Diapositiva 1

1
VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA
FLOWS
Gerardo DI MARTINO Antonio IODICE Daniele
RICCIO Giuseppe RUELLO
Università degli Studi di Napoli Federico
II Dipartimento di Ingegneria Elettronica e
delle Telecomunicazioni
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OUTLINE

• Introduction
• Fractal Models
• SAR Raw Signal Simulation
• Fractal Imaging
• Conclusions

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Introduction
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Information Content in SAR Images
ERS-1 --- Pixel Spacing 20m
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Information Content in SAR Images
TerraSAR-X --- Pixel Spacing 3m
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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

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Fractal Models
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Introduzione
Geometrical Models
Urban Areas
Natural Scenes
Fractal Geometry
Classical Geometry
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The fractional Brownian motion (fBm)
fBm parametrs
H Hurst Coefficient 0ltHlt1 D2-H s Standard
deviation at unitary distance m1-H
D is the fractal dimension tx-x
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FBm Model
The fBm is a continuous, not-differentiable,
not-stationary process. Its autocorrelation
function is
It depends on x, x e t. The structure function
(the rms of increments at distance t)
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FBm Model
Spectral Characterization The spectrum evaluation
requires space frequency, or space scale
techniques, leading to
Where the specrume parameters are related with H
and s
0lt H lt1 1lt a lt3
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Fractional Gaussian noise (fGn)
It is defined as the derivative of the fBm
process. The fBm process is not derivable,
therefore a regularization is needed
Such a process can be seen as a distribution and
it can be derived as follows
By adopting the following f function
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FGn Model
Scales smaller than the resolution cell do not
contribute to the SAR signal formation e Dx
If e ltlt t the fGn autocorrelation function is
The structure function turns out to be
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FGn Model
Spectrum Evaluation
The fGn is a stationary process, therefore we can
evaluate its spectrum as the derivative of its
autocorrelation function
If e ltlt 2p/k the spectrum is
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SAR Raw Signal Simulation
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Key Tool for Disaster Monitoring

• To solve the inverse problem use is made of
  solvers of the corresponding direct problem

SAR SIMULATOR
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SAR Raw Signal Simulation
Reflectivity function
SAR unit response
1. Scene description 2. Electromagnetic
scattering model 3. SAR raw signal formation
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The Simulator
z(x,y)
SAR RAW SIGNAL SIMULATOR
e, s
SAR PROCESSOR
zmic
Sensor parameters
We need both a macroscopic and a microscopic
description of the scene. We also need the
electromagnetic parameters relevant to the scene.
SAR simulated image
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Digital Elevation Model
3D representation of the Vesuvio volcano area.
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Simulation Details
Sensor Parameters
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
Background
Dielectric Constant 4
Conductivity S/m 0.1
Hurst coefficient 0.8
s m1-H 0.16
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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
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Simulated SAR image
Simulation of the area with aa lava flow
Resol. 1.69m x 3.99m azimuth x ground range
Multilook 8 x 4
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Simulated SAR image
Simulation of the area with pahoehoe lava flow
Resol. 1.69m x 3.99m azimuth x ground range
Multilook 8 x 4
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Fractal Imaging
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SAR Imaging
Is the SAR image of a fractal surface fractal?
Can we retrieve the fractal parameters of the
observed scene from SAR images?
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Imaging Model
By using the SPM for the scattering evaluation
(ipotesi di piccole pendenze), the image
intensity is expressed as
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)
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First Results
The image i(x,y) has the same characterization of
the fGn process, with mean a0 and standard
deviation a1sDxH-1
We can evaluate the structure funcion and the
spectrum of the image
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Results
SAR image can be considered a fractal with H
ranging from -1 and 0.
It means that a Hausdorff - Besicovitch fractal
dimension can not be defined
The SAR image is a self-affine Gaussian
stationary process, NOT fractal
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Procedure Rationale
fBm Synthesis (Weierstrass-Mandelbrot function)
s H
Profile
Image
Reflectivity Evaluation (SPM model)
Spectrum and Variogram Estimation
Spectrum and Variogram Estimation
Comparison with theory
Comparison with theory
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Surface Synthesis
Simulated pahoehoe lava flow.
Simulated aa lava flow.
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Results Azimuth cuts
Image Theoretical Spectrum
Image Estimated Spectrum
aa lava flow
Surface Theoretical Spectrum
Surface Estimated Spectrum
Image Theoretical Spectrum
Image Estimated Spectrum
pahoehoe lava flow
Surface Theoretical Spectrum
Surface Estimated Spectrum
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Results Range cuts
Image Theoretical Spectrum
Image Estimated Spectrum
aa lava flow
Surface Theoretical Spectrum
Surface Estimated Spectrum
Image Theoretical Spectrum
Image Estimated Spectrum
pahoehoe lava flow
Surface Theoretical Spectrum
Surface Estimated Spectrum
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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 .
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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