6th Diva user workshop

Transcription

1 6th Diva user workshop Theory and 2-D version C. Troupin M. Ouberdous, A. Barth & J.-M. Beckers GeoHydrodynamics and Environment Research, University of Liège, Belgium modb.oce.ulg.ac.be/ Roumaillac, October 8 12, 2012

2 Diva: interpolation (gridding) of in situ data [J.W. Gregory (1916), Cyrenaica, The Geography Journal, 47 (5), ]

3 Diva: interpolation (gridding) of in situ data

4 What is Diva? Data Interpolating Variational Analysis What is Diva? a method to produce gridded field a set of scripts and Fortran programs What is not Diva? a plotting tool a black-box a numerical model! gridding and plotting are different tasks

5 DIVA: Data-Interpolating Variational Analysis N d data points d i gridded field ϕ(x j, y j ) d i Formulation: minimize cost function J[ϕ] + NX min J[ϕ] = µ i [d i ϕ(x i, y i)] 2 dataanalysis mist Z i=1 ` ϕ : ϕ + α1 ϕ ϕ + α 2 0ϕ dd eld regularity D

6 DIVA: Data-Interpolating Variational Analysis N d data points d i gridded field ϕ(x j, y j ) d i Formulation: minimize cost function J[ϕ] + NX min J[ϕ] = µ i [d i ϕ(x i, y i)] 2 dataanalysis mist Z i=1 ` ϕ : ϕ + α1 ϕ ϕ + α 2 0ϕ dd eld regularity D

7 Four ways to use Diva 1 Command line (batch processing) 2 Ocean Data View 3 Diva-on-web 4 Matlab package

8 Four ways to use Diva 1 Command line (batch processing) 2 Ocean Data View 3 Diva-on-web 4 Matlab package [R. Schlitzer (2009), Ocean Data View User s Guide, Version 4.2]

9 Four ways to use Diva 1 Command line (batch processing) 2 Ocean Data View 3 Diva-on-web 4 Matlab package gher-diva.phys.ulg.ac.be/ web-vis/ diva.html

10 Four ways to use Diva 1 Command line (batch processing) 2 Ocean Data View 3 Diva-on-web 4 Matlab package Package available at modb.oce.ulg.ac.be/ mediawiki/ upload/ divaformatlab.zip Requires executables of Diva (mesh and analysis)

11 A little bit of history Code development ( ): 2D-analysis ( ): 3D-analysis ( ): Variational Inverse Method (VIM) (Brasseur, 1991, JMS, JGR) cross-validation (Brankart and Brasseur, 1996, JAOT) error computation (Brankart and Brasseur, 1998, JMS; Rixen et al., 2000, OM) set of bash scripts (divamesh, divacalc,... ) Fortran executables parameters optimization tools Matlab/Octave scripts for plotting superposition of 2D layers automated treatment and optimization stability constraint (Ouberdous et al.) advanced error computation (Troupin et al., 2012, OM)

12 A little bit of history 4D-analysis ( ): Web tools 2010 (on-going) General: start from ODV spreadsheet detrending (with J. Carstensen, DMU) NetCDF 4-D climatology files On-line analysis (Barth et al., 2010, Adv. Geosci.) gher-diva.phys.ulg.ac.be/ web-vis/ diva.html Climatology viewer: gher-diva.phys.ulg.ac.be/ web-vis/ clim.html transition to Fortran95 multivariate approach data transformation tools 4-D graphical interface implementation of source/decay terms... user-driven developments

13 A simple example with a few points Zero background influence around data point variable radius of influence distance between data points separation by obstacles confidence in measurements

14 A simple example with a few points Square domain, data at center influence around data point 1 variable radius of influence distance between data points separation by obstacles confidence in measurements

15 A simple example with a few points Square domain, data at center influence around data point 1 variable radius of influence distance between data points separation by obstacles confidence in measurements

16 A simple example with a few points Two symmetric data points influence around data point variable radius of influence distance between data points separation by obstacles confidence in measurements

17 A simple example with a few points Two symmetric data points influence around data point variable radius of influence distance between data points separation by obstacles confidence in measurements

18 A simple example with a few points Two symmetric data points influence around data point variable radius of influence distance between data points separation by obstacles confidence in measurements

19 A simple example with a few points Physical barrier influence around data point variable radius of influence distance between data points separation by obstacles confidence in measurements

20 A simple example with a few points Data at center influence around data point 1 variable radius of influence distance between data points separation by obstacles confidence in measurements

21 Analysis parameters are related to data Non-dimensional version: L = length scale = L (1) D = L 2 D (2)

22 Analysis parameters are related to data Non-dimensional version: L = length scale = L (1) D = L 2 D (2) J[ϕ] = + NX µ il 2 [d i ϕ(x i, y i)] 2 i=1 Z D ϕ : ϕ + α1l 2 ϕ ϕ + α0l 4 ϕ 2 d D

23 Analysis parameters are related to data Non-dimensional version: L = length scale = L (1) D = L 2 D (2) J[ϕ] = + NX µ il 2 [d i ϕ(x i, y i)] 2 i=1 Z D ϕ : ϕ + α1l 2 ϕ ϕ + α0l 4 ϕ 2 d D α 0 L for which data-analysis misfit regularity term: α 0L 4 = 1 α 1 influence of gradients: α 1L 2 = 2ξ, ξ = 1 µ il 2 weight on data: µ il 2 = 4π signal noise i

24 Analysis parameters are related to data Non-dimensional version: L = length scale = L (1) D = L 2 D (2) J[ϕ] = + NX µ il 2 [d i ϕ(x i, y i)] 2 i=1 Z D ϕ : ϕ + α1l 2 ϕ ϕ + α0l 4 ϕ 2 d D α 0 L for which data-analysis misfit regularity term: α 0L 4 = 1 α 1 influence of gradients: α 1L 2 = 2ξ, ξ = 1 µ il 2 weight on data: µ il 2 = 4π signal noise i

25 Analysis parameters are related to data Non-dimensional version: L = length scale = L (1) D = L 2 D (2) J[ϕ] = + NX µ il 2 [d i ϕ(x i, y i)] 2 i=1 Z D ϕ : ϕ + α1l 2 ϕ ϕ + α0l 4 ϕ 2 d D α 0 L for which data-analysis misfit regularity term: α 0L 4 = 1 α 1 influence of gradients: α 1L 2 = 2ξ, ξ = 1 µ il 2 weight on data: µ il 2 = 4π signal noise i

26 Analysis parameters are related to data Non-dimensional version: L = length scale = L (1) D = L 2 D (2) J[ϕ] = + NX µ il 2 [d i ϕ(x i, y i)] 2 i=1 Z D ϕ : ϕ + α1l 2 ϕ ϕ + α0l 4 ϕ 2 d D Coefficients α 0, α 1 and µ i related to 1 Correlation length L 2 Signal-to-noise λ 3 Observational noise standard deviation ɛ 2 i

27 Main analysis parameters Correlation length L: Measure of the influence of data points Estimated by a least-square fit of the covariance function Covariance data covariance data used for fitting fitted curve Signal-to-noise ratio λ: Measure of the confidence in data Estimated with Generalized Cross Validation techniques Θ Distance (km) GCV Random CV CV λ

28 Parameters: smoothness vs. data influence Data C Over-smoothed field C 44 o N o N L = 0.3 λ = o N o N o N 40 o N o N o N o N 0 o 2 o E 4 o E 6 o E 8 o E 10 o E 12 o E o N 0 o 2 o E 4 o E 6 o E 8 o E 10 o E 12 o E 12

29 Parameters: smoothness vs. data influence Data Too strong influence of data C C 44 o N o N L = 1 λ = o N o N o N 40 o N o N o N o N 0 o 2 o E 4 o E 6 o E 8 o E 10 o E 12 o E o N 0 o 2 o E 4 o E 6 o E 8 o E 10 o E 12 o E 12

30 Parameters: smoothness vs. data influence Data C Good balance C 44 o N o N L = 2 λ = o N o N o N 40 o N o N o N o N 0 o 2 o E 4 o E 6 o E 8 o E 10 o E 12 o E o N 0 o 2 o E 4 o E 6 o E 8 o E 10 o E 12 o E 12

31 Minimization with a finite-element method Field regularity plate bending problem finite-element solver 60 o N 64 o N 45 o N 57 o N 62 o N 44 o N 54 o N 60 o N 58 o N 43 o N 42 o N 51 o N 56 o N 41 o N 3 o W 0 o 3 o E 6 o E 9 o E 12 o E 54 o N 5 o E 10 o E 15 o E 20 o E 25 o E 40 o N 14 o E 16 o E 18 o E Advantages: boundaries taken into account numerical cost (almost independent on data number) no a posteriori masking

32 Minimization with a finite-element method XN e Triangular FE only covers sea: J[ϕ] = J e(ϕ e) (3) 8 < s In each element: ϕ e(r e) = q T e s(r e) with : q (4) in (3) + variational principle r e N de e=1 shape functions connectors position (4) X J e(q e) = q T e K eq e 2q T e g e + µ id i (5) i=1 where j Ke g local stiffness matrix vector depending on local data

33 Minimization with a finite-element method N X d On the whole domain: J(q) = q T Kq 2q T g + µ id i (3) i=1 Minimum: q = K 1 g (4) q = K 1 g (5) Stiffness matrix Connectors (new unknowns) Charge vector Mapping of data on FEM transfer operator T 2 g = T 2(r)d Solution at any location transfer operator T 1 ϕ(r) = T 1(r)q Results obtained at any location ϕ = T 1(r)K 1 T 2(r)d