation transfer is rather technical, a precise terminology is used to designate the various reflectance concepts. The following papers provide detailed definitions on the terms employed here.

- Nicodemus, F. E., J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, (1977)
*Geometrical Considerations and Nomenclature for Reflectance*, US Department of Commerce, National Bureau of Standards, NBS Monograph No.**160**, Washington, DC. - Martonchik, J. V., C. J. Bruegge, and A. Strahler (2000) 'A Review of Reflectance Nomenclature Used in Remote Sensing',
*Remote Sensing Reviews*,**19**, 9-20. - Verstraete, Michel M. and Bernard Pinty (2000) 'Environmental Information Extraction from Satellite Remote Sensing Data', in
*Geophysical Monograph*No.**114**, Edited by P. Kasibhatla, M. Heiman, P. Rayner, N. Mahowald, R. Prinn, Ronald and D. Hartley, American Geophysical Union, Washington, D.C., p. 125-137.

#### Sign convention for the zenith and azimuth angles:

of the various test cases, illumination zenith angles are counted from the local vertical and are always reported as a positive value. Illumination azimuth angles are arbitrarily counted from the positive**For the purpose of describing the illumination conditions**`x`-axis direction towards the positive`y`-axis direction (of the scene at hand) and are also given as positive values. For heterogeneous scenes graphical depictions will be provided to clarify these nomenclatures., the Observation Zenith Angle (OZA) is reported as follows:**For the purpose of RT model computations and results submission**- In the principal plane, results corresponding to forward scattering (when the observer is on the other side of the local vertical with respect to the Sun) are reported with a positive OZA. Hence, the relative azimuth (between the Sun and the observer) is ±180 deg.
- Conversely, results corresponding to backward scattering conditions (when the observer is on the same side of the local vertical as the Sun) are reported with a negative OZA. In this case, the relative azimuth is 0 deg. The hot-spot effect in the retro-reflection direction thus occurs always for negative OZA.
- In the cross plane, OZA is arbitrarily counted positively when the relative azimuth is −90 or 270 deg.
- In the cross plane, OZA is arbitrarily counted negatively when the relative azimuth is +90 deg.

*Example:* graphical display of the BRF in the principal plane.

#### Leaf angle distribution functions:

The spatial orientation of a leaf is described by the direction of its normal Ω_{L}(θ_{L},ϕ_{L}) to the upper surface, where θ_{L} is the inclination angle of the leaf normal, and ϕ_{L} is the azimuthal angle of the outward normal. Consider a horizontally homogeneous leaf layer of unit thickness at height `z`, and let the sum of areas of all leaves (or parts thereof) whose normals fall within an incremental solid angle around the direction Ω_{L} be `ĝ`_{L} (`z`,Ω_{L}). Here all leaves are assumed to face upward, so that all leaf normals are confined to the upper hemisphere (2π^{+}), and `s*`_{L}(`z`) is the total one sided area of all leaves in this horizontal layer. The leaf-normal distribution (LND) function `g`_{L}(`z`,Ω_{L}) = `ĝ`_{L}(`z`,Ω_{L}) ⁄ `s*`_{L}(`z`), denotes the fraction of total leaf area in the horizontal layer of unit thickness at height `z` whose normals fall within unit solid angle around the direction Ω_{L}, and must satisfy the following normalization criterion:

A variety of leaf angle distribution functions have been published in the literature. Within RAMI (and RAMI4PILPS), however, azimuthal invariance of the leaf normal distribution is assumed:

For the purpose of standardization and comparison the following formulae for `g*` are recommended:

##### Distributions from Bunnik (1978):

where:

`ag`= 1.0,`bg`= 1.0,`cg`= 1.0, and`dg`= 0.0 for planophile distributions`ag`= 1.0,`bg`= −1.0,`cg`= 1.0, and`dg`= 0.0 for erectophile distributions`ag`= 0.0,`bg`= 0.0,`cg`= 0.0, and`dg`= 1.0 for uniform distributions

##### Distributions from Goel and Strebel (1984):

where Γ is the Γ function and 0 < θ_{L} < π ⁄ 2 is the leaf inclination angle. Furthermore, normalization, requires that:

In addition,

- μ = 2.531 and ν = 1.096 for planophile distributions
- μ = 1.096 and ν = 2.531 for erectophile distributions
- μ = 1.066 and ν = 1.853 for uniform distributions

The correlation between the distribution functions of Bunnik and those of Goel and Strebel are given by the latter to be 0.9992 in the uniform case, and 0.9989 for both the planophile and erectophile cases.

##### References:

- Bunnik , N. J. J. (1978) 'The Multispectral Reflectance of Shortwave Radiation of Agricultural Crops in Relation With Their Morphological and Optical Properties', in Mededelingen Landbouwhogeschool, Wageningen, The Netherlands, 175 pages.
- Goel, Narendra S. and Strebel, D. E. (1984) 'Simple Beta Distribution Representation of Leaf Orientation in Vegetation Canopies',
*Agronomy Journal*,**76**, 800-803.

#### The parametric RPV (Rahman-Pinty-Verstraete) model:

The RPV model is a parametric model that allows to adequately represent surface anisotropy patterns. Through its mathematical formulation the RPV model splits a BRF field into an amplitude component (ρ_{0}) and an associated angular field describing the anisotropic behaviour of the surface under investigation. The latter is represented by the product of three separate functions accounting for both the illumination and viewing directions. Thus, overall, the RPV model formulates the BRF (or HDRF) of a surface as:

where

where θ and θ_{0} are the observation and illumination zenith angles, respectively. In the above formulation the relative azimuth angle ϕ is zero when the source of illumination is behind the sensor. The angular function `M`_{I}, known as the modified Minnaert function, permits the mathematical representation of the overall shape of the angular field through the parameter `k`. Specifically, `k` is close to 1.0 for a quasi-Lambertian surface (very limited angular variations of the spectral BRF field), `k` is lower than 1.0 when a bowl-shaped pattern dominates (the spectral BRF values increase with the view zenith angle in the orthogonal plane), and `k` is greater than 1.0 when a bell-shaped pattern dominates (i.e., the spectral BRF values decrease with the view zenith angle in the orthogonal plane). The other angular functions serve to add more complexity/flexibility to the anisotropy classes described above; they allow for the accounting of asymmetrical shapes due to the possible imbalance between the backward and forward scattering regions, as well as, for the backscatter enhancement due to the hot spot effect. The function `F`_{HG} is based on the Henyey-Greenstein function, and the parameter Θ establishes the degree of forward (positive Θ) or backward (negative Θ) scattering, depending on the sign. The `H` function allows to model the hotspot with its parameter ρ_{c}.

In order to verify/facilitate the output/implementation of the RPV model the following 5 files are provided:

`READ.ME`

- A short plain text file containing copyright information, useful addresses and bibliographic references.
`main1.f`

- A FORTRAN test programme to demonstrate the use of
`rpv.f`

and to generate a standard test data set. `rpv.f`

- The FORTRAN function to compute a bidirectional reflectance factor with the RPV model.
`rpv-in.dat`

- A typical input file required by
`main1.f`

. `rpv-out.dat`

- The output corresponding to the input in
`rpv-in.dat`

, when the verbose option is selected.

**NB:** It is important to read the angular definition statements in files `rpv.f`

and `main1.f`

. Furthermore, RAMI participants that have already downloaded the set of 5 RPV files available on the FAPAR website of the JRC should know that the files below are identical with the exception that they now enable the RPV model 1) to handle float angles in input, 2) to handle RPV input parameters with more than 2 decimal places, and 3) to handle 4 RPV-parameters in the input file.

For your convenience, the files above are packed into a zipped archive (`rpv.zip`

), which can be downloaded here.

On Linux these files can be compiled using a fortran compiler like `gfortran`

, `f95`

, etc. A statement like:

gfortran main1.f rpv.f -o rpv

whill thus generate an executable called `rpv`

. Next by typing:

rpv > rpv-out.dat.new

the `rpv`

binary will read the information stored in the file `rpv-in.dat`

and then generate an output file called `rpv-out.dat.new`

. The content of this file should be identical that of the `rpv-out.dat`

file provided above.

**NB:** With some fortran compilers - like `f77`

- small differences may arise in the floating point operations such that the BRFs in `rpv-out.dat.new`

and `rpv-out.dat`

may not be always exactly identical.

To ensure that the RPV model is encoded properly (beware of the azimuths) the following table provides graphical evidence as to the output of the RPV model for a variety of input parameter combinations:

ρ_{0} |
0.075 | 0.05 | 0.75 | 0.10 | 0.15 | 0.70 |

k |
0.55 | 0.95 | 0.95 | 0.60 | 0.080 | 0.95 |

Θ | −0.25 | −0.10 | 0.15 | −0.2 | −0.05 | 0.10 |

ρ_{c} |
0.075 | 0.05 | 0.75 | 0.10 | 0.15 | 0.70 |

RPV BRF output in principal plane (graph) |
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RPV BRF output in
entire hemisphere (graph) |
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RPV BRF output in entire hemisphere (data) |

Note: all graphs and datasets in this table were generated with the *Anisview* GUI tool written by Peter Vogt with contributions from M. M. Verstraete - both of the EC's JRC.

The following references describe the RPV model in detail or provide additional relevant information on to the use of this model:

- Rahman, H., M. M. Verstraete, and B. Pinty (1993) 'Coupled surface-atmosphere reflectance (CSAR) model. 1. Model description and inversion on synthetic data',
*Journal of Geophysical Research*, 98, 20,779-20,789. - Rahman, H., B. Pinty, and M. M. Verstraete (1993) 'Coupled surface-atmosphere reflectance (CSAR) model. 2. Semi-empirical surface model usable with NOAA Advanced Very High Resolution Radiometer data',
*Journal of Geophysical Research*, 98, 20,791-20,801. - Engelsen, O., B. Pinty, M. M. Verstraete, and J. V. Martonchik (1996) 'Parametric Bidirectional Reflectance Factor Models: Evaluation, Improvements and Applications',
*EC Joint Research Centre*, Technical Report No. EUR 16426 EN, 114 p.

#### RT model technicalities:

If applicable to the participating RT model, **cyclic boundary conditions** must be applied to all test cases unless specified otherwise on the individual measurement description pages. In the case of scenes with topography, this implies a repetition of the topographic features at scales equal to the scene dimension.

By default RT simulations are carried out with respect to a **reference plane**. Only those portions of the incoming and exiting radiation that pass through this reference plane are to be considered in the various measurements. Unless specified otherwise, the default reference plane covers the entire test case area (known as the "scene") and is located at the top-of-the-canopy height, that is, just above the highest structural element in the scene. The spatial extend of the reference plane can be envisaged as the (idealised) boundaries of the IFOV of a perfect sensor looking at a 'flat' surface located at the height level of the reference plane.

#### Statistical measures

##### Local angular model deviation, δ_{m}(θ_{v})

A criterion to quantify inter-model variability is the distance between BRF fields generated under identical geophysical and geometrical conditions. Specifically the local angular deviation, is computed to estimate how a given model `m`, behaves with respect to an ensemble of other models, at the specific exiting angle (θ_{v}) for an ensemble of `i` illumination conditions (out of a total of `N`_{θ0} illumination conditions), the structural scenario `s` (out of a total of `N`_{scenes} test cases), and the wavelength λ (out of a total of `N`_{λ} wavelengths). The deviation is obviously normalized for all simulations of the simulated BRF fields (`N` = `N`_{θ0} + `N`_{scenes} + `N`_{λ} + `N`_{models}):

Similar metrics can be designed to examine the model discrepancies for each geometrical condition of illumination and/or observation, scene and wavelength. They can all be derived following the generic form of the above equation. For instance the appropriate metrics for analyzing the model discrepancies separately for each wavelength would be:

##### Global angular model deviations, δ_{m}

An estimate of the global angular model deviation can be obtained by integrating over all available viewing conditions of the local angular model deviation: