In terms of fundamental theorem of surfaces, a surface is uniquely defined by the first fundamental coefficients and the second fundamental coefficients. The first fundamental coefficients are used to express how the surface inherits the natural inner product of R3, in which R3 is the set of triples (x, y, z) of real numbers. The first fundamental coefficients of a surface yield information about some geometric properties of the surface, by which we can calculate lengths of curves, angles of tangent vectors, areas of regions, and geode-sics on the surface. Those geometric properties and objects that can be determined only in terms of the first fundamental coefficients of a surface are called the intrinsic geometric properties. The collection of these geometric properties and objects forms the subject of intrinsic geometry of a surface, which studies such properties that do not depend on the shape of the surface, but depend only on measurements that we can carry out while staying on a surface itself. The second fundamental coefficients reflect the local warping of the surface, namely its deviation from tangent plane at the point under consideration.

If a surface is a graph of a function z = u(x,y), {Uij} are the sampled values of u at sampling points {(x;, yj)}, and {ffjj} are interpolations in terms of the sampled values {Uj}. Let u°- = uj and h represent grid cell size, then finite difference of basic equations of surface modeling based on the fundamental theorem of surfaces (SMTS) on the »th iterative step could be formulated as

»,+1 j 2», j + », - 1 j _ frl \ f »,+1j »i- 1 j ^11> ,j 2h h2

FD f

j 2h

are boundary conditions; E" ■ ,F"-, and G" - are the first fundamental

' <j' 'j 'j coefficients; L"j and are the second fundamentai coefficients; (rj^ntj, (r2^ntj, (ryntj, and (r22) "j are coefficients calculated in terms of the first and second fundamental coefficients.

If computational domain is normalized to [0,1] x [0,1], matrix formulation of the basic equations could be expressed as


M-1,1 , •••, M ,m, u"m}, ■ ■ ■, Umm)T; M represents lattice number in direction x and in direction y ; h = 1/(M + 1); A1 and B1 " are, respectively, matrix of left-hand item and vector of right-hand item of eqn [3]; A2 and B2 " are respectively left-hand-item matrix and right-hand item vector of eqn [4].

the following least squares algorithm can be developed to make the simulated values equal to sampled values at the sampling points:

where C(k,(i — 1)?M + j) = 1 and D (k) = u j, which means that the sampled value is U j at the kth sampling point (xhyj).

For sufficiently large A, the algorithm [5] can be transferred into unconstrained least squares approximation, which is termed SMTS,


Numerical tests and the real-world validation of SMTS demonstrate that, (1) however high the oscillation on boundary may be, the impact of the oscillation on the interior area of the computational domain can be eliminated after sufficient SMTS iteration steps (in other words, SMTS is able to eliminate impacts of boundary oscillation, which gives a solution to error propagation); (2) SMTS is able to repair the peak-value truncation that usually exists in surface modeling of the classical methods; (3) grid spacing has little impact on SMTS accuracy, which means that SMTS can be used to deal with multiscale issues; and (4) SMTS accuracy does depend little on sampling interval but mostly on locational representativeness of sampled points.

The QR decomposition method is used to transform the matrix expression of SMTS into two algebric equations, whose coefficient matrixes are respectively the upper triangular matrix and the lower triangular matrix. The spatial real-time simulation using STMS is abstracted into dynamically adding information points and reducing information points. Numerical tests show that the spatial real-time simulation under the dynamically adding information points only needs to conduct matrix decomposition to the matrix formed by adding information points. The spatial real-time simulation under dynamically reducing information points only needs to calculate submatrix of the matrix before reducing information points.

A management system of ecological modelbase (MSEM) with 3055 models has been developed in order to find a solution for direct modeling problem. Usually, there are two ways to develop MSEM. One is the model management techniques, including database approach, structured modeling approach, object-oriented approach, and knowledge-based approach. Another one is model management in GIS software such as ModelBuilder in ArcGIS. The first way can efficiently manage the models, but requires great code creation to handle spatial data. The second way can utilize GIS to manipulate spatial data, but it does not support building complex mathematical models. Therefore, an object-oriented framework for MSEM is developed, in which models are abstracted to model class and model instance. Model class and model instance are represented as a objects. Spatial data and mathematical equation are parsed by Model Engine that is composed of mathematical library and STMS component. Integration of STMS and MSEM would solve the direct modeling problem existing in current GIS.

SMTS finds solutions to the error problem, real-time problem, direct modeling problem, and multiscale problem of the current GIS. However, SMTS had a huge computation cost and very slow computational speed because it must solve a partial differential equation set for simulating each lattice of a surface, which made SMTS difficult to be widely applied.

The basic equations of SMTS are evidently elliptic. For the solution of elliptic partial differential equations, multigrid has been generally accepted as the fastest numerical method. Multigrid is thus employed to solve the partial differential equation set of SMTS and the YUE-HASM approach is developed.

We formulate

under consideration of simulation step length (or grid cell size) h. Then, formulation [6] can be expressed as,

If a coarser grid H = 2h is introduced, a two-grid method can be developed. Each iteration step of the





^ > r

) c

) O

S3 C C) C

) > Ç > r

s < c

^ o


> ^ 3-S







) C \

O (

J Ç 1 c

N r

^ n




Level 0 Level 1 Level 2

R0 P0

Iteration 1

Figure 1 Information fusion on multi-grids: (a) information on the fine grid O is transferred to the coarse grid 0; (b) information on the coarse grid 0 is transferred to the fine grid O; and (c) The structure of W-cycle (• represents iteration, R transfer information from the finer grid to the coarser grid, P transfer information from the coarser grid to the finer grid, ■ direct solution).


Figure 2 Comparison between surface simulated by YUE-HASM and surfaces simulated by classical methods: (a) the original Gaussian synthetic surface; (b) YUE-HASM surface; (c) SPLINE surface; (d) TIN surface; (e) KRIGING surface; and (f) IDW surface.

two-grid method consists of a presmoothing, a coarse grid correction, and a postsmoothing part. The pre-smoothing, coarse grid correction and postsmoothing parts form a two-grid cycle, which is repeated until expected accuracy is reached. Multigrid is the extension from two-grid levels ($h,$2h) to a sequence of levels, .. .,$hmax), which ends with the coarsest grid $hmax (Figure 1).

Numerical tests (Figure 2) and the real-world simulation of climate change trend and ecosystem change in Jiangxi province of China (Figure 3) demonstrated that computing time of YUE-HASM is proportional to the first power of the total number of grid cells, while SMTS

computing time was proportional to the third power of the total number of grid cells. YUE-HASM has highly accelerated computational speed, especially for simulation with huge computational work. In the meanwhile, YUE-HASM has greatly increased simulation accuracy.

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