Marching method

This method is known as the marching method. The accuracy of the procedure and the correctness of the program can be verified by testing it with analytically soluble Volterra equations, for example, the test problems with nonsingular convolution kernels listed on pp. 505-507 of Brunner and van der Houwen s book (1986). 

The essence of KW is that multi-point central differences are used as derivatives along most of the t scale, with some asymmetric expressions necessarily added at the ends. Rather than using the time-marching method that is common to all the methods described in previous sections, KW puts all the approximations into one large system of equations, and solves the lot. It turns out that this results in a fortuitous stability [141]. 

These equations are of parabolic type, and may be solved by a forward marching technique. The upstream profile U(a o, y) must be specified, and the free-stream pressure distribution P ix) must be known. F(xo, y) is then determined by Eq. (11a). The numerical problems are straightforward but not a trivial aspect of a successful method. Implicit schemes have been most successful, although explicit marching methods can be used if the wall region is treated separately. 

J. A. Sethian, in Level set methods and fast marching methods, Cambridge University Press, Cambridge, 1999. 

J. A. Sethian. Evolution, implementation, and application of level set and fast marching methods for advancing fronts. J. Comput Phys., 169[2] 503-555,2001. 

Denton, J. D.," An improved time-marching method for turbomachinery flow calculation", Trans. ASME, J. Eng. for Power, Vol. 105, 1983, pp 514-524. 

The related fast marching methods (Sethian 1999) have also been applied to image segmentation. These methods are faster, but are designed for problems in which the speed function never changes sign, so that the front is always moving forward or backward. 

Popovici, A.M. and Sethian, ).A. (1998) Three-dimensional travel-time computation using the fast marching method. Proc. SPIE, 3453, 82-93. 

In a forward-marching method such as Euler s method, we are more interested in the total error propagation over multiple usage of the algorithm than in the local one-step error. If we let Cj be the error between the approximate solution, rcj, and exact solution, x ti) to the differential equation, then... 

Sethian, A.J., 1999, Level Set Methods and Fast Marching Methods Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge University Press. 

There is an interesting exception to the above, where the time-marching method was not used. Anderson and Moldoveanu (1984) simulated the steady state of a small electrode embedded flush in the bottom of a wide but low channel with laminar flow of the electrolyte through the channel. Because the channel was wide, that dimension could be ignored, leaving x, the direction of the flow and z, the vertical dimension (assuming the channel lying flat). The diffusion-convection equation, at steady state, then is... 

To solve for the dimensionless temperatures 0, and 0 and the fraction remaining Y over the entire grid, a marching method is used in which the Newton-Raphson algorithm solves for 0 ,j j i j each point until the grid is completely specified. 

Mapping 3D Geo-Bodies Based on Level Set and Marching Methods... 

Summary. In this chapter, a simplified method for mapping objects based on the level set method is introduced. Level set and marching methods are used to map connected volumes within 3D seismic data. The simpler marching method solves the stationary problem stated by the level set formulation. The evolution of the object, from a seed point to the boundary, is described by a differential equation. 

Experience tells us that the explicitness often leads to numerical instability. This can be improved by implicitly moving the front, as will be explained in the level set formulation in Subsection 2.1. The underlying theory presented is based on the implicit level set and marching methods, further introduced in Section 2. The work on level sets was initiated by Sethian and Osher in... 

The outline of this chapter is as follows. Theoretical aspects are discussed in Section 2, where in particular the level set method and the marching method are introduced. Finally, Section 3 provides numerical results concerning one selected real-world model problem from seismic data analysis. 

The aim of this chapter is to develop an algorithm for mapping geo-bodies in spatial and temporal domains. To this end, we use level set and marching methods for the mapping. The basic steps of the overall procedure are shown in the flowchart of Figure 2. In this section, we explain two intermediate steps, concerning the marching method and the level set method, respectively. 

Fig. 2. Flowchart of the suggested procedure for the mapping algorithm, from raw seismic data to segmented geo-body output. The output result from the marching method may directly be apphed to the different analysis steps in the rightmost block. Otherwise, it is input to the refining level set method.

If the initial approximation from the (faster) marching method is not satisfactory, then the (slower) level set method is evoked. This gives room for a more complicated velocity function, and so a refined and more accurate solution is obtained. The output from the level set block is a geo-body, being available for different interpretation and analysis steps through the user. These interpretations may, for instance, involve decisions concerning the quality of the output geo-body for oil reservoir modelhng. 

The level set method may be viewed as an initial-value formulation, which requires solving a partial differential equation. The marching method computes the stationary solution of the problem by solving a differential equation. These equations are based on dynamic theory, and hence improve the stability of the segmentation solution. 

The next two subsections discuss some of the theory concerning the level set method and the marching method. This is done in reverse order to the flowchart of Figure 2, because the marching method solves the stationary case, and so this can be viewed as a special case solved by the level set method. 

The marching method. Figure 6, is designed to find the solution to the mapping problem through solving a differential equation in spatial dimensions. The basic principles are similar to the case of the level set method, and therefore we also work with a function containing information about the front, denoted as t). 

The Stationary Case. The marching method is used to solve the stationary case of the level set formulation. When moving from the original level set formulation to the stationary formulation, we are able to speed up the computations. The price to pay is that the velocity function F we work with has to be strictly monotone when solving the simplified scenario. 

This chapter presents a procedure for mapping geological bodies from seismic data. Theoretical aspects of the mapping procedure, concerning the level set method and marching method, are discussed. The overall method is adapted to applications from seismic data interpretation, where particular attention is placed on the design of the velocity function. Nmnerical examples, involving real-world seismic data taken from Barents Sea, support the utility of our method. 

J.A. Sethian (1996) Level Set Methods and Fast Marching Methods. Cambridge University Press. 

S.K. Richardsen and T. Randen (2004) Mapping 3D geo-bodies based on level set and marching methods. This volume. 

---