The Natural Radiation Environment

Natural targets are usually illuminated by the whole hemisphere of the sky, and thus receive direct solar flux and scattered sky light. Interactions at the surface result in a proportion of this incident radiation being reflected, either directly from the surface or after multiple interactions within the surface if the material is translucent to the incoming radiation. Natural targets are generally not perfectly diffuse (Lambertian) reflectors, and thus the intensity of the reflected flux varies with the angle with which it leaves the surface. Consequently, the radiation environment comprises two hemispherical distributions of electromagnetic radiation, one incoming and one outgoing, and it is the interaction between these two which constitutes the focus of interest in field spectroscopy. The radiation geometry of the field environment is shown the figure below.

In this figure the positions of the primary source of irradiation (the Sun) and the sensor are each defined by two angles, firstly the angle from the vertical (the zenith angle, theta) and secondly the angle measured in the horizontal plane from a reference direction (the azimuth angle, phi). Ignoring skylight, the energy from the Sun and the energy reflected to the sensor can be thought of as being confined to two slender elongated cones, each subtending a small angle at the target surface, termed solid angles and measured in steradians (sr). If these solid angles are infinitesimally small, the reflectance of the target can be defined as:

(1)

where dL is the reflected radiance per unit solid angle and dE is the irradiance per unit solid angle, and the subscripts i and r denote incident and reflected rays respectively Both the radiance and the irradiance vary in zenith and azimuth; hence to specify completely the reflectance field at the target, the reflectance must be measured at all possible source/sensor positions, resulting in the bidirectional reflectance distribution function (BRDF), (f). Although the BRDF is a useful theoretical concept it has drawbacks for practical applications, notably the requirement that dE be measured at the target surface and that both dE and dL be measured over infinitessimally small solid angles. For practical purposes, therefore, an alternative measure is needed to represent the directional reflectance of natural surfaces. Simplification is made to the concept of BRDF in two ways. First, the solid angle is increased to be large enough to contain measurable quantities of energy. Second, dE at the target is estimated, either from the global irradiance measured a short distance above the target, or by estimating it from the amount of energy reflected by a calibrated reflectance panel.

In the first method, an upward-looking spectrometer with a cosine-corrected receptor is used (that is, a sensor which shows no dependence upon the zenith or azimuth angle of the incident flux). This results in a property termed the bidirectional reflectance factor (BRF):

(2)

where dE is the irradiation as measured by the upward-looking sensor and k is a correction factor relating the signal from the cosine-corrected receptor to that expected from a perfectly diffuse white panel. The BRF may also be determined by comparing the reflected radiance from the target to that from a reflectance panel specified to be perfectly diffuse, completely reflecting and viewed under the same irradiation conditions and in the same geometry as the target. Because in practice a perfectly reflecting panel does not exist, a correction is made to account for the spectral reflectance of the panel. Thus:

(3)

where dLt is the radiance of the target and dLp is the radiance of the panel under the same specified conditions of illumination and viewing, and k is the panel correction factor. Note that k is also dependent upon the angular configuration, as perfectly Lambertian standard panels are impossible to achieve in practice. For a perfectly diffuse surface, the BRF (R) may be related to the BRDF as follows:

(4)

The term 'bidirectional' in this context refers to the two angles involved, one for the source position and one for the sensor position. In the above equations, all terms are also dependent upon wavelength, but for clarity the subscript for wavelength has not been shown.

The use of BRF instead of BRDF to represent the spectral reflectance of natural targets involves several further assumptions:

1. The angular field-of-view of the sensor is as small as possible (less than 10 degrees).
2. The reflectivity panel must fill the field-of-view of the sensor.
3. There should be no change in the irradiation amount or its spectral distribution between measurement of dLt and dLp (or dE).
4. Direct solar flux dominates the irradiation field. That is, the Sun is assumed to shine out of a black sky and skylight is ignored.
5. The sensor responds in a linear fashion to changes in radiant flux.
6. The reflectance properties of the standard panel are known and invariant over the course of the measurements.

Of these assumptions, that which is always violated in the field situation is the absence of sky light, which results in field measurements of BRF being made under an irradiance distribution which may be significantly different from the slender elongated cone referred to above. In recognition of this fact, the term hemispherical-directional reflectance factor (HDRF) is used to refer to BRF measured in the field, and hemispherical-conical reflectance factor (HCRF) is also used in order to emphasize that the reflected radiance is also measured over a finite solid angle and thus is not strictly directional.

Both methods of measuring BRF in the field result in a value of reflectance which is dependent upon solar elevation and azimuth and on viewing elevation and azimuth. Clearly, estimating the complete BRDF of a surface would involve a great many measurements, which in the field situation would be subject to error due to changes in solar position with time. Consequently, although it must be accepted that the BRDF represents the fullest statement possible of the spectral reflectance of a natural target, in practice it must be approximated by the BRF, which is turn, is often only approximated by being sampled, either over a limited range of wavelengths, or over a limited set of source positions, or over a limited set of sensor positions. Often, the rationale behind the choice of strategy for sampling the BRF is not stated. In some cases it may be self-evident that only a particular range of wavelengths is required, perhaps to match a particular remote sensor for example. In other cases, the choice of nadir viewing, or observations around local solar noon, may be made on logistic grounds with little or no reference to the degree to which the subsequent data represent the true BRDF of a target.

The terminology for reflectance measurements introduced by Nicodemus and co-workers (Nicodemus, 1970; 1976; Nicodemus et al., 1977) has been updated by Martonchik et al. (2000) and Schaepman-Strub et al. (2006). Four configurations are achievable in practice under field conditions (see figure below). Of these, the two most commonly measured are the hemispherical-conical reflectance factor and the bi-hemispherical reflectance factor (aka albedo).

	Biconical reflectance factor . In which the incident flux is contained within a slender cone and the reflected flux is measured using a sensor with an apertured field-of-view.
	Conical-hemispherical reflectance factor . In which the incident flux is contained within a slender cone and the total reflected flux is measured.
	Bi-hemispherical reflectance factor or albedo. In which the reflected flux into the total hemisphere is expressed as a proportion of that incident on the surface from the whole hemisphere (sun + sky).
	Hemispherical-conical reflectance factor (HCRF) . In which the incident flux originates from the whole hemisphere (sun + sky), and the reflected flux is measured using a sensor with an apertured field-of-view. This is the most common field spectral measurement in remote sensing.

Practical measurement configurations used in field spectroscopy (after Schaepman-Strub et al., 2006).

In the figure above the yellow shaded regions represent the incident flux and the green shaded regions represent the reflected flux.