Regularization techniques for ill-posed inverse problems in data assimilation

Optimal state estimation from given observations of a dynamical system by data assimilation is generally an ill-posed inverse problem. In order to solve the problem, a standard Tikhonov, or L2, regularization is used, based on certain statistical assumptions on the errors in the data. The regularization term constrains the estimate of the state to remain close to a prior estimate. In the presence of model error, this approach does not capture the initial state of the system accurately, as the initial state estimate is derived by minimizing the average error between the model predictions and the observations over a time window. Here we examine an alternative L1 regularization technique that has proved valuable in image processing. We show that for examples of flow with sharp fronts and shocks, the L1 regularization technique performs more accurately than standard L2 regularization.

On discrimination algorithms for ill-posed problems with an application to magnetic tomography

In this paper we explore classification techniques for ill-posed problems. Two classes are linearly separable in some Hilbert space X if they can be separated by a hyperplane. We investigate stable separability, i.e. the case where we have a positive distance between two separating hyperplanes. When the data in the space Y is generated by a compact operator A applied to the system states ∈ X, we will show that in general we do not obtain stable separability in Y even if the problem in X is stably separable. In particular, we show this for the case where a nonlinear classification is generated from a non-convergent family of linear classes in X. We apply our results to the problem of quality control of fuel cells where we classify fuel cells according to their efficiency. We can potentially classify a fuel cell using either some external measured magnetic field or some internal current. However we cannot measure the current directly since we cannot access the fuel cell in operation. The first possibility is to apply discrimination techniques directly to the measured magnetic fields. The second approach first reconstructs currents and then carries out the classification on the current distributions. We show that both approaches need regularization and that the regularized classifications are not equivalent in general. Finally, we investigate a widely used linear classification algorithm Fisher's linear discriminant with respect to its ill-posedness when applied to data generated via a compact integral operator. We show that the method cannot stay stable when the number of measurement points becomes large.

L1-regularisation for ill-posed problems in variational data assimilation

We consider four-dimensional variational data assimilation (4DVar) and show that it can be interpreted as Tikhonov or L2-regularisation, a widely used method for solving ill-posed inverse problems. It is known from image restoration and geophysical problems that an alternative regularisation, namely L1-norm regularisation, recovers sharp edges better than L2-norm regularisation. We apply this idea to 4DVar for problems where shocks and model error are present and give two examples which show that L1-norm regularisation performs much better than the standard L2-norm regularisation in 4DVar.

Pronoun ambiguity resolution in Greek: evidence from monolingual adults and children

A large body of psycholinguistic research has revealed that during sentence interpretation adults coordinate multiple sources of information. Particularly, they draw both on linguistic properties of the message and on information from the context to constrain their interpretations. Relatively little however is known about how this integrative processor develops through language acquisition and about how children process language. In this study, two on-line picture verification tasks were used to examine how 1st, 2nd and 4th/5th grade monolingual Greek children resolve pronoun ambiguities during sentence interpretation and how their performance compares to that of adults on the same tasks. Specifically, we manipulated the type of subject pronoun, i.e. null or overt, and examined how this affected participants’ preferences for competing antecedents, i.e. in the subject or object position. The results revealed both similarities and differences in how adults and the various child groups comprehended ambiguous pronominal forms. Particularly, although adults and children alike showed sensitivity to the distribution of overt and null subject pronouns, this did not always lead to convergent interpretation preferences.

Resolution of sharp fronts in the presence of model error in variational data assimilation

We show that the four-dimensional variational data assimilation method (4DVar) can be interpreted as a form of Tikhonov regularization, a very familiar method for solving ill-posed inverse problems. It is known from image restoration problems that L1-norm penalty regularization recovers sharp edges in the image more accurately than Tikhonov, or L2-norm, penalty regularization. We apply this idea from stationary inverse problems to 4DVar, a dynamical inverse problem, and give examples for an L1-norm penalty approach and a mixed total variation (TV) L1–L2-norm penalty approach. For problems with model error where sharp fronts are present and the background and observation error covariances are known, the mixed TV L1–L2-norm penalty performs better than either the L1-norm method or the strong constraint 4DVar (L2-norm)method. A strength of the mixed TV L1–L2-norm regularization is that in the case where a simplified form of the background error covariance matrix is used it produces a much more accurate analysis than 4DVar. The method thus has the potential in numerical weather prediction to overcome operational problems with poorly tuned background error covariance matrices.

Very large inverse problems in atmosphere and ocean modelling

For the very large nonlinear dynamical systems that arise in a wide range of physical, biological and environmental problems, the data needed to initialize a numerical forecasting model are seldom available. To generate accurate estimates of the expected states of the system, both current and future, the technique of ‘data assimilation’ is used to combine the numerical model predictions with observations of the system measured over time. Assimilation of data is an inverse problem that for very large-scale systems is generally ill-posed. In four-dimensional variational assimilation schemes, the dynamical model equations provide constraints that act to spread information into data sparse regions, enabling the state of the system to be reconstructed accurately. The mechanism for this is not well understood. Singular value decomposition techniques are applied here to the observability matrix of the system in order to analyse the critical features in this process. Simplified models are used to demonstrate how information is propagated from observed regions into unobserved areas. The impact of the size of the observational noise and the temporal position of the observations is examined. The best signal-to-noise ratio needed to extract the most information from the observations is estimated using Tikhonov regularization theory. Copyright © 2005 John Wiley & Sons, Ltd.

Inverse problems in dynamic cognitive modeling

Inverse problems for dynamical system models of cognitive processes comprise the determination of synaptic weight matrices or kernel functions for neural networks or neural/dynamic field models, respectively. We introduce dynamic cognitive modeling as a three tier top-down approach where cognitive processes are first described as algorithms that operate on complex symbolic data structures. Second, symbolic expressions and operations are represented by states and transformations in abstract vector spaces. Third, prescribed trajectories through representation space are implemented in neurodynamical systems. We discuss the Amari equation for a neural/dynamic field theory as a special case and show that the kernel construction problem is particularly ill-posed. We suggest a Tikhonov-Hebbian learning method as regularization technique and demonstrate its validity and robustness for basic examples of cognitive computations.

Inverse problems in neural field theory

We study inverse problems in neural field theory, i.e., the construction of synaptic weight kernels yielding a prescribed neural field dynamics. We address the issues of existence, uniqueness, and stability of solutions to the inverse problem for the Amari neural field equation as a special case, and prove that these problems are generally ill-posed. In order to construct solutions to the inverse problem, we first recast the Amari equation into a linear perceptron equation in an infinite-dimensional Banach or Hilbert space. In a second step, we construct sets of biorthogonal function systems allowing the approximation of synaptic weight kernels by a generalized Hebbian learning rule. Numerically, this construction is implemented by the Moore–Penrose pseudoinverse method. We demonstrate the instability of these solutions and use the Tikhonov regularization method for stabilization and to prevent numerical overfitting. We illustrate the stable construction of kernels by means of three instructive examples.

Probabilistic downscaling of remote sensing data with applications for multi-scale biogeochemical flux modeling

Upscaling ecological information to larger scales in space and downscaling remote sensing observations or model simulations to finer scales remain grand challenges in Earth system science. Downscaling often involves inferring subgrid information from coarse-scale data, and such ill-posed problems are classically addressed using regularization. Here, we apply two-dimensional Tikhonov Regularization (2DTR) to simulate subgrid surface patterns for ecological applications. Specifically, we test the ability of 2DTR to simulate the spatial statistics of high-resolution (4 m) remote sensing observations of the normalized difference vegetation index (NDVI) in a tundra landscape. We find that the 2DTR approach as applied here can capture the major mode of spatial variability of the high-resolution information, but not multiple modes of spatial variability, and that the Lagrange multiplier (γ) used to impose the condition of smoothness across space is related to the range of the experimental semivariogram. We used observed and 2DTR-simulated maps of NDVI to estimate landscape-level leaf area index (LAI) and gross primary productivity (GPP). NDVI maps simulated using a γ value that approximates the range of observed NDVI result in a landscape-level GPP estimate that differs by ca 2% from those created using observed NDVI. Following findings that GPP per unit LAI is lower near vegetation patch edges, we simulated vegetation patch edges using multiple approaches and found that simulated GPP declined by up to 12% as a result. 2DTR can generate random landscapes rapidly and can be applied to disaggregate ecological information and compare of spatial observations against simulated landscapes.

On integral equation and least squares methods for scattering by diffraction gratings

In this paper we consider the scattering of a plane acoustic or electromagnetic wave by a one-dimensional, periodic rough surface. We restrict the discussion to the case when the boundary is sound soft in the acoustic case, perfectly reflecting with TE polarization in the EM case, so that the total field vanishes on the boundary. We propose a uniquely solvable first kind integral equation formulation of the problem, which amounts to a requirement that the normal derivative of the Green's representation formula for the total field vanish on a horizontal line below the scattering surface. We then discuss the numerical solution by Galerkin's method of this (ill-posed) integral equation. We point out that, with two particular choices of the trial and test spaces, we recover the so-called SC (spectral-coordinate) and SS (spectral-spectral) numerical schemes of DeSanto et al., Waves Random Media, 8, 315-414 1998. We next propose a new Galerkin scheme, a modification of the SS method that we term the SS* method, which is an instance of the well-known dual least squares Galerkin method. We show that the SS* method is always well-defined and is optimally convergent as the size of the approximation space increases. Moreover, we make a connection with the classical least squares method, in which the coefficients in the Rayleigh expansion of the solution are determined by enforcing the boundary condition in a least squares sense, pointing out that the linear system to be solved in the SS* method is identical to that in the least squares method. Using this connection we show that (reflecting the ill-posed nature of the integral equation solved) the condition number of the linear system in the SS* and least squares methods approaches infinity as the approximation space increases in size. We also provide theoretical error bounds on the condition number and on the errors induced in the numerical solution computed as a result of ill-conditioning. Numerical results confirm the convergence of the SS* method and illustrate the ill-conditioning that arises.

Nonlinear set-membership estimation: a support vector machine approach

In this paper a support vector machine (SVM) approach for characterizing the feasible parameter set (FPS) in non-linear set-membership estimation problems is presented. It iteratively solves a regression problem from which an approximation of the boundary of the FPS can be determined. To guarantee convergence to the boundary the procedure includes a no-derivative line search and for an appropriate coverage of points on the FPS boundary it is suggested to start with a sequential box pavement procedure. The SVM approach is illustrated on a simple sine and exponential model with two parameters and an agro-forestry simulation model.

On instabilities in data assimilation algorithms

Data assimilation algorithms are a crucial part of operational systems in numerical weather prediction, hydrology and climate science, but are also important for dynamical reconstruction in medical applications and quality control for manufacturing processes. Usually, a variety of diverse measurement data are employed to determine the state of the atmosphere or to a wider system including land and oceans. Modern data assimilation systems use more and more remote sensing data, in particular radiances measured by satellites, radar data and integrated water vapor measurements via GPS/GNSS signals. The inversion of some of these measurements are ill-posed in the classical sense, i.e. the inverse of the operator H which maps the state onto the data is unbounded. In this case, the use of such data can lead to significant instabilities of data assimilation algorithms. The goal of this work is to provide a rigorous mathematical analysis of the instability of well-known data assimilation methods. Here, we will restrict our attention to particular linear systems, in which the instability can be explicitly analyzed. We investigate the three-dimensional variational assimilation and four-dimensional variational assimilation. A theory for the instability is developed using the classical theory of ill-posed problems in a Banach space framework. Further, we demonstrate by numerical examples that instabilities can and will occur, including an example from dynamic magnetic tomography.

Aerosol retrieval experiments in the ESA Aerosol_cci project

Within the ESA Climate Change Initiative (CCI) project Aerosol_cci (2010–2013), algorithms for the production of long-term total column aerosol optical depth (AOD) datasets from European Earth Observation sensors are developed. Starting with eight existing pre-cursor algorithms three analysis steps are conducted to improve and qualify the algorithms: (1) a series of experiments applied to one month of global data to understand several major sensitivities to assumptions needed due to the ill-posed nature of the underlying inversion problem, (2) a round robin exercise of "best" versions of each of these algorithms (defined using the step 1 outcome) applied to four months of global data to identify mature algorithms, and (3) a comprehensive validation exercise applied to one complete year of global data produced by the algorithms selected as mature based on the round robin exercise. The algorithms tested included four using AATSR, three using MERIS and one using PARASOL. This paper summarizes the first step. Three experiments were conducted to assess the potential impact of major assumptions in the various aerosol retrieval algorithms. In the first experiment a common set of four aerosol components was used to provide all algorithms with the same assumptions. The second experiment introduced an aerosol property climatology, derived from a combination of model and sun photometer observations, as a priori information in the retrievals on the occurrence of the common aerosol components. The third experiment assessed the impact of using a common nadir cloud mask for AATSR and MERIS algorithms in order to characterize the sensitivity to remaining cloud contamination in the retrievals against the baseline dataset versions. The impact of the algorithm changes was assessed for one month (September 2008) of data: qualitatively by inspection of monthly mean AOD maps and quantitatively by comparing daily gridded satellite data against daily averaged AERONET sun photometer observations for the different versions of each algorithm globally (land and coastal) and for three regions with different aerosol regimes. The analysis allowed for an assessment of sensitivities of all algorithms, which helped define the best algorithm versions for the subsequent round robin exercise; all algorithms (except for MERIS) showed some, in parts significant, improvement. In particular, using common aerosol components and partly also a priori aerosol-type climatology is beneficial. On the other hand the use of an AATSR-based common cloud mask meant a clear improvement (though with significant reduction of coverage) for the MERIS standard product, but not for the algorithms using AATSR. It is noted that all these observations are mostly consistent for all five analyses (global land, global coastal, three regional), which can be understood well, since the set of aerosol components defined in Sect. 3.1 was explicitly designed to cover different global aerosol regimes (with low and high absorption fine mode, sea salt and dust).

Improving flood models using LiDAR: progress and problems

Airborne scanning laser altimetry (LiDAR) is an important new data source for river flood modelling. LiDAR can give dense and accurate DTMs of floodplains for use as model bathymetry. Spatial resolutions of 0.5m or less are possible, with a height accuracy of 0.15m. LiDAR gives a Digital Surface Model (DSM), so vegetation removal software (e.g. TERRASCAN) must be used to obtain a DTM. An example used to illustrate the current state of the art will be the LiDAR data provided by the EA, which has been processed by their in-house software to convert the raw data to a ground DTM and separate vegetation height map. Their method distinguishes trees from buildings on the basis of object size. EA data products include the DTM with or without buildings removed, a vegetation height map, a DTM with bridges removed, etc. Most vegetation removal software ignores short vegetation less than say 1m high. We have attempted to extend vegetation height measurement to short vegetation using local height texture. Typically most of a floodplain may be covered in such vegetation. The idea is to assign friction coefficients depending on local vegetation height, so that friction is spatially varying. This obviates the need to calibrate a global floodplain friction coefficient. It’s not clear at present if the method is useful, but it’s worth testing further. The LiDAR DTM is usually determined by looking for local minima in the raw data, then interpolating between these to form a space-filling height surface. This is a low pass filtering operation, in which objects of high spatial frequency such as buildings, river embankments and walls may be incorrectly classed as vegetation. The problem is particularly acute in urban areas. A solution may be to apply pattern recognition techniques to LiDAR height data fused with other data types such as LiDAR intensity or multispectral CASI data. We are attempting to use digital map data (Mastermap structured topography data) to help to distinguish buildings from trees, and roads from areas of short vegetation. The problems involved in doing this will be discussed. A related problem of how best to merge historic river cross-section data with a LiDAR DTM will also be considered. LiDAR data may also be used to help generate a finite element mesh. In rural area we have decomposed a floodplain mesh according to taller vegetation features such as hedges and trees, so that e.g. hedge elements can be assigned higher friction coefficients than those in adjacent fields. We are attempting to extend this approach to urban area, so that the mesh is decomposed in the vicinity of buildings, roads, etc as well as trees and hedges. A dominant points algorithm is used to identify points of high curvature on a building or road, which act as initial nodes in the meshing process. A difficulty is that the resulting mesh may contain a very large number of nodes. However, the mesh generated may be useful to allow a high resolution FE model to act as a benchmark for a more practical lower resolution model. A further problem discussed will be how best to exploit data redundancy due to the high resolution of the LiDAR compared to that of a typical flood model. Problems occur if features have dimensions smaller than the model cell size e.g. for a 5m-wide embankment within a raster grid model with 15m cell size, the maximum height of the embankment locally could be assigned to each cell covering the embankment. But how could a 5m-wide ditch be represented? Again, this redundancy has been exploited to improve wetting/drying algorithms using the sub-grid-scale LiDAR heights within finite elements at the waterline.