1. Napoli, 11.11.2008 – USEReST 2008 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

2. Napoli, 11.11.2008 – USEReST 2008 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS OUTLINE Introduction Fractal Models SAR Raw Signal Simulation Fractal Imaging Conclusions

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

4. Napoli, 11.11.2008 – USEReST 2008 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS ERS-1 --- Pixel Spacing: 20m Information Content in SAR Images

5. Napoli, 11.11.2008 – USEReST 2008 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS TerraSAR-X --- Pixel Spacing: 3m Information Content in SAR Images

6. Napoli, 11.11.2008 – USEReST 2008 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

7. Napoli, 11.11.2008 – USEReST 2008 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Fractal Models

8. Napoli, 11.11.2008 – USEReST 2008 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Geometrical Models Natural Scenes Urban Areas Fractal Geometry Classical Geometry Introduzione

9. Napoli, 11.11.2008 – USEReST 2008 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS HHurst Coefficient 0<H<1D=2-H sStandard deviation at unitary distance[m 1-H ] fBm parametrs D is the fractal dimension ; t =x-x ’ The fractional Brownian motion (fBm)

10. Napoli, 11.11.2008 – USEReST 2008 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS 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 t ):

11. Napoli, 11.11.2008 – USEReST 2008 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Spectral Characterization The spectrum evaluation requires space – frequency, or space – scale techniques, leading to : Where the specrume parameters are related with H and s: FBm Model 0< H <1 1<  <3

12. Napoli, 11.11.2008 – USEReST 2008 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS 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: By adopting the following  function

13. Napoli, 11.11.2008 – USEReST 2008 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS If  <<  the fGn autocorrelation function is : FGn Model The structure function turns out to be: Scales smaller than the resolution cell do not contribute to the SAR signal formation  x

14. Napoli, 11.11.2008 – USEReST 2008 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Spectrum Evaluation FGn Model The fGn is a stationary process, therefore we can evaluate its spectrum as the derivative of its autocorrelation function: If  << 2  k the spectrum is :

15. Napoli, 11.11.2008 – USEReST 2008 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS SAR Raw Signal Simulation

16. Napoli, 11.11.2008 – USEReST 2008 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

17. Napoli, 11.11.2008 – USEReST 2008 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS 1. Scene description 2. Electromagnetic scattering model 3. SAR raw signal formation Reflectivity function SAR unit response SAR Raw Signal Simulation

18. Napoli, 11.11.2008 – USEReST 2008 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS The Simulator SAR RAW SIGNAL SIMULATOR z(x,y) z mic 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. , 

19. Napoli, 11.11.2008 – USEReST 2008 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Digital Elevation Model 3D representation of the Vesuvio volcano area.

20. Napoli, 11.11.2008 – USEReST 2008 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Simulation Details platform height514 [ km] platform velocity7.6 [km/sec] look-angle20 [degrees] azimuth antenna dimension4.7 [ m] range antenna dimension7 [ m] carrier frequency9.65 [GHz] pulse duration25 [ microsec] chirp bandwith100 [ Mhz] sampling frequency110 [ Mhz] pulse repetition frequency4500 [ Hz] Lava parametersaa Pahoehoe Dielectric Constant820 Conductivity [S/m]0.011 Hurst coefficient0.70.9 s [m 1-H ]0.250.05 Sensor Parameters Background Dielectric Constant4 Conductivity [S/m]0.1 Hurst coefficient0.8 s [m 1-H ]0.16

21. Napoli, 11.11.2008 – USEReST 2008 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 rangeMultilook 8 x 4

22. Napoli, 11.11.2008 – USEReST 2008 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Simulation of the area with aa lava flow Simulated SAR image Resol. 1.69m x 3.99m azimuth x ground rangeMultilook 8 x 4

23. Napoli, 11.11.2008 – USEReST 2008 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Simulation of the area with pahoehoe lava flow Simulated SAR image Resol. 1.69m x 3.99m azimuth x ground rangeMultilook 8 x 4

24. Napoli, 11.11.2008 – USEReST 2008 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Fractal Imaging

25. Napoli, 11.11.2008 – USEReST 2008 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Is the SAR image of a fractal surface fractal? SAR Imaging Can we retrieve the fractal parameters of the observed scene from SAR images?

26. Napoli, 11.11.2008 – USEReST 2008 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; a 0 and a 1 are the the coefficients of the McLaurin series expansion of i(x,y) for small values of p(x,y) and q(x,y)

27. Napoli, 11.11.2008 – USEReST 2008 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 a 0 and standard deviation a 1 s  x H-1 We can evaluate the structure funcion and the spectrum of the image:

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

29. Napoli, 11.11.2008 – USEReST 2008 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Procedure Rationale fBm Synthesis (Weierstrass-Mandelbrot function) Reflectivity Evaluation (SPM model) Spectrum and Variogram Estimation Comparison with theory sHsH ProfileImage

30. Napoli, 11.11.2008 – USEReST 2008 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Surface Synthesis Simulated pahoehoe lava flow.Simulated aa lava flow.

31. Napoli, 11.11.2008 – USEReST 2008 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Results: Azimuth cuts Image 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

32. Napoli, 11.11.2008 – USEReST 2008 VOLCANO MONITORING VIA FRACTAL MODELING OF LAVA FLOWS Results: Range cuts Image 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

33. Napoli, 11.11.2008 – USEReST 2008 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.

34. Napoli, 11.11.2008 – USEReST 2008 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