Prediction of permittivity using received backscatter values on Greenland - PowerPoint Presentation

Slide1

Prediction of permittivity using received backscatter values on Greenland

Kevin and Kyra Moon

EE 670

December 1, 2011Slide2

BackgroundMotivation

Problem

Theoretical model for backscatterSimulationsEstimatorsMLMAPExample of estimatorsResultsConclusion

OutlineSlide3

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

BackgroundSlide4

In the highest

part of Greenland, the snow never melts

Called the dry snow zoneUsed frequently for calibration purposesHowever, some annual variation in the backscatter has been detected which is consistent from year to yearBackgroundSlide5

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 variationSlide6

We decided to test if received backscatter values could predict changes in permittivityThe 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

ProblemSlide7

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 Slide8

The equations for our model were

(temperature to density)

(this is approximately linear)

(density to permittivity)

really complicated (several lines of equations)

Theoretical ModelSlide9

Theoretical ResultsSlide10

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

SimulationSlide11

Randomly generated temperatures using the histogramNormalized the histogram

Calculated the

cumulative distribution functionGenerated uniformly distributed random numbers between 0 and 1Assigned 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

SimulationSlide12

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 noiseSimulationSlide13

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)

EstimationSlide14

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)Slide15

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 LikelihoodSlide16

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 LikelihoodSlide17

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)Slide18

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-posterioriSlide19

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)Slide20

Results at 13 dB of SNRSlide21

MAP has superior performance to ML because there is more information availableHowever, 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