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Mathematical Applications in Remote Sensing

This field is crucial for understanding the Earth and other celestial bodies by studying the energy they emit or reflect. Electromagnetic waves, such as visible light, infrared, and radio waves, are characterized by properties like wavelength, frequency, and amplitude. These properties are mathematically related through equations that allow scientists to calculate and predict the behavior of these waves. For instance, the speed of light (c) is the product of wavelength (λ) and frequency (ν), expressed as c = λν. Additionally, laws such as Wien's displacement law and the Stefan-Boltzmann law provide mathematical frameworks to understand how the temperature of an object affects the radiation it emits.

Figures (9)

Figure 6.2 One-dimensional sinusoidal waves show the relationship among wavelength, frequency, and speed. The wave with the shortest wavelength has the highest frequency. Amplitude is one-half the height of the wave from peak to trough.
Figure 6.3 Portions of the electromagnetic spectrum are shown in order of decreasing frequency and increasing wavelength. (credit “Cosmic ray": modification of work by NASA; credit “PET scan": modification of work by the National Institute of Health; credit “X-ray": modification of work by Dr. Jochen Lengerke; credit “Dental curing": modification of work by the Department of the Navy; credit “Night vision": modification of work by the Department of the Army; credit “Remote": modification of work by Emilian Robert Vicol; credit “Cell phone": modification of work by Brett Jordan; credit “Microwave oven": modification of work by Billy Mabray; credit “AM radio": modification of work by Dave Clausen)
Figure 6.4 Radio and cell towers are typically used to transmit long-wavelength electromagnetic radiation. Increasingly, cell towers are designed to blend in with the landscape, as with the Tucson, Arizona, cell tower (right) disguised as a palm tree. (credit left: modification of work by Sir Mildred Pierce; credit middle: modification of work by M.O. Stevens)
Figure 6.5 This schematic depicts how amplitude modulation (AM) and frequency modulation (FM) can be used to transmit a radio wave.
Figure 6.6 Interference fringe patterns are shown for light passing through two closely spaced, narrow slits. The spacing of the fringes depends on the wavelength, with the fringes being more closely spaced for the shorter-wavelength blue light. (credit: PASCO)
Figure 6.7 A vibrating string shows some one-dimensional standing waves. Since the two end points of the string are held fixed, only waves having an integer number of half-wavelengths can form. The points on the string between the end points that are not moving are called the nodes.
Figure 6.8 Two-dimensional standing waves can be visualized on a vibrating surface. The surface has been sprinkled with a powder that collects near the nodal lines. There are two types of nodes visible: radial nodes (circles) and angular nodes (radii).
Figure 6.9 The spectral distribution (light intensity vs. wavelength) of sunlight reaches the Earth's atmosphere as UV light, visible light, and IR light. The unabsorbed sunlight at the top of the atmosphere has a distribution that approximately matches the theoretical distribution of a blackbody at 5250 °C, represented by the blue curve. (credit: modification of work by American Society for Testing and Materials (ASTM) Terrestrial Reference Spectra for Photovoltaic Performance Evaluation)
Figure 6.10 Blackbody spectral distribution curves are shown for some representative temperatures.

Mathematical applications in remote sensing involve using equations to understand and predict how electromagnetic waves behave. The speed of light (c) is a constant and is the product of wavelength (λ) and frequency (ν), shown by the equation c = λν. This relationship helps scientists calculate how these waves move and interact with objects in space.

Wien's displacement law is another key concept. It states that the peak wavelength of radiation emitted by an object is inversely proportional to its temperature. This means hotter objects emit shorter wavelengths, like blue light, while cooler objects emit longer wavelengths, like red light.

The law is expressed as λ_peak = b / T, where b is a constant and T is the temperature. This helps scientists determine the temperature of distant objects, like stars, based on the light they emit. The Stefan-Boltzmann law builds on this by showing that the total energy radiated by an object is proportional to the fourth power of its temperature.

The equation is M = σT⁴, where M is the energy radiated and σ is the Stefan-Boltzmann constant. This law is crucial for calculating how much energy objects, like planets or stars, emit based on their temperature.

Key Points

  • Wien's displacement law states that the black-body radiation curve for different temperatures will peak at different wavelengths that are inversely proportional to the temperature.
  • The Stefan–Boltzmann law states that the total energy radiated per unit surface area per unit time by an ideal absorber/emitter or black body is directly proportional to the fourth power of the black body's temperature.

Terms

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