6th diva user workshop - uliege.be · roumaillac (france), october 8–12, 2012 diva workshop. a...

6th Diva user workshopTheory and 2-D version

C. Troupin ([email protected]), M. Ouberdous,A. Barth & J.-M. Beckers

GeoHydrodynamics and Environment Research,University of Liège, Belgiumhttp:// modb.oce.ulg.ac.be/

Roumaillac, October 8–12, 2012

Roumaillac (France), October 8–12, 2012 Diva workshop

http://modb.oce.ulg.ac.be/

Diva: interpolation (gridding) of in situ data

[J.W. Gregory (1916), Cyrenaica, The Geography Journal, 47 (5), 321-342]


Diva: interpolation (gridding) of in situ data


What is Diva?

DataInterpolatingVariationalAnalysis

What is Diva?a method to produce gridded fielda set of scripts and Fortran programs

What is not Diva?a plotting toola black-boxa numerical model

! gridding and plotting are different tasks


DIVA: Data-Interpolating Variational Analysis

Nd data points di • → gridded field

di

ϕ(xj , yj)

Formulation: minimize cost function J [ϕ]

min J [ϕ] =

NXi=1

µi [di − ϕ(xi, yi)]2

data�analysis mis�t

+

ZD

`∇∇ϕ : ∇∇ϕ+ α1∇ϕ ·∇ϕ+ α0ϕ

2´ dD �eld regularity


Four ways to use Diva

1 Command line (batch processing)2 Ocean Data View3 Diva-on-web4 Matlab package



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]



1 Command line (batch processing)2 Ocean Data View3 Diva-on-web4 Matlab package

http:// gher-diva.phys.ulg.ac.be/ web-vis/ diva.html


http://gher-diva.phys.ulg.ac.be/web-vis/diva.html


1 Command line (batch processing)

2 Ocean Data View

3 Diva-on-web

4 Matlab package

Package available athttp:// modb.oce.ulg.ac.be/ mediawiki/ upload/ divaformatlab.zip

Requires executables of Diva (mesh and analysis)


http://modb.oce.ulg.ac.be/mediawiki/upload/divaformatlab.zip

A little bit of history

Code development (1990-1996):

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)

2D-analysis (2006-2007):

set of bash scripts (divamesh, divacalc, . . . )Fortran executablesparameters optimization toolsMatlab/Octave scripts for plotting

3D-analysis (2007-2008):

superposition of 2D layersautomated treatment and optimizationstability constraint (Ouberdous et al.)advanced error computation (Troupin et al., 2012, OM)


A little bit of history

4D-analysis (2008-2009):

start from ODV spreadsheetdetrending (with J. Carstensen, DMU)NetCDF 4-D climatology files

Web tools On-line analysis (Barth et al., 2010, Adv. Geosci.)http:// gher-diva.phys.ulg.ac.be/ web-vis/ diva.htmlClimatology viewer:http:// gher-diva.phys.ulg.ac.be/ web-vis/ clim.html

2010 (on-going)transition to Fortran95multivariate approachdata transformation tools4-D graphical interfaceimplementation of source/decay terms. . .

General: user-driven developments


http://gher-diva.phys.ulg.ac.be/web-vis/diva.html

http://gher-diva.phys.ulg.ac.be/web-vis/clim.html

A simple example with a few points

Zero background

influence around data pointvariable radius of influencedistance between datapointsseparation by obstaclesconfidence inmeasurements



Square domain, data at center

+1

−5 0 5−5

0

5

0

0.2

0.4

0.6

0.8

1influence around data pointvariable radius of influencedistance between datapointsseparation by obstaclesconfidence inmeasurements



Two symmetric data points

−1 +1

−5 0 5−5

0

5

−1

−0.5

0

0.5




Physical barrier

−1 +1

−5 0 5−5

0

5

−1

−0.5

0

0.5




Data at center

+1

−5 0 5−5

0

5

0

0.2

0.4

0.6

0.8



Analysis parameters are related to data

Non-dimensional version:

L = length scale → ∇̃ = L∇ (1)

→ D = L2D̃ (2)





→ D = L2D̃ (2)

J̃ [ϕ] =

NXi=1

µiL2[di − ϕ(xi, yi)]

2

+

ZD̃

“∇̃∇̃ϕ : ∇̃∇̃ϕ+ α1L

2∇̃ϕ · ∇̃ϕ+ α0L4ϕ2”

dD̃





→ D = L2D̃ (2)

J̃ [ϕ] =

NXi=1


2

+

ZD̃

“∇̃∇̃ϕ : ∇̃∇̃ϕ+ α1L

2∇̃ϕ · ∇̃ϕ+ α0L4ϕ2”

dD̃

α0 →L for which data-analysis misfit ' regularity term: α0L4 = 1

α1 → influence of gradients: α1L2 = 2ξ, ξ = 1

µiL2 →weight on data: µiL

2 = 4πsignal

noisei





→ D = L2D̃ (2)

J̃ [ϕ] =

NXi=1


2

+

ZD̃

“∇̃∇̃ϕ : ∇̃∇̃ϕ+ α1L

2∇̃ϕ · ∇̃ϕ+ α0L4ϕ2”

dD̃

Coefficients α0, α1 and µi related to

1 Correlation length L

2 Signal-to-noise λ

3 Observational noise standard deviation ε2i


Main analysis parameters

Correlation length L:

Measure of the influence of datapoints

Estimated by a least-square fit ofthe covariance function

Signal-to-noise ratio λ:

Measure of the confidence indata

Estimated with GeneralizedCross Validation techniques

0 100 2000

0.03

0.06

0.09

0.12

Distance (km)

Cov

aria

nce

data covariancedata used for fittingfitted curve

0.01 0.1 1 9.19 32.42 100

0.6653

0.6764

0.6964

0.7525

λ

Θ

GCVRandom CVCV


Parameters: smoothness vs. data influence

Data

0o 2oE 4oE 6oE 8oE 10oE 12oE 36oN

38oN

40oN

42oN

44oN

° C

12

13

14

15

16

17

Over-smoothed field


38oN

40oN

42oN

44oN

L = 0.3°

λ = 0.01

° C

12

13

14

15

16

17



Data


38oN

40oN

42oN

44oN

° C

12

13

14

15

16

17

Too strong influence of data


38oN

40oN

42oN

44oN

L = 1°

λ = 300

° C

12

13

14

15

16

17



Data


38oN

40oN

42oN

44oN

° C

12

13

14

15

16

17

Good balance


38oN

40oN

42oN

44oN

L = 2°

λ = 1

° C

12

13

14

15

16

17


Minimization with a finite-element method

Field regularity→plate bending problem→ finite-element solver

3oW 0o 3oE 6oE 9oE 12oE

51oN

54oN

57oN

60oN

5oE 10oE 15oE 20oE 25oE

54oN

56oN

58oN

60oN

62oN

64oN

14oE 16oE 18oE 40oN

41oN

42oN

43oN

44oN

45oN

Advantages:

boundaries taken into account

numerical cost (almost independent on data number)

no a posteriori masking



Triangular FE only covers sea: J [ϕ] =

NeXe=1

Je(ϕe) (3)

In each element: ϕe(re) = qeT s(re) with

8<:s → shape functionsq → connectorsre → position

(4)(4) in (3) + variational principle

Je(qe) = qeT Keqe − 2qe

T ge +

NdeXi=1

µidi (5)

where

Ke → local stiffness matrixg → vector depending on local data



On the whole domain: J(q) = qT Kq− 2qT g +

NdXi=1

µidi (3)

Minimum: q = K−1g (4)

q = K−1 g (5)

Stiffness matrix

Connectors (new unknowns)

Charge vector

Mapping of data on FEM→ transfer operator T2 → g = T2(r)d

Solution at any location→ transfer operator T1 → ϕ(r) = T1(r)q

Results obtained at any location→ ϕ = T1(r)K−1T2(r)d


6th diva user workshop - uliege.be · roumaillac (france), october 8–12, 2012 diva workshop. a...

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