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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. OUTLINE. Introduction Fractal Models

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## VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS

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**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**OUTLINE**• Introduction • Fractal Models • SAR Raw Signal Simulation • Fractal Imaging • Conclusions VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS**Introduction**VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS**Information Content in SAR Images**ERS-1 --- Pixel Spacing: 20m VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS**Information Content in SAR Images**TerraSAR-X --- Pixel Spacing: 3m VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS**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**Fractal Models**VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS**Introduzione**Geometrical Models Urban Areas Natural Scenes Fractal Geometry Classical Geometry VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS**The fractional Brownian motion (fBm)**fBm parametrs H Hurst Coefficient 0<H<1 D=2-H s Standard deviation at unitary distance [m1-H] D is the fractal dimension ; t=x-x’ VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS**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): VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS**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 1< a <3 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS**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 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS**FGn Model**Scales smaller than the resolution cell do not contribute to the SAR signal formation e =Dx If e << t the fGn autocorrelation function is : The structure function turns out to be: VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS**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 << 2p/k the spectrum is : VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS**SAR Raw Signal Simulation**VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS**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**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**The Simulator**z(x,y) e, s SAR RAW SIGNAL SIMULATOR 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 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS**Digital Elevation Model**3D representation of the Vesuvio volcano area. VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS**Simulation Details**SensorParameters VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS**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**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 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS**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 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS**Fractal Imaging**VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS**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**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) VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS**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: VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS**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**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 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS**Surface Synthesis**Simulated aa lava flow. Simulated pahoehoe lava flow. VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS**Results: Azimuth cuts**ImageTheoreticalSpectrum 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 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS**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 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS**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**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 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS

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