1. Granular Avalanche Modeling Methodology Working Group 2 October 2006

4. Atenquique, Mexico 1955

6. Volcan Colima, Mexico

7. San Bernardino Mountain: Waterman Canyon

8. Guinsaugon. Phillipines, 02/16/06 Heavy rain sent a torrent of earth, mud and rocks down on the village of Guinsaugon. Phillipines, 02/16/06 A relief official says 1,800 people are feared dead.

9. Model Topography and Equations(2D) ground geophysical mass Upper free surface F s (x,t) = s(x,y,t) – z = 0, Basal material surface F b (x,t) = b(x,y) – z = 0 Kinematic BC: Elevation data from public and private DEMs - different sources and different formal resolutions. z is the direction normal to the hillside

10. Model System-Basic Equations The equations for a continuum incompressible medium are: semi-empirical relationship between the stress tensor T and u are derived from Coulomb theory Boundary conditions for stress:

11. Model System-Depth Average Theory Depth average the continuity equation: where

12. Basic model System of 3 PDEs for (h, hv x hv y ) in space and time (x,y,t) Also need to provide initial mass M, location of this mass, initial velocity.

13. Abstracting Y = F(X,θ) + ε model Uncertain parameters θ = (φ int, φ bed, M, x 0, v 0, θ rest ) And would like to include uncertainty in topography

14. An Aside on M It is those rare very large flows that cause enormous damage and loss of life.

15. An Aside on friction

16. TITAN2D (choose grid ↔ speed)  Use adaptive meshing for computational efficiency  Large scale computations to produce realistic simulations of mass flows  Integrated with GIS to obtain terrain data (massaging required)  Need to manipulate DEM grids to computational grids  Integrated with multi-scale visualization tools  Runs efficiently on a range of computers – laptops to large clusters  Code is GRID enabled for remote access through a portal http://grid.ccr.buffalo.edu  All software (source code) freely available for use at http://www.gmfg.buffalo.edu

17. Other Computer Models (fast) Flow3D – similar integration of terrain. Code simulates a frictional block sliding downhill under gravity (can’t run up over an embankment) LaharZ – combines terrain data with statistical estimate (based on historical data) and potential energy of the mass, to estimate the volumetric flow from one terrain block into the next

18. ‘Field’ Data Table top experiments, reasonably controlled. But scaling up doesn’t work! In the field, geologists can measure flow depth (i.e., the “h” after flow stops) at selected sites on the deposit field [take core samples] In some instances they can estimate flow speed at locations, by examining run-up near bends Both are highly prone to error Rare event: geologists on site during a flow!

19. Inclined Plane (Pile Height Contours) Uniform mesh Adaptive mesh (3 levels of refinement)

20. Table Top Experiment

21. Comparison to experiment

23. Effect of different initial volumes Left – block and Ash flow on Colima, V =1.5 x 10 5 m 3 Right – same flow -- V = 8 x10 5 m 3

24. Real Topography (Little Tahoma, WA) Tahoma peak (deposit area extent) Tahoma peak, Mount Rainier (debris avalanche, 1963)

25. The 2005 Vazcún Valley Lahar 12 February 2005. Vazcún Valley, north-east flank of Volcán Tungurahua, Ecuador Small ash-rich lahar Volume: 50,000m 3 (calculated from field observations) to 70,000m 3 (calculated). Velocity: 7m/s and 3m/s Photo: Defensa Civil

26. Flow Thickness At El Salado Baths As measured 5 months later, flow depth was 3.00 m. Two-phase code gives max flow depths of 3.5m at this section. Model flow depth is within 50cm of agreement.

27. Our Halting Early Attempts I Generate a response surface f p = ∑ α θ p + e(α, θ) Vary M, v 0 Latin hypercube for initial set of runs Next θ by maximizing variance point, until little further change in variance. Then up the order of the polynomial. “Truth” surface – 10,000 runs on reasonable spatial grid, cross product grid on θs

28. Hazard map Conditioned on a large event occurring!! Probability that flow thickness will exceed 1 m.

29. Our Halting Early Attempts II

30. INPUT UNCERTAINTY PROPAGATION Model inputs are often uncertain –range data and distributions may be estimated –need to propagate this to range and distributions on desired outputs Polynomial Chaos (PC) –Assume that the field variables are functions of a random variable(s)  –Expand in terms of “orthogonal polynomials”  and use orthogonality to obtain the coefficients – Fourier series like process. Provably accurate methodology. –Complicated for highly nonlinear problems

31. Quantifying Uncertainty -- Approach Polynomial Chaos (PC): approximate pdf with sum of finite number of orthogonal polynomials  i Multiply by  m and integrate to use orthogonality Coupling among all the equations for the coefficients Wiener 34 Xiu and Karniadakis’02

32. Example

33. Polynomial Chaos Quadrature –Instead of Galerkin projection, integrate by quadrature weights –Leads to a method that has the simplicity of MC sampling and cost of PC –Can directly compute all moment integrals –Degrades for large number of random variables

34. NISP / Polynomial Chaos Quadrature Replace integration with quadrature and interchange order of time and stochastic dimension integration → ”smart” sampling