Guillaume: 1 revision

2008-12-03T10:39:05Z

1 revision

192.168.20.177 at 12:37, 4 November 2008

2008-11-04T12:37:38Z

New page

[[Image:SmoothBathimNesting.gif|300px|thumb|Animation consisting of three frames showing respectively i) the father bathymetry, ii) the son bathym, iii) and the interpolated bathym.]]
This program creates a new sub-model bathimetry to allow a smooth transition in the boundary between the coarser model and the high resolution model. See also [[SmoothBatimCoefs]].

== Quick start ==
#Create the father and son bathymetries (you may use [[Mohid GIS]]).
#Create the SmoothCoeficients [[Grid Data | griddata file]] (you may use [[SmoothBatimCoefs]]).
#Create and edit the options ''SmoothBatimNesting.dat'' file and save it in the same path as the executable.
#Run the executable.

== Options file ==
Here's what the options file ''SmoothBatimNesting.dat'' looks like:
FATHER_BATIM : ..\..\WestIberiaTide\GridData.dat
SON_BATIM : ..\GridData_2.dat
SMOOTH_COEF : SmoothCoef.dat
NEW_SON_BATIM : SmoothData.dat

== Smoothing algorithm ==
Let it be <math>X \in \mathcal{F}\equiv\left\{F,\,S,\,I\right\}</math>, as an element of the father, the son and the interpolated bathymetries discrete fields. Let it be <math>I_X : \mathbb{N}^2 \longrightarrow \mathbb{R}^2 </math> as the respective index mapping function. Let it be <math> X^{\prime}: \mathbb{R}^2 \longrightarrow \mathbb{R}</math>, a continuous map, so that <math>X \equiv X^\prime\circ I_X,\;\forall X \in \mathcal{F}</math>. Let <math>r_X \subset \mathbb{R}^2</math> define the work domain of <math>X^\prime \in \mathcal{F}^\prime</math>. In particular, <math>X^\prime</math> verifies <math>X^\prime(\overline{r_X})\equiv 0</math>. The son grid is the same as the interpolated grid, meaning <math>I_S = I_I</math>. However, the father field is completely independent from the other fields, meaning that <math>r_{FS}\equiv\left(r_F \cap r_S\right) \subseteq \mathbb{R}^2</math>. The smoothing algorithm is a simple linear interpolation of the father and son fields:
:<math>I^\prime = \left(1 - C^\prime\right) F^\prime + C^\prime\,S^\prime</math>
where the smoothing coefficient is given by <math>C^\prime: \mathbb{R}^2 \longrightarrow \left[0\;1\right]</math>. Thus, the smoothing coefficient gridded counterpart <math>C</math> is defined; and its index mapping function <math>I_C</math> is equal to the son's; meaning <math>I_C = I_S</math>. Let us define the grid <math>G \subseteq \mathbb{N}^2</math> such that <math>I_S(G)\subseteq r_S</math> and <math>I_S(G)\rightarrow r_S</math>. Thus the smoothing coefficient <math>C</math> needs can be defined in <math> G </math>. Concretely speaking, <math>G \mapsto (I_S(G),C(I_S(G)))</math> is the smoothing coefficient's [[Grid Data]] file described above, thus it leaves to the modeller the choice of the smoothing coefficient field.

=== Implementation ===
Let us define a collection of four-vertices polygons taken from the father grid:
:<math>\begin{matrix} V : & \mathbb{N}^2 \longrightarrow \mathbb{R}^8 \\ & \left(i,\,j\right) \mapsto (I_{F\;i\,j},\,I_{F\;i+1\,j},\,I_{F\;i\,j+1},\,I_{F\;i+1\,j+1})\end{matrix} </math>
where <math>D\left(V_{i\,j}\right)\subset \mathbb{R}^2</math> is the subdomain defined by the vertices of <math>V_{i\,j}</math>.
Let us define an auxiliary grid of the son grid given by:
:<math>\forall\; _{ij} \in \mathbb{N}^2,\;\tilde{I}_{S\;i\,j}=\frac{1}{4}\left(I_{S\;i\,j}+I_{S\;i+1\,j}+I_{S\;i\,j+1} +I_{S\;i+1\,j+1}\right)</math>.
Let us define a new mapping into a subset of <math>F</math>'s image:
:\<math>\forall\;_{i\,j} \in \mathbb{N}^2\quad \exists !\;_{u\,v} \in \mathbb{N}^2\; :\quad \tilde{I}_{S\;i\,j} \in D(V_{u\,v}) \wedge \tilde{F}_{i\,j}=F_{u\,v} </math>.
Here's the implemented smoothing algorithm:
:<math>I_{i\,j} = \left(1 - C_{i\,j}\right) \tilde{F}_{i\,j} + C_{i\,j}\,S_{i\,j}</math>.

== Test image ==
[[Image:SmoothBathimNesting.gif|540px|thumb|center|Animation consisting of three frames showing respectively i) the father bathymetry, ii) the son bathym, iii) and the interpolated bathym.]]

[[Category:Tools]]

← Older revision	Revision as of 10:39, 3 December 2008
(No difference)

SmoothBatimNesting - Revision history

Guillaume: 1 revision

192.168.20.177 at 12:37, 4 November 2008