4.1. Link Analysis, Doppler Focusing, and Resolution
[55] The SHARAD instrument is designed to radiate 10 Watts to achieve a specular surface Signalto(Galactic)Noise Ratio (SNR) of 50.6 ± 1.25 dB. The SNR depends on a number of factors and can be expressed as
where P_{p} is transmitted power, G is antenna gain, λ is wavelength, Γ is surface power reflection coefficient, R_{0} is distance from spacecraft to surface, T_{g} is the limiting galactic noise temperature, K is Boltzmann's constant, B_{N} is bandwidth, and L is propagation loss.
[56] Table 3 shows the SNR at the point of AnalogtoDigital Conversion (ADC) using the uncertainty only in the parameters that are not surface model dependent. Additional signal power is subsequently obtained by range (G_{r}) and Doppler (G_{a}) focusing or gain. The range compression gain is the timebandwidth product (Bτ) of the 10MHz bandwidth/85 μsec chirp signal, Bτ = 10·10^{6} Hz × 85·10^{−6} sec = 850 (29.3 dB). Because the chirp signal is weighted (done in ground processing, nominally using a Hanning function) to reduce range sidelobes, the pulse compression is not ideal and loses up to 1.8 dB in gain. The actual compression gain in range, G_{r}, is then taken as 27.5 dB.
Table 3. Link Analysis for SHARADParameter and Units  Nominal Value (Linear)  Uncertainty (Linear)  Requirement (Linear)  Contribution to SNR, dB  Margin, dB 


Signal Contribution 
Peak power  10 W  ± 2.5 W  >10 W  10  −1.25^{a} 
Antenna gain  1    0  
Wavelength  15 m  negligible  15 m  23.5  
Surface Fresnel reflectivity  0.1  model dep.   −10  
64π^{2}  631.7    −28  
Range  320 km^{b}    −110  
Receiver gain max     87  
Losses  3.2   N/A  −5  
Signal power at ADC     −2.5 dBm^{c}  

Noise Contribution 
Boltzmann constant  1.38 × 10^{−23} J/K    −228.6  
Galactic noise temp., T_{g}  10^{4.9} K    49  
Noise bandwidth (MHz)  10  negligible   70  
Receiver gain max     87  
Losses  3.2    −5  
Noise power at ADC     2.4 dBm  
SNR at ADC     −4.9 dB  −1.25 
[57] The azimuth (alongtrack or Doppler) gain is assumed, conservatively, to be no larger than the coherent integration of signals returned from the first Fresnel zone. The spacecrafttosurface distance, R_{0}, can vary between approximately 255 and 320 km over an MRO orbit. This leads to a Fresnel zone diameter range of
The azimuth gain is given by
where V_{S/C} is spacecraft horizontal velocity and L_{syn} is the synthetic aperture length, taken as the diameter of the Fresnel zone. Note that 1/PRF is the pulse repetition interval, PRI, in seconds, so
where L_{p} is the distance traveled by the spacecraft between pulses. Thus G_{a} can be interpreted simply as the number of pulses available for integration within a Fresnel zone. The realization of G_{a} is based on the premise that coherent integration can be performed for returned pulses from a ground patch of Fresnelzone size.
[58] The focused signaltonoise ratio, SNR_{f}, is then logarithmically the sum:
The dynamic range for a subsurfacesignaltonoise ratio of 3 dB is then 47.6 ± 1.25 dB. The penetration depth (maximum depth for a 3dB detection of a subsurface reflector) for the radar depends on (1) the path loss, a function of f, and integrated values of tanδ and ɛ_{r}; (2) the reflection loss at the subsurface reflector; (3) the transmission losses at shallower dielectric interfaces in the signal path, and (4) the volume scattering loss. The last two losses are not accounted for in equation (1). For the very simple model of a dielectric interface separating a dry, porous (30%) material above and the same material with icefilled pore spaces below, the penetration depth ranges from 200 m to 1500 m for a tanδ and ɛ_{r} range from 0.03 and 9 to 0.004 and 5, respectively.
[59] Without Doppler focusing, the horizontal ground resolution can be taken as ranging from the diameter of a Fresnel zone (∼3 km) to the pulselimited diameter [2√(C_{0}R_{0}/B), C_{0} = freespace velocity], about twice this value. The Fresnelzone resolution will be realized for relatively smooth surfaces. For more typical surfaces, the pulselimited diameter is more realistic. This is the crosstrack resolution range for SHARAD given in Table 2.
[60] Synthetic aperture focusing can improve the alongtrack (azimuth) resolution without sacrificing the SNR. The azimuth spatial resolution, ρ_{a}, is given by approximately (300km spacecraft altitude)
where for simplicity we have ignored the small difference between spacecraft and ground velocities. The azimuth resolution improvement C_{a}, is given approximately by
which is of course the ratio of the Fresnel zone diameter to the azimuth spatial resolution.
[61] The azimuth signal is highly oversampled by the 700 Hz PRF. As long as the spatial extent of the pulses involved in onboard coherent summing does not exceed the azimuth resolution, then there is no resolution degradation. The number of pulses that could be summed in 750 m is
The actual onboard presumming range is 1 to 32 samples (pulses), so this implies that even with the maximum amount of onboard presumming, additional coherent summing will have to be done on the ground to achieve the full SNR. The azimuth focusing process will, by design, integrate pulses that are oversampled along the ground track for the realizable Doppler spectrum. Note that before presumming on board, phase compensation with respect to the 20 MHz center frequency can be carried out to compensate estimated phase changes in the reflected signal due to the spacecraft radial velocity component and the surface slope. Information about these two quantities for specific data takes is uploaded to the spacecraft in the form of polynomial coefficients. The utility of the slope correction needs to be evaluated, since it carries uncertainties related to the spacecraft position predictions and surface topography, and it makes assumptions about the scattering properties of possible subsurface interfaces.
[62] It is of course possible to construct longer synthetic aperture lengths, and thus obtain finer azimuth resolution, for rougher surfaces that provide backscatter beyond a Fresnel zone distance of spacecraft motion. This could reduce surface clutter, particularly from compact scatterers. If a subsurface reflector is relatively smooth in this case, then the improved azimuth resolution would not be realized for the subsurface feature, but the ratio of power from subsurface reflections to surface scatterers should increase. The SHARAD azimuth (alongtrack) resolution given in Table 2 reflects a range about the nominal value of 750 m derived from the Fresnel zone synthetic aperture length (equation (7)).