The Principles of Field Spectroscopy

Radiometry

Field spectroscopy, as a technique, is based upon the science of radiometry. Radiometry, as defined by Manolakis et al., is “the science that governs the quantification of electromagnetic energy”. Effectively, it is the science to which we can ascribe physical units - which describe both the amount of energy and the directionality of that energy – to measurements of light.

The basic quantity of radiometry is energy, usually described in units of Watts (W), and the basic concept is the flux (Φ), which characterises the energy and also the directionality of the flow of energy.

Radiometry as a science is a very broad field, and we encourage browsers to look at our reading list for more information on the topic. For the purposes of this introduction, we will focus only on the four quantities of pertinence to field spectroscopy – irradiance, radiant exitance, radiance, and reflectance – and provide a brief but comprehensive overview of them.

Irradiance and Radiant Exitance

Irradiance is the radiant flux received or flowing through a real or imaginary area. Effectively, this means the total energy (in W) received by an area (usually, $m^2$). Irradiance is given the symbol E. In field spectroscopy, and more generally in radiometry, the irradiance is the totality of the flux from the entire hemisphere that is received by the area. This, irradiance is properly defined as the flux per unit area integrated across the hemisphere (0 – 180°) in both zenith (θ) and azimuth (ϕ). Mathematically defined, this is given by:

$$E =\int_{0}^{2\pi} \int_{0}^{\pi} \frac{d\Phi}{dA}. cos(θ) sin(θ) dθ dφ$$

While irradiance is the flux received by a surface, the radiant exitance is the analogous concept describing the flux emitted by a surface, integrated across the entire hemisphere that is emitted by the area. It is given the symbol $M_{e}$.

In field spectroscopy, if we are using a spectrometer in passive mode – i.e., using the Sun or another natural light source for illumination, and not an active source, such as a lamp – the irradiance component of measurements is often referred to as downwelling irradiance which is simply the power per unit area received by the Earth’s surface from the Sun.

It is important to note that both irradiance and radiant exitance describe the distribution of flux with respect to position only, and are not dependent on the direction of ray propagation.

Radiance

As mentioned, a fundamental concept in radiometry is the characterisation of the directionality of energy flow. To better understand what this means in terms of radiance, we will first introduce the concept of the solid angle.

Consider an area, $S$, that exists outside of a sphere. If lines were drawn from the outside perimeter of $S$ to the centre point of the sphere, they would pierce a portion of the sphere wall, which could be described as an area, $A$. The solid angle, $\omega$, is defined as the ratio of the area on the sphere wall, $A$, over the radius of the sphere squared, $r^2$:

$$\omega = \frac{A}{r^2}$$

This is similar to the definition of the radian, which is defined as the angle subtended at the centre of a circle by an arc ($s$) that is equal to the circle’s radius i.e s / r, but extended to three dimensional space. The unit of the solid angle is the steradian, $sr$, and – by considering the geometry of a sphere – there are 4$\pi$ steradians in a sphere, and 2$\pi$ steradians in one hemisphere.

Radiance ($L$) is defined as the flux projected per unit solid angle leaving or falling upon a surface area, $A$. By inclusion of the solid angle, the direction of ray propagation is considered. This last point is critical, because it allows the flux received by an optical system (such as the human eye, or a field spectrometer) which is looking at the surface at a particular angle and at a particular field of view to be characterised. Mathematically defined, radiance is given by:

$$L = \frac{d^2 \Phi}{dA d\omega}$$

If radiance is known, the radiant exitance can be determined – with the proviso that the surface from which the radiance is emitted has specific properties which will be described below –1:

$$M_{e} = \pi L$$

In field spectroscopy, you may see radiance in general described as upwelling radiance, i.e. the radiance as measured under passive acquisition when viewing an object on the Earth’s surface.

Reflectance

Reflectance (or albedo) is the primary radiometric quantity with which field spectroscopy – and remote sensing in general – concerns itself with, and can be broadly defined as the ratio of the energy leaving a surface to that arriving at the surface. The most commonly used symbol for reflectance is $\rho$.

Considering the discussion above regarding radiance, radiant exitance, and irradiance, this definition can be extended to give the total reflectance, i.e., the ratio of all energy leaving the surface integrated across the hemisphere, to all incoming energy to the surface integrated across the hemisphere:

$$ \rho_{total} = \frac{M_{e}}{E}$$

If we have measured radiance, then we can also determine the reflectance via conversion to $M_{e}$ as detailed in the previous section:

$$ \rho_{total} = \frac{\pi L}{E}$$

In field spectroscopy, if we take simultaneous measurements of irradiance and radiance to determine reflectance, we are employing what is known as a cos-conical configuration.

As mentioned, the conversion from $L$ to ${M_{e}}$ rests on an important assumption regarding the properties of the surface being mentioned, namely, that we assume it has Lambertian, or ideal diffuse, reflectance properties. With ideal reflectance, a surface, regardless of which angle it is viewed from, will have the same measured radiance. Objects and surfaces can have two other forms of reflectance:

  • Specular – where all incoming energy is reflected into a single outgoing direction.
  • Diffuse – where incoming energy is reflected at many angles.

Most objects show characteristics of both, and without full understanding of the how angular changes in the incoming energy in both azimuth and zenith affect the reflectance properties of the object, it can be difficult to infer total reflectance from a surface.

Alternatively, rather than take measurements of both $M_{e}$ and $E$ simultaneously, reflectance measurements of a surface can be taken if the target radiance is compared to that of a Lambertian reference panel.

The most common material used for reference panels is Spectralon®

Mathematically expressed, $\rho$ measured in this way – where $\kappa$ is the absolute, characterised reflectance of the panel used – is described by:

$$\rho = \frac{L_{target}}{L_{panel}} \kappa $$

This is known as a bi-conical configuration, and is the principal method by which reflectance measurements are taken using the facility’s field spectrometers.

Summary

To conclude, the radiometric quantities which we primarily deal with in field spectroscopy are:

  • Spectral irradiance – the radiant flux received or flowing through a real or imaginary area, integrated across the entire hemipshere received by the area, symbol $E$, and commonly expressed in units $W m^{-2} nm^{-1}$.
  • Spectral radiant exitance – the flux emitted by a surface, integrated across the entire hemisphere that is emitted by the area, symbol $M_{e}$, and commonly expressed in units $W m^{-2} nm^{-1}$.
  • Spectral radiance – the flux projected per unit solid angle leaving or falling upon a surface area, symbol $L$, and commonly expressed in units $W m^{-2} sr^{-1} nm^{-1}$.
  • Spectral reflectance – the ratio of the energy leaving a surface to that arriving at the surface, symbol $\rho$, and commonly expressed as either a ratio from 0 to 1 or as a percentage from 0 to 100%.

Most field spectroscopy is focused on determining the latter of these quantities, and subsequent sections will focus on how to measure this quantity. However, it should be noted that the facility provides instruments and instrument configurations which can be used to measure all of these quantities, as well as additional quantities such as spectral transmission and spectral absorption. For more details, please contact us at fsf@ed.ac.uk.

  1. The surface emits over 2$\pi$ steradians, but integrating over that solid angle with the third cosine effect yields a factor of 2, for a total $pi$ factor. ↩