- 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
- 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
- If the atmospheric microwave absorption coefficient Î³ varies with height z as Î³ = Ï„ Î²exp (-Î²z), Ï„ being the optical thickness of
- 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
- 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
- Identify the nature of the spatial filters whose kernels are given below:(i) (ii)
- (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
- 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
- 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
- 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
- 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
- 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
- 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,
- 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
- 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
- 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
- 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
- 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,
- 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.
- 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
- 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
- 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
- (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
- 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 +
- 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
- 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
- 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
- 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
- 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
- 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.
- 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
- 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
- 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
- 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,
- 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
- 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
- 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),
- 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
- 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
- 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
- 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
- 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
- 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
- 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,
- 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.
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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].
- 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