Thin plate splines with tension

The Data Interpolating Variational Analysis (Diva) is a method designed to interpolate irregularly-spaced, noisy data onto any desired location, in most cases on regular grids. It is the combination of a particular methodology, based on the minimisation of a cost function, and a numerically efficient method, based on a finite-element solver. The cost function penalises the misfit between the observations and the reconstructed field, as well as the regularity or smoothness of the field. The method bears similarities to the smoothing splines, where the second derivatives of the field are also penalised.The intrinsic advantages of the method are its natural way to take into account topographic and dynamic constraints (coasts, advection, . . . ) and its capacity to handle large data sets, frequently encountered in oceanography. The method provides gridded fields in two dimensions, usually in horizontal layers. Three-dimension fields are obtained by stacking horizontal layers.In the present work, we summarize the background of the method and describe the possible methods to compute the error field associated to the analysis. In particular, we present new developments leading to a more consistent error estimation, by determining numerically the real covariance function in Diva, which is never formulated explicitly, contrarily to Optimal Interpolation. The real covariance function is obtained by two concurrent executions of Diva, the first providing the covariance for the second. With this improvement, the error field is now perfectly consistent with the inherent background covariance in all cases.A two-dimension application using salinity measurements in the Mediterranean Sea is presented. Applied on these measurements, Optimal Interpolation and Diva provided very similar gridded fields (correlation: 98.6%, RMS of the difference: 0.02). The method using the real covariance produces an error field similar to the one of OI, except in the coastal areas.

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“…The first term of (1) models "tensions" on a thin plate, hence the term spline in tension (Schweikert, 1966;Franke, 1985),…”

Section: Formulationmentioning

confidence: 99%

Generation of analysis and consistent error fields using the Data Interpolating Variational Analysis (DIVA)

et al. 2012

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“…Exceptions were when the number of knots was quite small ( The tension effects are limited compared with the results that can be obtained using thin plate splines with tension (see [FR85] In the case given in [MF92] where the data was specifically generated to reflect the curvature of the underlying surface, the knots computed by this algorithm tend to be gathered in regions where the density of data points is greatest. Figure 1 gives the results in one case.…”

Section: Some Resultsmentioning

confidence: 99%

Least Squares Surface Approximation to Scattered Data Using Multiquadric Functions

Franke¹,

Hagen²,

Nielson³

1992

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“…Sa généralisation aux fonctions à deux variables a été donnée, d'une façon empirique, par Franke [6]. L'étude des splines sous tension dans un cadre d'espaces fonctionnels « de type Sobolev » a été faite dans [2].…”

Section: Interpolation Par Splines Sous Tensionunclassified

Estimation de l'erreur pour l'interpolation par des splines de type plaque mince sous tension

Bouhamidi

2006

Comptes Rendus. Mathématique

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Thin plate splines with tension

Cited by 61 publications

References 10 publications

Generation of analysis and consistent error fields using the Data Interpolating Variational Analysis (DIVA)

Generation of analysis and consistent error fields using the Data Interpolating Variational Analysis (DIVA)

Least Squares Surface Approximation to Scattered Data Using Multiquadric Functions

Estimation de l'erreur pour l'interpolation par des splines de type plaque mince sous tension

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