Transmission

yv is a pointing-up unit vector. At the point r, the normal of the surface can be

calculated by its surface slope,

) then the surface normal is expressed as,

)

We denote the direction of incident ray of light is , according to the angle of incidence equals to the angle of reflection. We can obtain the reflected ray of light,

Ni

3.5.2 Transmission

The expression for transmission of light is not as simple as reflection of light.

We have to consider two things. First, the direction of transmitted ray is only related with the surface normal and the direction of incident ray. Second, the transmission between two different mediums obeys Snell’s Law.

Assuming that the direction of incident ray is in a medium with index of refraction , and the transmitted ray is in the medium with index of , so we can

Ni

ni n_t

calculate the angle between incident ray and surface normal,

sinθ_i = 1−(N_i ⋅N_s)² = N_i×N_s , (38) and the angle of the transmitted ray will be

s t

t = N ×N

θ

sin . (39) According to the Snell’s law, we can calculate the direction of refracted ray.

3.5.2 Fresnel Reflectivity and Transtivity

When light proceeds through media, parts of it will transmit from one medium to another. The other parts of it will be reflected. So theoretically no light is lost during the proceeding. In other words, reflectivity R and transtivity T are related by this following constraint,

=1 + T

R . (40) However the derivation of the expression for R and T is based on the electromagnetic theory of dielectrics. We do not attempt to introduce where it came from. We consult some charts in [10] which referenced some papers in surface wave optics, we directly use the following Fresnel term in our algorithm, which stands for R :

Chapter 4

Implementation and Results

We implement our system on a PC with 1.6GHz AMD Sempron Processor, 2.0 GB of system RAM, and a GeFroce 6200 PCI-X with 128MB of video memory. In average, it takes 0.29s to generate each frame. The conjugated gradient method solver for simulating the tsunami surface movement takes most of the computing time.

Though we only store non-zero parts of the n-by-n symmetric and positive-definite matrix in the conjugated gradient method, it indeed reduces the complexity from to ( n depends on the number of vertex). Solving conjugated gradient method is still time-consuming. Then we implement environment mapping in our shader program on GPU to make the surface reflection more physically real.

) (n²

O O(n)

In the following two sections, we will demonstrate how we simulate the propagation of tsunami wave in our system. In Section 4.1, we show the difference of water surface when different ocean topographies are applied for simulation. In Section 4.2, we show the simulations with different initial tsunami amplitudes. And the boat appearing in the scene is about 60 meters in length.

4.1 Different ocean topographies

Figure 4.1 is a height map which we download from a weather website [14]. It is an ocean topography in the direction of northeast of Taiwan. The brighter pixels mean the closer to the sea level.

Figure 4.1 The height map of the underwater terrain in the northeast of Taiwan In Figures 4.2 - 4.6, we only demonstrate the tsunami surface propagating on the ocean surface by setting Figure 4.1 as the ocean topography at different frames.

Figure 4.2 Frame No. 99 (Ocean256.raw)

Figure 4.3 Frame No. 199 (Ocean256.raw)

Figure 4.4 Frame No. 299 (Ocean256.raw)

Figure 4.5 Frame No. 399 (Ocean256.raw)

Figure 4.6 Frame No. 499 (Ocean256.raw) Figure 4.7 A height map edited by using image processing tool [13].

Figure 4.7 The edited height map

In Figures 4.8-4.12, we demonstrate the corresponding movement of surface.

Figure 4.8 Frame No. 99 (DEM256.raw)

Figure 4.9 Frame No. 199 (DEM256.raw)

Figure 4.10 Frame No. 299 (DEM256.raw)

Figure 4.11 Frame No. 399 (DEM256.raw)

Figure 4.12 Frame No. 499 (DEM256.raw)

Through these simulations, we can find out that the generated tsunami surface with real ocean topography is much rougher than the one with edited height map.

The reason is the depth of the ocean determines the velocity of wave. So the initial ocean wave propagating velocities on the surface with real ocean topography are changing. The depth varies so abruptly in the edited height map that the initial ocean wave propagating velocities differ a lot in that area. Then we can observe the surface generating much higher waves at the position.

The following Figures 4.13a-f and Figures 4.14a-f are the simulation of tsunami waves with applying ocean surface.

Figure 4.13a Frame No. 90 Figure 4.13b Frame No. 160

Figure 4.13c Frame No. 260 Figure 4.13d Frame No. 360

Figure 4.13e Frame No. 500 Figure 4.13f Frame No. 560

Figure 4.14a Frame No. 1 Figure 4.14b Frame No. 40

Figure 4.14c Frame No. 90 Figure 4.14d Frame No. 160

Figure 4.14e Frame No. 260 Figure 4.14e Frame No. 360

4.2 Different initial tsunami amplitudes

This section we describe the differences by using different initial amplitudes of tsunami waves.

Figures 4.15a-c show that the initial amplitude of tsunami wave is 15 meters, and Figures 4.16a-c show that the initial amplitude of tsunami wave is 1 meter. We can see that, the simulated tsunami surface is different apparently with different initial amplitude values. We also can notice that different amplitudes result in different wave velocities. So at the same frame, the wave crests are at different positions. Higher amplitude results in higher speed.

Figure 4.15a Frame No. 150 (amplitude 15m) Figure 4.16a Frame No. 150 (amplitude 1m)

Figure 4.15b Frame No. 300 Figure 4.16b Frame No. 300

Figure 4.15c Frame No. 540 Figure 4.16c Frame No. 540

Chapter 5

Conclusion and Future Works

We have presented a simulation system for tsunami wave propagation. By combining the shallow water equations which are often used in ocean engineering to simulate the movement of tsunami waves and the rendering techniques developed in computer graphics to render the 3D scene. We can obtain more physically-correct tsunami wave propagation. The shallow water equation is simplified by integrating the three-dimensional Navier-Stokes equations in z direction. It can simulate the wave movement correctly when the ratio of depth of water to the wave length is rather small, just like tsunami waves. Then we apply a more stable and efficient numerical method, called Semi-Lagrangian method, on SWE and it allows that we can maintain the stability of the simulation system with larger time steps. Conjugated gradient method is then used to solve the linear system because it can reduce convergence time by searching the optimized solution along the conjugated direction. Perlin’s noise function is introduced and we generated several fractal surfaces. By blending these surfaces, the final ocean surface can be generated. With environment mapping and physically-based lighting model, we can obtain the final simulation results of tsunami wave propagation.

In the future, we may apply the statistical wave models and the Fast Fourier Transforms to generate much more realistic ocean surface. This method is accepted by most oceanographers and described in oceanographic literature. It is based on the

ability to decompose the wave height field as a sum of sine and cosine waves, and the decomposition uses FFT computationally. Besides, we can divide parts of the ocean surface especially at crests into 3D grids and apply 3D Navier-Stokes equations on these grids. So we may obtain much more detail on the crests and simulate other parts of the surface by SWE. This hybrid simulation structure may bring us more realistic result and it will not spend a lot of time such as simulating the whole scene by 3D Navier-Stokes equations.

References

[1] Erleben., Spopring., Henriksen., and Dohlmann. Physics-based animation.

Charles River Media. Graphics Series. 2005

[2] Foster, N., and Fedkiw, R. 2001. Practical animation of liquids. In Proceedings of SIGGRAPH 2001, ACM Press / ACM SIGGRAPH, E. Fiume, Ed., Computer Graphics Proceedings, Annual Conference Series, ACM, 23-30.

[3] Fournier, A., and Reeves, W. T., 1986. A simple model of ocean waves.

Computer Graphics (SIGGRAPH ’86), 20, 75-84.

[4] Gonzato, J. C., and Saëc, B. L. 2000. On modeling and rendering ocean scenes.

The Journal of Visualization and Computer Animation 11, 1, 27-37.

[5] Irving, G., guendelman, E., Losasso, F., Fedkiw, R. 2006. Efficient Simulation of Large Bodies of Water by Coupling Two and Three Dimensional Techniques.

ACM Transactions on Graphics 25.

[6] Jason, L. M. 2005. Real-Time Synthesis and Rendering of Ocean Water. ATI Research Technical Report, April 2005.

[7] Losasso, F., Gibou, F., and Fedkiw, R. 2004. Simulating water and smoke with an octree data structure. ACM Trans. Graph. (SIGGRAPH Proc.)23, 457-462.

[8] O’Brien, J. F., and Hodgins, J. K. 1995. Dynamic simulation of splashing fluids.

In Comput. Anim. ’95, 198-205.

[9] Perlin, K. An image synthesizer. 1985. In Computer Graphics(SIGGRAPH ’85 Proceedings), B. A. Barsky, Ed. 1985, vol.19(3), 287-296.

[10] Tessendorf, J., Simulating ocean waters. 2001. In SIGGRAPH course notes (course 47).

[11] Xudong, Y., Xuexian, P., Liang, Z., and Sikun, L. 2005. GPU-based real-time simulation and rendering of unbounded ocean surface. Computer Aided Design and Computer Graphics, 2005. Ninth International Conference.

[12] Yaohua, H., Luiz, V., Xin, T., Baining, g., and Harry, S. 2006. Realistic, real-time rendering of ocean waves. Journal of Visualization and Computer Animation, Volume 17. 59-67.

[13] Adobe System. Adobe Photoshop.

[14] http://ilmuse.gov.tw/~epaper/200609/05.htm