Physical Principles Of Remote Sensing 3rd Edition W. G. Rees - Solutions

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When is Equation (2.46) valid?
The Planck formula can be inverted to obtain the brightness temperature from the spectral radiance as follows: Show that, for observations at a wavelength of 10.7 ?m, K1 and K2 have values of 851 W m-2 sr-1 ?m-1 and 1347 K respectively. Hence calculate the brightness temperature of a body whose
Show that the amplitude of fully developed speckle has a Rayleigh distribution. Assume that the signal is composed of a very large number of components, each of which has the same amplitude but a phase randomly selected from the range 0 to 2π.
If the atmospheric microwave absorption coefficient Î³ varies with height z as Î³ = Ï„ Î²exp (-Î²z), Ï„ being the optical thickness of the entire atmosphere, show that the brightness temperature measured by a passive microwave
The NASA Team algorithm for calculating sea-ice concentration from SSM/I data defines the polarisation ratio PR asand thegradient ratioGR aswhere, in both of these expressions,TfP represents the at-satellite brightness temperature of P-polarised radiation at a frequency of f GHz. It is known from
ASAR operates at a frequencyof 5.3GHz. Its satellite platformis 775kmabove the Earth’s surface and has a speed relative to the surface of 6.73 kms-1. The swath width is 100 kmand the incidence angle at the centre of the swath is 23°. The along-track resolution is 30m. Estimate (i) The
Identify the nature of the spatial filters whose kernels are given below:(i) (ii) (iii)1 3 1 1 3 1
(i) Estimate appropriate values, corresponding to those in Table 11.1, for printed paper as a data storage medium. (ii) Estimate the potential data storage requirements from a 5-year spaceborne remote sensing mission.Table 11.1 Medium/device Storage capacity (GB) Volume needed to store 1 TB
Use the data of Table 7.1 to show that the effective area of a rectangular or circular paraboloidal antenna is roughly equal to its geometrical area.Table 7.1 HPBW (degrees) D (dB) Antenna type Sidelobes (dB) Notes Monopole Short dipole Half-wave dipole Six-element Yagi (iso) (iso) (none) (iso)
A laser profiler is operated at a wavelength of 1μm from an altitude of 10000 m, and views at 45° to the nadir. Estimate the range corrections needed to account for the dry atmosphere and the water vapour component if the atmosphere contains 50 mm of precipitable water. The atmospheric pressure
An airborne scanning laser profiler transmits pulses at a frequency f0. Its field of view is scanned in a direction perpendicular to the aircraft€™s velocity through an angle ±Î¸ either side of the nadir direction, at a scanning frequency fs. The aircraft€™s
Explain how it is possible for the horizontal resolution of a radar altimeter to exceed the limit set by diffraction at the antenna, provided that the pulse duration tpis short enough, and derive the result that the resolution is given by 2(cHtp)1/2, where c is the speed of light and H is the range
A radar altimeter emits a pulse whose variation of power with time is Gaussian with a length of 3.00 ns between 1/e points. The pulse is reflected from a surface whose height distribution is Gaussian with a range of 1.00 m between the 1/e points. Calculate the time for the reflected pulse to rise
Prove Equation (8.16).Equation 8.16 dP, P,(t+2(h – H)/c)f(h)dh dt 0-
For microwave radiation propagating vertically through the Earth€™s atmosphere, the delay to a pulse as a result of the dry component of the atmosphere, expressed as an equivalent distance, is 2.33 meters irrespective of frequency. The corresponding figure for the water vapour component
A chirp signal, in which the angular frequency rises uniformly from Ï‰0€“Î”Ï‰/2 to Ï‰0+ Î”Ï‰/2 over a time T, can be written as exp(iÏ†(t)), where the phase Ï†(t) is given byfor |t| ‰¤ T/2. This
A radar transmits a power of 4 kW at a wavelength of 5 cm and over a range of 800 km. If the radar can detect a received signal of 10 - 16 W, calculate the required gain of the antenna, and hence estimate the area of the antenna, if it is to be able to detect values of σ0 down to –25 dB from an
A very simplified model of the radar back scattering coefficient of a sea surface is σ0 = A + Bcos 2ψ where ψ is the angle between the wind direction and the radar look azimuth, and A and B are as follows for Ku band scattering at 40° incidence angle, HH-polarisation:A = 0.8v - 30,B = 3.5 -
A SAR operates at a wavelength of 6 cm and is used for interferometric SAR observations. In position M, the radar has (x, y) coordinates (-400000, +800000) m, while in position S its coordinates are (-400100, +800000) m. A scatterer A located near the origin has coordinates (x, y).(i) Show that the
Prove Equations (10.2) to (10.5).Equation 10.2Equation 10.3Equation 10.4Equation 10.5 Av = 2GM RE R GM Av
Consider a satellite in orbit about a spherically symmetric planet. If the orbit is circular, the sub-satellite point travels uniformly along its circular track. If the orbit is not circular, however, the sub-satellite track is still circular, but the sub-satellite point moves along it at a
A satellite is in a circular sun-synchronous orbit with an inclination of 98.2°. It makes exactly 233 orbits of the Earth in exactly 16 days, and crosses the equator southbound at 0930h local time. Calculate the local time at which it crosses latitude 52° north in the southbound direction.
The nodal period of a satellite in a circular orbit of radius a and inclination i about the Earth is given by
The satellite Jason is in a circular orbit with a semimajor axis of 7714 km and an inclination of 66°.(i) Show that this orbit is suitable for altimetry.(ii) Comment on the suitability of the orbit for measuring ocean tides.
How many uniformly spaced geostationary satellites would be needed to ensure that all points on the Earth’s surface with latitudes less than 66.5° can be seen at an elevation angle of at least 10°?
A single band of an image has a histogram with a Gaussian distribution of standard deviations. Show that the theoretical limit to which the data can be compressed using Huffman coding is log2 Ïƒ + 2:05 exp(-x²/20?)dx = ov2n, x’exp(-x²/2o²)dx: = o³/2n. 0-
(i) Show that if A, S and E are, respectively, isotropic smoothing (averaging), sharpening (high-boost) and edge-detection (high-pass) filters described in the spatial domain, and I is the identity operator, the following are possible relationships:(a) S = aI + (1 - a)A,(b) E = b(I - S).Discuss the
The relationship between spectral radiance LÎ»and brightness temperature T in a thermal infrared band of a satellite sensor can be expressed asFor band 6 of the Landsat-7 ETM + instrument, the constants have values K1 = 666.1 Wm-2 sr-1 mm-1 and K2 = 1282.7 K. Use the satellite metadata
A passive microwave radiometer operating at 37 GHz has an effective area of 0.3 m2, and can detect a change of 0.9 K in its antenna temperature. It is operated from an altitude of 800 km to observe a single ice floe (whose brightness temperature is 253 K) surrounded by water (brightness temperature
At 10 GHz and a certain incidence angle, the apparent brightness temperatures of sea water, first-year ice and multi-year ice are 80 K, 252 K and 200 K respectively. At 37 GHz these figures become 119 K, 253 K and 168 K. If a microwave radiometer measures a brightness temperature of 180 K at both
A phased array is designed to operate at a wavelength of 6 cm. The antenna elements are spaced at intervals of 4 cm. (i) Show that if the beam is steered more than 30° away from the normal (θ = 0) direction it will respond to radiation from two directions instead of just one. (ii) Show
Verify the solution given in the text for the two-stream scattering model.
The cross-section of a glass prism is an equilateral triangle. The refractive index of the glass is 1.601 at a free-space wavelength of 0.4 μm and 1.569 at 0.7 μm. Show that if white light is incident on the prism at an angle of 40˚ to the surface normal, the spectrum from 0.4 to 0.7 μm will be
Sunlight is incident on a rough surface at an angle of 450 to the normal. Calculate the brightness temperature of the surface at wavelengths of 4 μm and 10 μm, assuming that the surface is a perfect Lambertian scatterer (i.e. one for which the diffuse albedo is 1) at both wavelengths. Ignore
Derive Equations (6.12) and (6.13).Equation 6.12Equation 6.13 F = Fo cos(ot – zVoCp/2K) exp(-zVoCp/2K) Fo |T = cos(ot — zVоСр/2K — п/4) еxp(-?VоСp/2K).
A two-look TIR radiometer views the same point on the Earth€™s surface at nadir and at 60 degrees from nadir. The brightness temperature of the detected radiation is T1and T2respectively. By assuming that the surface is a black body of temperature Tb, the atmosphere has a uniform
The output of the band 6 sensor of the Landsat-7 ETM + instrument, as represented by an integer pixel value in an image, is the nearest integer to 16.03(L + 3.2) + 1, where L is the at-satellite spectral radiance measured in units (W m–2 sr–1 µm–1) as the graph. Band 6 is a TIR band, and the
A surface shows daily temperature fluctuations with amplitude 23 K, lagging 2.3 hours behind the fluctuations in incoming solar radiation. Use the model of thermal inertia developed in this chapter to estimate (i) The value of ρcK,(ii) The amplitude of the variations in the incoming solar
Assume that the atmospheric absorption coefficient Î³ varies with height h asand use the simple model of limb-sounding geometry derived in Section 6.7.4 to show that the reduction in intensity I is given bywhere h0 is the height at which the radiation path passes closest to the
Show that Doppler broadening will dominate over pressure broadening of a spectral line for a gas at temperature T and pressure p provided that the wavelength of the spectral line is less than kT/pσ, where sis the collision cross-section defined in Equation (3.84).
The attenuation coefficient of a typical tropospheric aerosol is 0.1 km-1 at sea level, and the total optical thickness of the aerosol for a vertical path through the atmosphere is 0.2. By assuming that the density, and hence the attenuation coefficient, of the aerosol obeys a negative exponential
Estimate the illuminance at the Earth€™s surface due to solar illumination when the Sun is 450above the horizon. Use Equation (5.1), but approximate the function V(Î») shown in Figure 5.1 by a function that is 1 for Î» between 0.51 Î¼m and 0.61
A vertical aerial photograph reveals a tall building. The foot of one corner of the building has (x, y) coordinates (30.5, 62.0) (both measured in mm from the lower left-hand corner of the negative), and the top of the same edge appears at (19.0, 58.0). The corresponding coordinates for an adjacent
Show that Equation (2.31) can be approximated by Equation (2.33) at sufficiently low frequencies. exp(x) ≈ 1 + x when |x| << 1.
Show that the real and imaginary parts of the refractive index of a non-magnetic material are given by
A commonly used approximation to the absorption length is
Sea water has an electrical conductivity σ of typically about 4 S m-1. The real part of its dielectric constant at radio frequencies of 100 MHz and below is 88.2. By assuming that the imaginary part is given by Equation (3.21.2), find the absorption length of electromagnetic waves at 100 MHz and
Randomly polarised radiation at a wavelength of 3 cm is incident on a plane water surface at an angle of 83o to the normal. Calculate the Stokes vector, and hence polarisation state, of the reflected radiation. The dielectric constant of water at 3 cm is 63.1 – 32.1i.
Show that the diffuse albedo for specular reflection of perpendicularly polarised radiationand confirm that this expression has the correct limiting forms as the refractive index n tends to 1 and to infinity. Зп? — 2п—1 3(п+1)?
A rough surface has a BRDF proportional to cos(θ0)cos(θ1), where θ0 and θ1 are respectively the angles of the incident and scattered radiation measured from the surface normal. The BRDF has no azimuthal dependence. Show that, if the albedo for normally incident radiation is 1, the diffuse
Consider scattering form a surface with r.m.s. height variation of λ/2, where λ is the wavelength. Show that the stationary phase model can describe this scattering provided that the incident angle is less than about 60o and m < 0.60, whereas the scalar model requires m < 0.25, where m is
A typical pack of freshly fallen snow has a density of 100 kg m-3 and consists of ice crystals which can be modelled as spheres of radius 0.5 mm. Show that if the snow pack is sufficiently deep it would be expected to have a visible-band albedo close to 1, and estimate the depth necessary for this
Prove Equation 3.29.vvg = c2,
According to the Minnaert model of surface scattering, the BRDF R is given bywhere A and a are constants; Î¸0 and Î¸1 are the angles between the incident and scattered rays, respectively, and the surface normal; and the BRDF has no azimuthal dependence. Show that the
If scattering is insignificant, show that the radiative transfer equation may be written asand if hf = kT this can be rewritten asBy comparing these two equations, show thatExplain its significance. 2hf3 c2(ehf ]kt – 1) dLf dz -- L; = Ya аТь %3 7a(Т — Ть). dz
The absorption cross-section at 20 GHz of a water droplet with a radius of 48 μm is about 2 × 10-11 m2. Hence calculate the absorption coefficient of a cloud consisting of droplets of radius 48 μm with a number density of 2 million droplets per cubic metre. Would you expect this cloud to be
Assuming that the temperature T of the troposphere is given by the expression T = 288 - 0.0065z, where T is measured in kelvin and z is the altitude in meters, and that the pressure p in kPa is given by p = 101(T/288)5.25, calculate the density of the troposphere at an altitude of 10 km.
The meteorological visibility in fog is defined as the distance that gives an optical thickness of 4. If a typical fog has a mass density of 10-3 kg water per cubic metre and gives a visibility of 100 m, estimate the size of the water droplets.
A typical 35-mm camera with a standard lens and normal outdoor film can be assumed to have the following parameters: focal length f = 50 mm; spatial resolution l = 40 lp/mm; film format w = 25 x 35 mm. Estimate the performance of this system when used to obtain vertical aerial photographs from an
Use Equation (4.12) to show that, at low microwave frequencies, the absorption coefficient of a cloud of water droplets is given approximately by(i.e. independent of the droplet size), where ris the mass of water in kg per m3 of cloud. Assume that, for water at microwave frequencies,
Use the data of Table 4.3 to estimate the effective raindrop radius, number density and sedimentation rate for rain rates of 1 and 100 mm/hr, assuming that all the drops are spherical and have the same radius.
An aerial photographic system is used for stereophotography. The film negative is a square of sides, the focal length is f, the baseline is band the photographs (which are acquired with the camera pointing vertically downwards) are obtained from a height H above sea level. The stereophotographs are
Show that the Fourier transform of the Gaussian functionis proportional toand interpret the result. - ю) f(1) = exp – 202 ioto o?o² exp 2
Calculate the ratio of the spectral radiances of black bodies at 300 K and 6000 K at (i) A wavelength of 0.1 μm, (ii) A wavelength of 1 μm, (iii) A frequency of 1000 GHz, (iv) A frequency of 1 GHz.
When radiation having a Stokes vector S = [S0, S1, S2, S3] is incident on an antenna that receives only linearly x-polarised radiation, the detected power is proportional to S.P, where P = [1,1,0,0]. Show how the detected power varies with the polarisation state for radiation of a given flux
The electric field of an electromagnetic wave in free space is given byEx = 0Ey = Ecos(ωt - kx)Ez = 2Esin(ωt - kx)where E = 1 kVm-1. Find (i) the direction of propagation,(ii) the polarisation,(iii) the magnetic field and(iv) the flux density of the radiation.

Physical Principles Of Remote Sensing 3rd Edition W. G. Rees - Solutions

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