Prediction of permittivity using received backscatter values on Greenland

Prediction of permittivity using received backscatter values on Greenland Kevin and Kyra Moon EE 670December 1, 2011

Outline • Background • Motivation • Problem • Theoretical model for backscatter • Simulations • Estimators • ML • MAP • Example of estimators • Results • Conclusion

Background • To get an “image” of the ground, a radar or satellite sends out an electromagnetic wave and measures the return it receives from the ground • The returned value is called “backscatter”, or . • There are many different factors affecting the brightness of • Roughness of surface • Conductivity of surface

Background • In the highestpart of Greenland, the snow never melts • Called the dry snow zone • Used frequently for calibration purposes • However, some annual variation in the backscatter has been detected which is consistent from year to year

Annual variation • This variation cannot be caused by melt because it does not drop below a specific threshold • Temperatures are typically between • However, it is possible that increasing temperatures do change the permittivity of the snow, thus changing the backscatter

Problem • We decided to test if received backscatter values could predict changes in permittivity • The answer to this would provide insight into possible causes for the annual variation • If backscatter cannot predict changes in permittivity, then it is likely there are other factors affecting the annual variation

Theoretical Model • We created a model relating permittivity to backscatter (at least for snow) • Because knowing the temperature helps us predict the permittivity more accurately, we found a relationship between temperature and permittivity • This model required an intermediate step relating temperature to snow density and snow density to permittivity

Theoretical Model • The equations for our model were • (temperature to density) • (this is approximately linear) • (density to permittivity) • really complicated (several lines of equations)

Theoretical Results

Simulation • We then ran a simulation to see if backscatter could predict permittivity. • We assumed that the underlying temperature data was weighted based on real data

Simulation • Randomly generated temperatures using the histogram • Normalized the histogram • Calculated the cumulative distribution function • Generated uniformly distributed random numbers between 0 and 1 • Assigned each random number the temperature value corresponding to the same index as the closest value of the cdf that was still less than the random number

Simulation • For a given temperature, the snow density, permittivity, and corresponding backscatter were calculated using the earlier equations • The backscatter was then corrupted with additive white Gaussian noise • This simulated real noise between the ground and the satellite receiver, including atmospheric and instrumental noise

Estimation • To estimate the actual permittivity using the noisy received backscatter , we used two decision rules • ML: We assumed each permittivity was equally likely • MAP: We assumed each permittivity was weighted according to the histogram (since permittivity is a function of temperature)

Maximum Likelihood (ML) • The maximum likelihood rule is • That is, we choose the value of permittivity which makes receiving most likely. • Since is a function of permittivity, this is equivalent to

Maximum Likelihood • The goal is to choose , because that will give us the correct permittivity • Note that , where is a Gaussian random variable with 0 mean and variance related to SNR (white noise) • Hence, • This is a Gaussian random variable

Maximum Likelihood • To maximize this probability, the ML rule tells us to minimize the distance between and • If the noise didn’t move too far from , then this will give us the correct backscatter • The permittivity corresponding to the estimated backscatter is chosen to be .

Maximum a-posteriori (MAP) • The maximum a-posteriori rule is • We no longer assume that every permittivity is equally likely • This makes more sense given the distribution of temperatures

Maximum a-posteriori • The derivation for MAP estimation is similar to that of ML • When we reach , • rather than just choosing which minimizes the distance , we choose which maximizes that constraint and is deemed likely by the histogram.

MAP vs ML example What MAP would estimate (this value is a lot more likely, even if the distance from received is further) True value What ML would estimate (minimize distance from received) Received value (or equivalently, permittivity or backscatter)

Results at 13 dB of SNR

Conclusions • MAP has superior performance to ML because there is more information available • However, neither estimator is a good predictor of permittivity based on received backscatter values • It is likely that the annual variation noticed in Greenland is caused by more than just changes in permittivity

Prediction of permittivity using received backscatter values on Greenland