Inverse Problems

A model of the multibeam echosounding process was developed. This model has now been used as the basis for the application of a model inversion technique, with the aim of analyzing midwater multibeam echosounder data, for fisheries applications.

The magnetic field around the Moon has been successfully observed at a nominal altitude of ∼100 km by the lunar magnetometer (LMAG) on the SELENE (Kaguya) spacecraft in a polar orbit since October 29, 2007. The LMAG mission has three main objectives: (1) mapping the magnetic anomaly of the Moon, (2) measuring the electromagnetic and plasma environment around the Moon and (3) estimating the electrical conductivity structure of the Moon. Here we review the instrumentation and calibration of LMAG and report the initial global mapping of the lunar magnetic anomaly at the nominal altitude. We have applied a new de-trending technique of the Bayesian procedure to multiple-orbit datasets observed in the tail lobe and in the lunar wake. Based on the nominal observation of 14 months, global maps of lunar magnetic anomalies are obtained with 95% coverage of the lunar surface. After altitude normalization and interpolation of the magnetic anomaly field by an inverse boundary value problem, we obtained full-coverage maps of the vector magnetic field at 100 km altitude and the radial component distribution on the surface. Relatively strong anomalies are identified in several basin-antipode regions and several near-basin and near-crater regions, while the youngest basin on the Moon, the Orientale basin, has no magnetic anomaly. These features well agree with characteristics of previous maps based on the

The availability and performance of laser-based adaptive optics (AO) systems are strongly dependent on the power and quality of the laser beam before being projected to the sky. Frequent and time-consuming alignment procedures are usually required in the laser systems with free-space optics to optimize the beam. Despite these procedures, significant distortions of the laser beam have been observed during the first two years of operation of the Gemini South multi-conjugate adaptive optics system (GeMS). A beam shaping concept with two deformable mirrors is investigated in order to provide automated optimization of the laser quality for astronomical AO. This study aims at demonstrating the correction of quasi-static aberrations of the laser, in both amplitude and phase, testing a prototype of this two-deformable mirror concept on GeMS. The paper presents the results of the preparatory study before the experimental phase. An algorithm to control amplitude and phase correction, based on phase retrieval techniques, is presented with a novel unwrapping method. Its performance is assessed via numerical simulations, using aberrations measured at GeMS as reference. The results predict effective amplitude and phase correction of the laser distortions with about 120 actuators per mirror and a separation of 1.4 m between the mirrors. The spot size is estimated to be reduced by up to 15% thanks to the correction. In terms of AO noise level, this has the same benefit as increasing the photon flux by 40%.

A new model describing laser noise in terms of non-Markovian stochastic phase jumps has been recently introduced. In this letter an exact expression for the correlation function for the electric field is obtained. This expression, valid for all possible values of the parameters of the stochastic process, reduces to the previously reported approximations. A procedure is described for the inverse problem of determining the stochastic parameters from measured data for real lasers.

The uncertain conductivity value of skull and brain tissue influences the accuracy of the electroencephalogram (EEG) inverse problem solution. Indeed, when the assumed conductivity in the numerical procedure is different from the actual conductivity then a source localization error is introduced. When using traditional least-squares minimization methods, the number of electrodes in the EEG cap does not influence the spatial resolution. A recently developed reduced conductivity dependence (RCD) methodology, based on the selection of electrodes, is able to increase the spatial resolution of the EEG inverse problem. This paper presents the implications of the RCD method when using a large number of electrodes in the EEG cap on the spatial resolution of the EEG inverse solutions. We show by means of numerical experiments that in contrast to traditional methods, the RCD method enables to increase the spatial resolution. The computations show that the EEG hardware should be modified with as large as possible electrodes.

—Hough transform (HT) is a typical method to detect or segment geometry objects from images. In this paper, we study the principle of Hough Transform and its mathematical expressions, and try to use a new approach based on Hough transform for quick line and circle detection in image processing. Our method accurately detected some simple graphics, such as straight line of different direction, circles of different detection, thickness and different number. The results show that our method is less memory consumption and calculated fast, which could be applied for line detection and segmentation in 3D ultrasonic image.

In this paper, the deterministic parameter identification procedure proposed in a companion paper is applied to suction-dependent elasto-plasticity problems. A mathematical model for such type of problems is firstly presented, then it is applied to the parameter identification using laboratory data. The identification procedure is applied in a second example to exploitation of a gas reservoir. The effects of the

Projection data obtained through optical techniques for tomographic measurements, such as interferometry for refractive-index-based measurements, are often incomplete. This is due to limitations in the optical system, data storage, and alignment and vignette issues. Algebraic iterative reconstruction techniques are usually favored for such incomplete projections. A number of iterative algorithms, based on additive and multiplicative corrections, are used with a known simulated phantom and noise source to assess the reconstruction performance of incomplete data sets. In addition, we present reconstructions using experimental data obtained from a coherent gradient sensing interferometer for a steady temperature field in a fluid medium. We tested the algorithms using the simulated data set for incompleteness conditions similar to those found in the experimental data, and the best-performing algorithm is identified.

With respect to the microwave imaging of the dielectric properties in an imaging region, the full derivation of a new inversion algorithm based on the contrast source inversion (CSI) algorithm and a finite-element method (FEM) discretization of the Helmholtz differential operator formulation for the scattered electromagnetic field is presented. The unknown dielectric properties are represented as nodal values on a two-dimensional (2D) arbitrary triangular mesh using linear basis functions. The use of FEM to represent the Helmholtz operator allows for the flexibility of having an inhomogeneous background medium, as well as the ability to accurately model any boundary shape or type: both conducting and absorbing. The resulting sparse and symmetric FEM matrix equation can be solved efficiently, and it is shown how its solution can be used to calculate the gradient operators required in the conjugate-gradient CSI update without storing the inverse of the FEM matrix. The inversion algorithm is applied to conductive-enclosures of various shapes and unbounded-region microwave tomography configurations where the 2D transverse magnetic (TM) approximation can be applied.

This paper concerns the reconstruction of an anisotropic conductivity tensor $\gamma$ from internal current densities of the form $J = \gamma\nabla u$, where $u$ solves a second-order elliptic equation $\nabla\cdot(\gamma\nabla u) = 0$ on a bounded domain $X$ with prescribed boundary conditions. A minimum number of such functionals equal to $n + 2$, where $n$ is the spatial dimension, is sufficient to guarantee a local reconstruction. We show that $\gamma$ can be uniquely reconstructed with a loss of one derivative compared to errors in the measurement of $J$. In the special case where $\gamma$ is scalar, it can be reconstructed with no loss of derivatives. We provide a precise statement of what components may be reconstructed with a loss of zero or one derivatives.

In this paper, we investigate the inverse problem of determining the potential of the Schrödinger equation from finite measurements on the boundary. It is well known that this is an ill posed problem in the sense of Hadamard. The stability estimate is proved under the assumption that the potentials have some a priori constraints. Based on this conditional stability result, we propose one kind of Tikhonov regularization and prove the convergence rate for the regularized solution.

Capability in accurately determining the permittivity of a dielectric specimen is significant for a number of reasons. In this study, a new method is proposed to estimate the texture of the test rod with negligible error. It is based on the assumption that the specimen rod possesses electrically small dimensions, under which the scattering integral is simplified. The obtained formula is suitable for a variety of measuring configurations and has been tested for a particular one, with satisfying results.

Bregman iterative algorithms have been extensively used for l1 and TV regularization problems,
allowing to obtain simple, fast and effective algorithms. In this paper, a coordinate descent algorithm for
geosounding inversion is modified by including a Bregman iterative procedure in the linearization.
Modeling results are presented for synthetic data, observing better convergence properties than the original
versions, avoiding the need of the continuation descent procedure.

In this paper, we derive a principal component regression (PCR) method for estimating the optical flow between frames of video sequences according to a pel-recursive manner. This is an easy alternative to dealing with mixtures of motion vectors due to the lack of too much prior information on their statistics (although they are supposed to be normal). The 2D motion vector estimation takes into consideration local image properties. The main advantage of the developed procedure is that no knowledge of the noise distribution is necessary. Preliminary experiments indicate that this approach provides robust estimates of the optical flow. KEY WORDS Motion estimation, principal component regression, and surveillance. 1.

The Marchenko inversion method of the one-dimensional non-relativistic quantum mechanics is generalised to perturbations. The change in the potential that corresponds to a finite change in the scattering matrix is obtained by formulating a Marchenko-like integral equation. The method used here may be generalisable to higher dimensions because the inversion is formulated in terms of scattering matrices, not only in terms of reflection coefficients.

Two new fast algorithms for computing the inverse modified discrete cosine transform (IMDCT) as used in the adaptive spectral entropy coder (ASPEC) [2] are proposed. A fixed-point error analysis is presented to determine the number of significant bits required for fixed-point implementations.

A numerical approximation of the Green's function equation based on a heat-flux formulation is given. It is derived by assuming as a functional form of the surface heat flux a stepwise variation with space and time. The obtained approximation is very important in investigation of the inverse heat conduction problems (IHCPs) because it gives a convenient expression for the temperature in terms of the heat flux components. Additionally, it is very important for the unsteady surface element (USE) method which is a modern boundary discretization method. Green's function approximate solution equation (GFASE) also creates 'naturally' fixed groups or modules of work elements called ''building blocks'' that may be added together to obtain space and time values of temperature. In the current case, they are subject to a partial heating by an applied surface heat flux. The ''building block'' solution can be derived by using the various analytical and numerical approaches available in heat conduction literature though the exact analysis is preferable, as discussed in the text. Poorly-convergent series deriving from Green's functions approach are replaced by closed-form algebraic solutions.

Many works related learning from examples to regularization techniques for inverse problems, emphasizing the strong algorithmic and conceptual analogy of certain learning algorithms with regularization algorithms. In particular it is well known that regularization schemes such as Tikhonov regularization can be effectively used in the context of learning and are closely related to algorithms such as support vector machines. Nevertheless the connection with inverse problem was considered only for the discrete (finite sample) problem and the probabilistic aspects of learning from examples were not taken into account. In this paper we provide a natural extension of such analysis to the continuous (population) case and study the interplay between the discrete and continuous problems. From a theoretical point of view, this allows to draw a clear connection between the consistency approach in learning theory and the stability convergence property in ill-posed inverse problems. The main mat...

We give and prove two new and fundamental properties of total-variationminimizing function regularization (TV regularization): edge locations of function features tend to be preserved, and under certain conditions are preserved exactly; intensity change experienced by individual features is inversely proportional to the scale of each feature. We give and prove exact analytic solutions to the TV regularization problem for simple but important cases. These can also be used to better understand the effects of TV regularization for more general cases. Our results explain why and how TV-minimizing image restoration can remove noise while leaving relatively intact larger-scaled image features, and thus why TV image restoration is especially effective in restoring images with larger-scaled features. Although TV regularization is a global problem, our results show that the effects of TV regularization on individual image features are often quite local. Our results give us a better understanding of what types of images and what types of image degradation are most effectively improved by TV-minimizing image restoration schemes, and they potentially lead to more intelligently designed TV-minimizing restoration schemes.

An ionospheric reconstruction method is developed that is capable of simultaneously utilizing information from all possible sources of ionospheric measurement. We describe the method and demonstrate its performance on ionospheric data collected by an over-the-horizon radar (vertical and backscatter ionograms) and total electron content data collected by receivers operated in the Caribbean by the Applied Research Laboratories of the University of Texas at Austin. For each data set the method produces a smooth three-dimensional ionospheric model that is consistent with all the measurements. The method is based on the Newton-Kontorovich method for nonlinear operator equations and Tikhonov's regularization technique for ill-posed problems. We demonstrate here the ability of our technique to reconstruct ionospheric spatial variations from global-scale to medium-scale irregularities or traveling ionospheric disturbances. Overall results of this study demonstrate that the technique suggested makes it possible to unite data from various ionospheric instruments to provide monitoring of the ionosphere over a large geographical area in a wide range of spatial scales.

L'Assimilation de Données est un cadre méthodologique utilisé en sciences de l'environnement pour la prédiction des systèmes complexes, que sont les modèles météorologiques, océanographiques, ou encore de qualité de l'air. L'Assimilation de Données opère en résolvant un systèmeà trois composantes: une premièreéquation décrit l'évolution temporelle du vecteur d'état; une secondeéquation décrit la relation entre le vecteur d'état et l'observation; enfin, une dernièreéquation décrit la condition initiale. Dans ce rapport, nous utilisons ce cadre mathématique pour l'étude des problèmes mal posés du traitement d'image, habituellement résolus par des techniques de régularisation. Pour cela, le problème est formulé au moyen du systèmeà trois composantes de l'Assimilation de Données. Comme illustration, cette approche est appliquée au calcul du flot optique.

This paper discusses the results of geoacoustic inversion carried out using explosive charge data from the Asian Seas International Acoustic Experiment (ASIAEX) East China Sea (ECS) Experiment. A multifrequency incoherent matched-field inversion processor and a genetic algorithm (GA) are used for the inversion. A multistep matched field inversion approach is presented, which makes use of the varying sensitivities of wave fields at various frequencies to reduce the inversion problem into a sequence of smaller inversions with fewer unknowns to estimate at each stage. Different parameters are estimated using data at different frequencies according to their sensitivities. Inversion results for different areas in the ECS region are summarized and compared with core data.

Rooms for music are often designed starting from a shape known from past experience to perform well acoustically. We reverse this design process by defining an acoustical signature and using it to generate novel room shapes. Our method builds off previous research on reconstructing room shapes from recorded impulse responses, but takes an instrumental rather than forensic approach, opening a range of unforeseen geometrical possibilities to the designer.

A custom-built sound signature design interface allows the user to adjust the location of image sources and listen to their impact on the convolution of an anechoic signal. Each image source is then used to derive a solution space for first and second-order reflections at a receiver position. As the user explores the solution space, planar surfaces are constructed at the reflection points, their perimeters expanded to make the image source audible to a larger receiver area. The user is free to select a particular surface geometry from the solution space based on aesthetic, functional or other criteria knowing that the acoustic contribution of the surfaces will remain consistent.

Along with presenting how we have applied acoustic reverse-engineering to a range of design projects, we outline a proof of concept study in which we compared objective parameters for a number of different room shapes derived from the same acoustical signature.

Literature on Unified Modeling Language (UML) and aspect-orientation often focus on improving the software development process [2]. This paper presents a systems engineering perspective that motivates an aspect-oriented software architectural analysis approach using ...

NP Complete (abbreviated as NPC) problems, standing at the crux of deciding whether P=NP, are among hardest problems in computer science and other related areas. Through decades, NPC problems are
treated as one class. Observing that NPC problems have different natures, it is unlikely that they will have the same complexity. Our intensive study shows that NPC problems are not all equivalent in computational complexity, and they can be further classified. We then show that the classification of NPC problems may depend on their natures, reduction methods, exact algorithms, and the boundary between P and NP. And a new perspective is provided: both P problems and NPC problems have the duality feature in terms of
computational complexity of asymptotic efficiency of algorithms. We also discuss about the NPC problems in real-life and shine some lights on finding better solutions to NPC problems.

An important component of any spatial temporal gradient motion estimation algorithm is the accuracy by which spatial gradients are calculated. When an image sequence is corrupted by noise, the problem of determining these spatial gradients becomes extremely difficult. This is immediately apparent, since the magnitude response of the derivative operator is j!j. In other words, the components of an image are amplified upon differentiation in proportion to their frequency value. Thus, high-frequency noise terms will dominate any low-frequency features in the differentiated image. If this corrupted differentiated image is then used within a spatiotemporal gradient motion estimator, the noise will erroneously influence the estimated motion vector. In this paper, the problem of estimating the spatial gradient is treated as an inverse problem with noise. Formulating the problem in this manner results in a recursive gradient estimator that suppresses the effects of noise.

This paper presents an efficient method for the design of complex fiber Bragg gratings. The method relies on the synthesis of the impulse response of the grating by means of a differential layer-peeling algorithm. The algorithm developed takes into account all the multiple reflections inside the grating, giving an exact solution to the inverse scattering problem. Its low algorithmic complexity enables the synthesis of long fiber gratings. The method is illustrated by designing several filters with interest for optical fiber communication systems: dispersionless bandpass filters and second-and third-order dispersion compensators.

The development of energy selective, photon counting X-ray detectors allows for a wide range of new possibilities in the area of computed tomographic image formation. Under the assumption of perfect energy resolution, here we propose a tensor-based iterative algorithm that simultaneously reconstructs the X-ray attenuation distribution for each energy. We use a multi-linear image model rather than a more standard "stacked vector" representation in order to develop novel tensor-based regularizers. Specifically, we model the multi-spectral unknown as a 3-way tensor where the first two dimensions are space and the 3 rd dimension is energy. This approach allows for the design of tensor nuclear norm regularizers, which like its two dimensional counterpart, is a convex function of the multi-spectral unknown. The solution to the resulting convex optimization problem is obtained using an alternating direction method of multipliers (ADMM) approach. Simulation results shows that the generalized tensor nuclear norm can be used as a stand alone regularization technique for the energy selective (spectral) computed tomography (CT) problem and when combined with total variation regularization it enhances the regularization capabilities especially at low energy images where the effects of noise are most prominent.

System-on-Chip designs require low cost integration of analog and digital blocks. Often, the analog requirements are not considered sufficiently early in the device design cycle, resulting in devices that are suboptimal for the analog components. This paper presents an innovative methodology for deriving comprehensive device specifications based upon a set of Figure-of-Merit circuits which account for both analog and digital requirements. By utilizing these specifications for device design, a more efficient codevelopment of mixed-signal processes, libraries and products is possible. The methodology is illustrated with an example based upon an advanced 120nm CMOS technology.

An inverse problem of construction of chemical structures with given value of property is considered. An algorithm for the solution of the inverse problem for the case of correlating equation "property-topological 'K"-indiCeS" is proposed, and examples of its application are given.

In this paper the problem of recovering a regularized solution of the Fredholm integral equations of the first kind with Hermitian and square-integrable kernels, and with data corrupted by additive noise, is considered. Instead of using a variational regularization of Tikhonov type, based on a priori global bounds, we propose a method of truncation of eigenfunction expansions that can be proved to converge asymptotically, in the sense of the L^2--norm, in the limit of noise vanishing. Here we extend the probabilistic counterpart of this procedure by constructing a probabilistically regularized solution without assuming any structure of order on the sequence of the Fourier coefficients of the data. This probabilistic approach allows us to use the statistical tools proper of time-series analysis, and in this way we attain a new regularizing algorithm, which is illustrated by some numerical examples. Finally, a comparison with solutions obtained by the means of the variational regulari...

Some iterative matrix relations for the geometric, kinematic and dynamic analysis of a Delta parallel robot are established in this paper. The prototype of this manipulator is a three degree of freedom spatial mechanism, which consists of a system of parallel chains. Supposing that the position and the translation motion of the platform are known, an inverse dynamic problem is solved using the virtual powers method. Finally, some recursive matrix relations and some graphs for the moments and the powers of the three active couples are determined.

The investigation of the earth's internal structure is of great interest in
areas as diverse as civil engineering, water management and oil
exploration. A common technique looks to estimate the electrical
conductivity of the subsurface, a proxy for the internal structure, from a
grid of surface voltage measurements - an example of an ill-posed
inverse problem. This paper uses a Bayesian modeling approach to
compare different prior model descriptions using Markov chain Monte
Carlo estimation. A range of simulation examples are considered.

Synthetic aperture radar (SAR) is a coherent active microwave imaging method. In remote sensing it is used for mapping the scattering properties of the Earth's surface in the respective wavelength domain. Many physical and geometric parameters of the imaged scene contribute to the grey value of a SAR image pixel. Scene inversion suffers from this high ambiguity and requires SAR data taken at different wavelength, polarization, time, incidence angle, etc.

Karst aquifers are characterized by extreme heterogeneity due to the presence of karst conduits embedded in a fractured matrix having a much lower hydraulic conductivity. The resulting contrast in the physical properties of the system implies that the system reacts very rapidly to some changes in the boundary conditions and that numerical models are extremely sensitive to small modifications in properties or positions of the conduits. Furthermore, one major issue in all those models is that the location and size of the conduits is generally unknown. For all those reasons, estimating karst network geometry and their properties by solving an inverse problem is a particularly difficult problem. In this paper, two numerical experiments are described. In the first one, 18,0 0 0 flow and transport simulations have been computed and used in a systematic manner to assess statistically if one can retrieve the parameters of a model (geometry and radius of the conduits, hydraulic conductivity of the conduits) from head and tracer data. When two tracer test data sets are available, the solution of the inverse problems indicate with high certainty that there are indeed two conduits and not more. The radius of the conduits are usually well identified but not the properties of the matrix. If more conduits are present in the system, but only two tracer test data sets are available, the inverse problem is still able to identify the true solution as the most probable but it also indicates that the data are insufficient to conclude with high certainty. In the second experiment, a more complex model (including non linear flow equations in conduits) is considered. In this example, gradient-based optimization techniques are proved to be efficient for estimating the radius of the conduits and the hydraulic conductivity of the matrix in a promising and efficient manner. These results suggest that, despite the numerical difficulties, inverse methods should be used to constrain numerical models of karstic systems using flow and transport data. They also suggest that a pragmatic approach for these complex systems could be to generate a large set of karst conduit network realizations using a pseudo-genetic approach such as SKS, and for each karst realization, flow and transport parameters could be optimized using a gradient-based search such as the one implemented in PEST.

The paper presents a cognitive architecture for solution of inverse kinematics problem (IKP) of 6-DOF elbow manipulator with spherical wrist by Locally Recurrent Neural Networks (LRNNs) and simulated the solution by using MATLAB/Simulink. This design is aimed to allow the manipulator system to perform complex movement operations by solving the Inverse Kinematic Problem (IKP) with LRNNs by using the position and orientation of end-effector which represent by wrist with 3-DOF. The Levenberg-Marquardt back propagation (LMBP) is used in the learning of LRNNs which offered the high computation and accuracy for solving IKP for manipulator. This model permits direct forward dynamics simulation, which accurately predicts wrist position, also present a solution to the inverse problem of determining set of joints angle to achieve a given command for posture of manipulator. The simulation of design achieved by connect the program with SimulinkMATLAB Ver. 2009b to calculate the forward and inverse kinematic and implement the movements manipulator. Satisfactory results are obtained, that explains the ability of implement the posture of 6-DOF manipulator by calculate the kinematic with LRNNs and implement high complex movements.

Stress inversions are a useful and popular tool for structural geologist and seismologist alike. These methods were first introduced by Wallace (1951) and Bott (1959) and subsequent studies continue to be based on their assumptions: the remote stress tensor is spatially uniform for the rock mass containing the faults and temporally constant over the history of faulting in that region, and the slip on each fault surface has the same direction and sense as the maximum shear stress resolved on that surface from the remote stress tensor. Furthermore, successful implementation requires that slip accumulates on faults of diverse orientation. Many studies employ these methods on isolated faults or on fault systems with limited ranges of orientations, which can lead to erroneous results. We propose a new method that incorporates the effects of mechanical interaction of the entire fault or fault system, and solves the complete mechanical problem rather than employing empirical relationships between slip and stress or strain (or strain rate). The method requires knowledge of the fault geometry and information on at least one slip vector component along portions of the known fault geometry. For example, if throw is known, the strike-slip component can be solved for. We test the method using a single synthetic fault with anisotropic roughness similar to that measured at fault outcrops. While the orientation of remote stress may be determined precisely, the lack of diverse fault orientations introduces a systematic error in the remote stress ratio. We further test the effect of diversity of fault orientations and find that Wallace–Bott type inversions do not perform as well for limited ranges of orientations when compared to the proposed method. Finally, we use published data from the 1999 Chi-Chi, Taiwan, earthquake, and find that the method using surface data only, and surface data with subsurface focal mechanisms, produce similar results. The resulting stress orientations are in good agreement with results from Wallace–Bott inversions. Furthermore, the slip distribution is in general agreement with kinematic slip inversions using coseismic surface deformation. Stress inversion methods using fault slip data can thus be improved upon, significantly in some cases, by solving a mechanical boundary value problem that takes into account the geometry of faults or fault systems. As a bonus, the solution provides the stress, strain, and displacement fields throughout the region and the slip distributions on the faults.► New method for stress inversion incorporates the effects of interaction of faults. ► Errors in tests cases are smaller than those of faultless (Wallace-Bott) inversions. ► Using slip data from Chelungpu fault reproduces slip distribution well. ► New method provides a useful alternative to faultless inversion methods.

Data assimilation systems allow for estimating surface fluxes of
greenhouse gases from atmospheric concentration
measurements. Good knowledge about fluxes is essential to
understand how climate change affects ecosystems and to
characterize feedback mechanisms.  Based on assimilation of more
than one year of atmospheric in-situ concentration measurements,
we compare the performance of two established data assimilation
models, CarbonTracker and TM5‑4DVar, for CO₂ flux
estimation. CarbonTracker uses an Ensemble Kalman Filter method
to optimize fluxes on ecoregions. TM5‑4DVar employs a 4-D
variational method and optimizes fluxes on a 6°×4°
longitude/latitude grid. Harmonizing the input data allows
analyzing the strengths and weaknesses of the two approaches by
direct comparison of the modelled concentrations and the
estimated fluxes.  We further assess the sensitivity of the two
approaches to the density of observations and operational
parameters such as the length of the assimilation time window.

Our results show that both models provide optimized CO₂
concentration fields of similar quality. In Antarctica
CarbonTracker underestimates the wintertime CO₂ concentrations,
since its 5-week assimilation window does not allow for adjusting
the far-away surface fluxes in response to the detected
concentration mismatch. Flux estimates by CarbonTracker and
TM5‑4DVar are consistent and robust for regions with good
observation coverage, regions with low observation coverage
reveal significant differences. In South America, the fluxes
estimated by TM5‑4DVar suffer from limited representativeness of
the few observations. For the North American continent, mimicking
the historical increase of measurement network density shows
improving agreement between CarbonTracker and TM5‑4DVar flux
estimates for increasing observation density.