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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
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
OUTLINE
• Introduction
• Fractal Models
• SAR Raw Signal Simulation
• Fractal Imaging
• Conclusions
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
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
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
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
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
Fractal Models
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
Geometrical Models
Natural Scenes Urban Areas
Fractal Geometry Classical Geometry
Introduzione
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
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)
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
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
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
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
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
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
);(
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
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
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
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
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
SAR Raw Signal Simulation
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
Key Tool for Disaster MonitoringTo solve the inverse problem use is made of solvers
of the corresponding direct problem
SARSIMULATOR
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
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
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
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.
,
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
Digital Elevation Model
3D representation of the Vesuvio volcano area.
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
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
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
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
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
Simulation of the area with aa lava flow
Simulated SAR image
Resol. 1.69m x 3.99m azimuth x ground range Multilook 8 x 4
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
Simulation of the area with pahoehoe lava flow
Simulated SAR image
Resol. 1.69m x 3.99m azimuth x ground range Multilook 8 x 4
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
Fractal Imaging
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
Is the SAR image of a fractal surface fractal?
SAR Imaging
Can we retrieve the fractal parameters of the observed scene from SAR images?
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
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)
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
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
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
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
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
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
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
Surface Synthesis
Simulated pahoehoe lava flow.Simulated aa lava flow.
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
Results: Azimuth cutsImage Theoretical Spectrum
Image Estimated Spectrum
Surface Theoretical Spectrum
Surface Estimated Spectrum
aa
lava flow
pahoehoe
lava flow
Image Theoretical Spectrum
Image Estimated Spectrum
Surface Theoretical Spectrum
Surface Estimated Spectrum
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
Results: Range cutsImage Theoretical Spectrum
Image Estimated Spectrum
Surface Theoretical Spectrum
Surface Estimated Spectrum
aa
lava flow
pahoehoe
lava flow
Image Theoretical Spectrum
Image Estimated Spectrum
Surface Theoretical Spectrum
Surface Estimated Spectrum
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
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 .
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11.11.2008 – USEReST 2008
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