VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS

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 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11. 2008 – USERe. ST 2008

OUTLINE • • • Introduction Fractal Models SAR Raw Signal Simulation Fractal Imaging Conclusions VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11. 2008 – USERe. ST 2008

Introduction VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11. 2008 – USERe. ST 2008

Information Content in SAR Images ERS-1 --- Pixel Spacing: 20 m VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11. 2008 – USERe. ST 2008

Information Content in SAR Images Terra. SAR-X --- Pixel Spacing: 3 m VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11. 2008 – USERe. ST 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. 2008 – USERe. ST 2008

Fractal Models VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11. 2008 – USERe. ST 2008

Introduzione Geometrical Models Natural Scenes Urban Areas Fractal Geometry Classical Geometry VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11. 2008 – USERe. ST 2008

The fractional Brownian motion (f. Bm) f. Bm parametrs H Hurst Coefficient 0<H<1 s Standard deviation at unitary distance D=2 -H [m 1 -H] D is the fractal dimension ; t=x-x’ VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11. 2008 – USERe. ST 2008

FBm Model The f. Bm 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): VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11. 2008 – USERe. ST 2008

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: 0< H <1 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS 1< a <3 Napoli, 11. 2008 – USERe. ST 2008

Fractional Gaussian noise (f. Gn) It is defined as the derivative of the f. Bm process. The f. Bm 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 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11. 2008 – USERe. ST 2008

FGn Model Scales smaller than the resolution cell do not contribute to the SAR signal formation e =Dx If e << t the f. Gn autocorrelation function is : The structure function turns out to be: VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11. 2008 – USERe. ST 2008

FGn Model Spectrum Evaluation The f. Gn is a stationary process, therefore we can evaluate its spectrum as the derivative of its autocorrelation function: If e << 2 p/k the spectrum is : VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11. 2008 – USERe. ST 2008

SAR Raw Signal Simulation VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11. 2008 – USERe. ST 2008

Key Tool for Disaster Monitoring To solve the inverse problem use is made of solvers of the corresponding direct problem SAR SIMULATOR VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11. 2008 – USERe. ST 2008

SAR Raw Signal Simulation Reflectivity function SAR unit response 1. Scene description 2. Electromagnetic scattering model 3. SAR raw signal formation VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11. 2008 – USERe. ST 2008

The Simulator z(x, y) e, s zmic SAR RAW SIGNAL SIMULATOR SAR PROCESSOR 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 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11. 2008 – USERe. ST 2008

Digital Elevation Model 3 D representation of the Vesuvio volcano area. VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11. 2008 – USERe. ST 2008

Simulation Details Sensor Parameters platform height 514 [ km] platform velocity 7. 6 [km/sec] look-angle Lava parameters aa Pahoehoe Dielectric Constant 8 20 20 [degrees] Conductivity [S/m] 0. 01 1 azimuth antenna dimension 4. 7 [ m] Hurst coefficient 0. 7 0. 9 range antenna dimension 7 [ m] s [m 1 -H] 0. 25 0. 05 carrier frequency 9. 65 [GHz] pulse duration 25 [ microsec] chirp bandwith 100 [ Mhz] sampling frequency 110 [ Mhz] pulse repetition frequency 4500 [ Hz] VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Background Dielectric Constant 4 Conductivity [S/m] 0. 1 Hurst coefficient 0. 8 s [m 1 -H] 0. 16 Napoli, 11. 2008 – USERe. ST 2008

Simulated SAR image Simulation of the area in absence of lava flows Resol. 1. 69 m x 3. 99 m azimuth x ground range Multilook 8 x 4 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11. 2008 – USERe. ST 2008

Simulated SAR image Simulation of the area with aa lava flow Resol. 1. 69 m x 3. 99 m azimuth x ground range Multilook 8 x 4 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11. 2008 – USERe. ST 2008

Simulated SAR image Simulation of the area with pahoehoe lava flow Resol. 1. 69 m x 3. 99 m azimuth x ground range Multilook 8 x 4 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11. 2008 – USERe. ST 2008

Fractal Imaging VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11. 2008 – USERe. ST 2008

SAR Imaging Is the SAR image of a fractal surface fractal? Can we retrieve the fractal parameters of the observed scene from SAR images? VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11. 2008 – USERe. ST 2008

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; a 0 and a 1 are the coefficients of the Mc. Laurin 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. 2008 – USERe. ST 2008

First Results The image i(x, y) has the same characterization of the f. Gn process, with mean a 0 and standard deviation a 1 s. Dx. H-1 We can evaluate the structure funcion and the spectrum of the image: VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11. 2008 – USERe. ST 2008

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 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11. 2008 – USERe. ST 2008

Procedure Rationale s H f. Bm Synthesis Profile (Weierstrass-Mandelbrot function) Reflectivity Evaluation Image (SPM model) Spectrum and Variogram Estimation Comparison with theory VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11. 2008 – USERe. ST 2008

Surface Synthesis Simulated aa lava flow. VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Simulated pahoehoe lava flow. Napoli, 11. 2008 – USERe. ST 2008

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 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Surface Theoretical Spectrum Surface Estimated Spectrum Napoli, 11. 2008 – USERe. ST 2008

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 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Surface Theoretical Spectrum Surface Estimated Spectrum Napoli, 11. 2008 – USERe. ST 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. 2008 – USERe. ST 2008

Future work • Full Extension to 2 D • Inclusion of a reliable lava flow model • Inclusion of a more appropriate speckle model (Kdistribution) in the simulation procedure • Inclusion of te speckle in the imaging analysis VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Napoli, 11. 2008 – USERe. ST 2008