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

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


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into the water. This is the Brewster effect.

      2.1.6 Coherency

      The coherency time Δt is defined as the period over which there is strong correlation of the field amplitude. More specifically, it is the time after which two waves at ν and ν + Δν are out of phase by one cycle; that is, it is given by:

      (2.18)equation

      The coherence length is defined as

      (2.19)equation

      Two waves or two sources are said to be coherent with each other if there is a systematic relationship between their instantaneous amplitudes. The amplitude of the resultant field varies between the sum and the difference of the two amplitudes. If the two waves are incoherent, then the power of the resultant wave is equal to the sum of the power of the two constituent waves. Mathematically, let E1(t) and E2(t) be the two component fields at a certain location. Then the total field is

      (2.20)equation

      The average power is

      (2.21)equation

      If the two waves are incoherent relative to each other, then images and P = P1 + P2. If the waves are coherent, then images. In the latter case, we have:

equation

      This is the case of optical interference fringes generated by two overlapping coherent optical beams. The bright bands correspond to where the energy is above the mean and the dark bands correspond to where the energy is below the mean.

      2.1.7 Group and Phase Velocity

      (2.22)equation

      If we have two waves characterized by (ω − Δω, k − Δk) and (ω + Δω, k + Δk), then the total wave is given by

      (2.23)equation

Graph depicts phase velocity.

      (2.24)equation

      As Δω and Δk are assumed to be small, then we can write

      (2.25)equation

      If the medium is nondispersive, then

      (2.26)equation

      This implies that

      (2.27)equation

      (2.28)equation

      However, if the medium is dispersive (i.e., ω is a nonlinear function of k), such as in the case of ionospheres, then the two velocities are different.

      2.1.8 Doppler Effect

Graph depicts group velocity.

      The relationship between νd and ν is

      (2.29)equation

      (2.30)equation

      which can be written as

      (2.31)equation