Introduction to the Physics and Techniques of Remote Sensing. Jakob J. van Zyl

vector

       H = magnetic vector

       B = induction vector

       μ0, ∈0 = permeability and permittivity of vacuum

       μr, ∈r = relative permeability and permittivity

       ∇. = divergence

       ∇x = curl

      Maxwell’s concept of electromagnetic waves is that a smooth wave motion exists in the magnetic and electric force fields. In any region where there is a temporal change of the electric field, a magnetic field appears automatically in that same region as a conjugal partner and vice‐versa. This is expressed by the above coupled equations.

      2.1.3 Wave Equation and Solution

      In homogeneous, isotropic, and nonmagnetic media, Maxwell’s equations can be combined to derive the wave equation:

      (2.7)

      where ∇² is the Laplacian. In the case of a sinusoidal field:

      (2.8)

      where

      (2.9)

      Usually μ_r = 1 and ∈_r varies from 1 to 80 and is a function of the frequency. The solution for the above differential equation is given by:

(2.10)

      where A is the wave amplitude, ω is the angular frequency, ϕ is the phase, and k is the wave vector in the propagation medium (speed of light in vacuum). The wave frequency ν is defined as ν = ω/2π.

Remote sensing instruments exploit different aspects of the solution to the wave equation in order to learn more about the properties of the medium from which the radiation is being sensed. For example, the interaction of electromagnetic waves with natural surfaces and atmospheres is strongly dependent on the frequency of the waves. This will manifest itself in changes in the amplitude (the magnitude of A in (2.10)) of the received wave as the frequency of the observation is changed. This type of information is recorded by multispectral instruments such as the LandSat Thematic Mapper and the Advanced Spaceborne Thermal Emission and Reflection Radiometer. In other cases, one can infer information about the electrical properties and geometry of the surface by observing the polarization (the vector components of A in (2.10)) of the received waves. This type of information is recorded by polarimeters and polarimetric radars. Doppler lidars and radars, on the other hand, measure the change in frequency between the transmitted and received waves in order to infer the velocity with which an object or medium is moving. This information is contained in the angular frequency ω of the wave shown in (2.10). The quantity kr − ωt + φ in (2.10) is known as the phase of the wave. This phase changes by 2π every time the wave moves through a distance equal to the wavelength λ. Measuring the phase of a wave therefore provides an extremely accurate way to measure the distance that the wave actually travelled. Interferometers exploit this property of the wave to accurately measure differences in the path length between a source and two collectors, allowing one to significantly increase the resolution with which the position of the source can be established or to measure slight displacements.

      2.1.4 Quantum Properties of Electromagnetic Radiation

Maxwell’s formulation of electromagnetic radiation leads to a mathematically smooth wave motion of fields. However, at very short wavelengths, it fails to account for certain significant phenomena when the wave interacts with matter. In this case a quantum description is more appropriate.

      The electromagnetic energy can be presented in a quantized form as bursts of radiation with a quantized radiant energy Q, which is proportional to the frequency ν:

      (2.11)

      where h = Planck’s constant = 6.626 × 10⁻³⁴ joule second. The radiant energy carried by the wave is not delivered to a receiver as if it is spread evenly over the wave, as Maxwell had visualized, but is delivered on a probabilistic basis. The probability that a wave train will make full delivery of its radiant energy at some place along the wave is proportional to the flux density of the wave at that place. If a very large number of wave trains are coexistent, then the overall average effect follows Maxwell’s equations.

      2.1.5 Polarization

      An electromagnetic wave consists of a coupled electric and magnetic force field. In free space, these two fields are at right angles to each other and transverse to the direction of propagation. The direction and magnitude of only one of the fields (usually the electric field) is sufficient to completely specify the direction and magnitude of the other field using Maxwell’s equations.

      The polarization of the electromagnetic wave is contained in the elements of the vector amplitude A of the electric field in Equation (2.10). For a transverse electromagnetic wave, this vector is orthogonal to the direction in which the wave is propagating, and therefore we can completely describe the amplitude of the electric field by writing A as a two‐dimensional complex vector:

(2.12)

Here we denote the two orthogonal basis vectors as for horizontal and for vertical. Horizontal polarization is usually defined as the state where the electric vector is perpendicular to the plane of incidence. Vertical polarization is orthogonal to both horizontal polarization and the direction of propagation, and corresponds to the case where the electric vector is in the plane of incidence. Any two orthogonal basis vectors could be used to describe the polarization, and in some cases the right‐ and left‐handed circular basis is used. The amplitudes, a_h and a_v, and the relative phases, δ_h and δ_v, are real numbers. The polarization of the wave can be thought of as that figure that the tip of the electric field would trace over time at a fixed point in space. Taking the real part of (2.12), we find that the polarization figure is the locus of all the points in the h‐v plane that have the coordinates E_h = a_h cos δ_h; E_v = a_v cos δ_v. It can easily be shown that the points on the locus satisfy the expression

(2.13)