Prediction of permittivity using received backscatter values on Greenland
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Prediction of permittivity using received backscatter
values on GreenlandKevin and Kyra Moon
EE 670December 1, 2011
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Background◦ Motivation◦ Problem
Theoretical model for backscatter Simulations Estimators
◦ ML◦ MAP
Example of estimators Results Conclusion
Outline
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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
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In the highest part 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
Background
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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
Annual variation
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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
Problem
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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
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The equations for our model were (temperature to density) (this is approximately linear)
(density to permittivity) really complicated (several lines of equations)
Theoretical Model
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Theoretical Results
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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
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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
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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
Simulation
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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)
Estimation
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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 (ML)
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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
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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 Likelihood
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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 (MAP)
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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.
Maximum a-posteriori
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MAP vs ML example
(or equivalently, permittivity or backscatter)
True value
Received value
What ML would estimate (minimize distance from received)
What MAP would estimate (this value is a lot more likely, even if the distance from received is further)
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Results at 13 dB of SNR
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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
Conclusions