napoli, 11.11.2008 – userest 2008 volcano monitoring via fractal modeling of lava flows gerardo di...

VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008

VOLCANO MONITORING VIA FRACTAL MODELING OF

LAVA FLOWS

Gerardo DI MARTINOAntonio IODICEDaniele RICCIO

Giuseppe RUELLO

Università degli Studi di Napoli “Federico II”

Dipartimento di Ingegneria Elettronica e delle Telecomunicazioni


OUTLINE

• Introduction

• Fractal Models

• SAR Raw Signal Simulation

• Fractal Imaging

• Conclusions


Introduction

VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008 ERS-1 --- Pixel Spacing: 20m

Information Content in SAR Images

VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008 TerraSAR-X --- Pixel Spacing: 3m

Information Content in SAR Images


Goals of the Work

•SAR image interpretation• SAR raw signal mechanism comprehrension

•Information preservation• Development of processing algorithms that preserve the information

•Information retrieval• Retrieval of the physical parameters required by the users


Fractal Models


Geometrical Models

Natural Scenes Urban Areas

Fractal Geometry Classical Geometry

Introduzione


d

ssxzxz

HH 22

2

2exp

2

1)()(Pr

H Hurst Coefficient 0<H<1 D=2-H

s Standard deviation at unitary distance [m1-H]

fBm parametrs

D is the fractal dimension ; =x-x’

The fractional Brownian motion (fBm)


HHHxx

sxzxzxxR

2222

2),(

The fBm is a continuous, not-differentiable, not-stationary process. Its autocorrelation function is:

FBm Model

It depends on x, x’ e

The structure function (the rms of increments at distance ):

HsV 22)( log2log2)(log HsV


oSS )(

Spectral Characterization

The spectrum evaluation requires space – frequency, or space – scale techniques, leading to :

DH 2821 HHH

sS21

1

)cos(2

0

Where the specrume parameters are related with H and s:

FBm Model

0< H <1 1< <3


xdxxxzxz )(

It is defined as the derivative of the fBm process. The fBm process is not derivable, therefore a regularization is needed:

Fractional Gaussian noise (fGn)

Such a process can be seen as a distribution and it can be derived as follows:

xdxxxzxz kkk 1)(

otherwise

xsex0

,01

By adopting the following function

xzxzxz

1

);(


If << the fGn autocorrelation function is :

FGn Model

22222 122);(

HHz HHsV

222 12)(

H

z HHsR

The structure function turns out to be:

Scales smaller than the resolution cell do not contribute to the SAR signal formation x


Spectrum Evaluation

FGn Model

12

22 1cos1sin212);(

Hz

kkHHskS

The fGn is a stationary process, therefore we can evaluate its spectrum as the derivative of its autocorrelation function:

If << 2k the spectrum is :

12

2 1sin21)(

Hzk

HHskS


SAR Raw Signal Simulation


Key Tool for Disaster MonitoringTo solve the inverse problem use is made of solvers

of the corresponding direct problem

SARSIMULATOR


1. Scene description

2. Electromagnetic scattering model

3. SAR raw signal formation

rrrxxg ;,

xrx ,,Reflectivity function

SAR unit response

rrrxxgrcjxrxdxdrrxh ,,

2exp ,, ,

SAR Raw Signal Simulation


The Simulator

SAR RAW SIGNAL SIMULATOR

z(x,y)

zmic

SAR PROCESSOR

SAR simulated image

Sensor parameters

We need both a macroscopic and a microscopic description of the scene.

We also need the electromagnetic parameters relevant to the scene.

,


Digital Elevation Model

3D representation of the Vesuvio volcano area.


Simulation Details

platform height 514 [ km]

platform velocity 7.6 [km/sec]

look-angle 20 [degrees]

azimuth antenna dimension 4.7 [ m]

range antenna dimension 7 [ m]

carrier frequency 9.65 [GHz]

pulse duration 25 [ microsec]

chirp bandwith 100 [ Mhz]

sampling frequency 110 [ Mhz]

pulse repetition frequency 4500 [ Hz]

Lava parameters aa Pahoehoe

Dielectric Constant 8 20

Conductivity [S/m] 0.01 1

Hurst coefficient 0.7 0.9

s [m1-H] 0.25 0.05

Sensor Parameters

Background

Dielectric Constant 4

Conductivity [S/m] 0.1

Hurst coefficient 0.8

s [m1-H] 0.16


Simulated SAR image

Simulation of the area in absence of lava flows

Resol. 1.69m x 3.99m azimuth x ground range Multilook 8 x 4


Simulation of the area with aa lava flow

Simulated SAR image



Simulation of the area with pahoehoe lava flow

Simulated SAR image



Fractal Imaging


Is the SAR image of a fractal surface fractal?

SAR Imaging

Can we retrieve the fractal parameters of the observed scene from SAR images?


Imaging Model

By using the SPM for the scattering evaluation (ipotesi di piccole pendenze), the image intensity is expressed as:

xpaaxi 10)(

Where p is the derivative of the surface; a0 and a1 are the the coefficients of the McLaurin series expansion of i(x,y) for small

values of p(x,y) and q(x,y)


First Results

The image i(x,y) has the same characterization of the fGn process, with mean a0 and standard deviation a1sxH-1

We can evaluate the structure funcion and the spectrum of the image:

);();( 21 xVaxV zI

);();( 21 xSaxS zI


Results

SAR image can be considered a fractal with H ranging from -1 and 0.

The SAR image is a self-affine Gaussian stationary process, NOT fractal

It means that a Hausdorff - Besicovitch fractal dimension can not be defined


Procedure Rationale

fBm Synthesis

(Weierstrass-Mandelbrot function)

Reflectivity Evaluation

(SPM model)

Spectrum and Variogram Estimation

Spectrum and Variogram Estimation

Comparison with theory

Comparison with theory

s H

Profile Image


Surface Synthesis

Simulated pahoehoe lava flow.Simulated aa lava flow.


Results: Azimuth cutsImage Theoretical Spectrum

Image Estimated Spectrum

Surface Theoretical Spectrum

Surface Estimated Spectrum

aa

lava flow

pahoehoe

lava flow

Image Theoretical Spectrum





Results: Range cutsImage Theoretical Spectrum




aa

lava flow

pahoehoe

lava flow

Image Theoretical Spectrum





Conclusions

A model-based approach for the monitoring of lava flows via SAR images was presented

A SAR simulator for new generation sensors provides a powerful instrument to drive detection techniques

A lava surface model was presented, based on a novel imaging model .


Future work

• Full Extension to 2D

• Inclusion of a reliable lava flow model

• Inclusion of a more appropriate speckle model (K-distribution) in the simulation procedure

• Inclusion of te speckle in the imaging analysis

napoli, 11.11.2008 – userest 2008 volcano monitoring via fractal modeling of lava flows gerardo di...

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