Добірка наукової літератури з теми "Inversion uncertainty"

SUMMARY SWinvert is a workflow developed at The University of Texas at Austin for the inversion of surface wave dispersion data. SWinvert encourages analysts to investigate inversion uncertainty and non-uniqueness in shear wave velocity (Vs) by providing a systematic procedure and specific actionable recommendations for surface wave inversion. In particular, the workflow encourages the use of multiple layering parametrizations to address the inversion's non-uniqueness, multiple global searches for each parametrization to address the inverse problem's non-linearity and quantification of Vs uncertainty in the resulting profiles. While the workflow uses the Dinver module of the popular open-source Geopsy software as its inversion engine, the principles presented are of relevance to analysts using other inversion programs. To illustrate the effectiveness of the SWinvert workflow and to develop a set of benchmarks for use in future surface wave inversion studies, synthetic experimental dispersion data for 12 subsurface models of varying complexity are inverted. While the effects of inversion uncertainty and non-uniqueness are shown to be minimal for simple subsurface models characterized by broad-band dispersion data, these effects cannot be ignored in the Vs profiles derived for more complex models with band-limited dispersion data. To encourage adoption of the SWinvert workflow, an open-source Python package (SWprepost), for pre- and post-processing of surface wave inversion data, and an application on the DesignSafe-Cyberinfrastructure (SWbatch), for performing batch-style surface wave inversions with Dinver using high-performance computing, have been developed and released in conjunction with this work. The SWinvert workflow is shown to provide a methodical procedure and a powerful set of tools for performing rigorous surface wave inversions and quantifying the uncertainty in the resulting Vs profiles.

Abstract. The rigorous quantification of uncertainty in geophysical inversions is a challenging problem. Inversions are often ill-posed and the likelihood surface may be multi-modal; properties of any single mode become inadequate uncertainty measures, and sampling methods become inefficient for irregular posteriors or high-dimensional parameter spaces. We explore the influences of different choices made by the practitioner on the efficiency and accuracy of Bayesian geophysical inversion methods that rely on Markov chain Monte Carlo sampling to assess uncertainty using a multi-sensor inversion of the three-dimensional structure and composition of a region in the Cooper Basin of South Australia as a case study. The inversion is performed using an updated version of the Obsidian distributed inversion software. We find that the posterior for this inversion has a complex local covariance structure, hindering the efficiency of adaptive sampling methods that adjust the proposal based on the chain history. Within the context of a parallel-tempered Markov chain Monte Carlo scheme for exploring high-dimensional multi-modal posteriors, a preconditioned Crank–Nicolson proposal outperforms more conventional forms of random walk. Aspects of the problem setup, such as priors on petrophysics and on 3-D geological structure, affect the shape and separation of posterior modes, influencing sampling performance as well as the inversion results. The use of uninformative priors on sensor noise enables optimal weighting among multiple sensors even if noise levels are uncertain.

Logging-while-drilling (LWD) resistivity responses are inevitably contaminated by Gaussian and non-Gaussian noise. Noise contamination can influence the stability and accuracy of the inversion of the data. In addition, the uncertainty of the bed-boundary positions can complicate the inversion. We have developed a novel efficient nonlinear inversion algorithm, called Huber inversion, to accurately estimate the layer-by-layer resistivity when LWD measurements were affected by Gaussian and non-Gaussian noise. Huber inversion combines the advantages of the [Formula: see text]- and [Formula: see text]-norm inversions, which are more robust than the traditional least-squares inversion algorithm. We use a multiple initial bed-boundary positions method to reduce the inversion uncertainty caused by uncertain bed-boundary positions. The initial bed-boundary positions could be restricted into defined ranges based on the investigation depth of LWD instruments, the geologic environment, and log data from adjacent wells. The slipping inversion window technique is also adopted to satisfy the real-time requirements and ensure that the inversion parameters are the global optima. Numerical simulations under different conditions demonstrate the high stability and superiority of the proposed method. Reliable inversion results can be used for routine petrophysical interpretation, accurate geosteering, and quantitative formation evaluation.

ABSTRACT. We assessed the effectiveness of seismic inversion in estimating the elastic properties of layers whose thickness represents a fraction of the wavelength. We used an approach that integrates a quantitative study of inversion uncertainties based on the stochastic Bayesian method and sensitivity analysis, considering the full waveform seismic response of the layer model. Three inversion input data combinations PP, PS, and joint PP-PS reflections provide comprehensive information for the analysis. Estimates of Vp, Vs, and the density of thin layers are sensitive to the intensity of the elastic property contrasts and incidence angle coverage. Results show that the elastic parameters of layers as thin as 1/16 of the peak wavelength can be estimated with low uncertainty if the input data contain incidence angles up to 40 degrees for the PP-PS case and up to 55 degrees for the PP case, when the elastic property contrast is not small.Keywords: stochastic inversion; reflectivity method; sensitivity analysis. Análise de Incertezas da Inversão Elástica Multicomponente de Camadas FinasRESUMO. Nós avaliamos a eficácia da inversão sísmica na estimativa das propriedades elásticas de camadas cuja espessura representa uma fração do comprimento de onda. Utilizamos uma abordagem que integra um estudo quantitativo de incertezas de inversão baseado no método estocástico Bayesiano e análise de sensibilidade, considerando a resposta sísmica completa da forma de onda do modelo de camadas. As análises foram realizadas em três combinações de dados de entrada para inversão: PP, PS e reflexões PP-PS conjuntas. As estimativas de Vp, Vs e densidade de camadas finas são sensíveis à intensidade dos contrastes de propriedades elásticas e à cobertura do ângulo de incidência. Os resultados mostram que os parâmetros elásticos de camadas tão finas quanto 1/16 do comprimento de onda de pico podem ser estimados com baixa incerteza se os dados de entrada contiverem ângulos de incidência de até 40 graus para o caso PP-PS e até 55 graus para o caso PP, quando o contraste da propriedade elástica não é pequeno.Palavras-chave: inversão estocástica; método da refletividade; análise de sensibilidade.

In Russian, numeral expressions can be made approximate through Approximative Inversion, whereby the noun and the numeral appear to exchange positions. Approximative Inversion has been analyzed as head movement, where a head containing the noun raises to the left of the numeral, but this leads to incorrect semantics. I propose that Approximative Inversion involves post-nominal generation of the numeral in a reduced relative structure, where it is associated with a feature marking speaker uncertainty. This feature triggers a round-number reading of the numeral, resulting in what appears to be number approximation due to speaker uncertainty.

Abstract. Combining measurements of atmospheric CO2 and its radiocarbon (14CO2) fraction and transport modeling in atmospheric inversions offers a way to derive improved estimates of CO2 emitted from fossil fuel (FFCO2). In this study, we solve for the monthly FFCO2 emission budgets at regional scale (i.e., the size of a medium-sized country in Europe) and investigate the performance of different observation networks and sampling strategies across Europe. The inversion system is built on the LMDZv4 global transport model at 3.75∘ × 2.5∘ resolution. We conduct Observing System Simulation Experiments (OSSEs) and use two types of diagnostics to assess the potential of the observation and inverse modeling frameworks. The first one relies on the theoretical computation of the uncertainty in the estimate of emissions from the inversion, known as “posterior uncertainty”, and on the uncertainty reduction compared to the uncertainty in the inventories of these emissions, which are used as a prior knowledge by the inversion (called “prior uncertainty”). The second one is based on comparisons of prior and posterior estimates of the emission to synthetic “true” emissions when these true emissions are used beforehand to generate the synthetic fossil fuel CO2 mixing ratio measurements that are assimilated in the inversion. With 17 stations currently measuring 14CO2 across Europe using 2-week integrated sampling, the uncertainty reduction for monthly FFCO2 emissions in a country where the network is rather dense like Germany, is larger than 30 %. With the 43 14CO2 measurement stations planned in Europe, the uncertainty reduction for monthly FFCO2 emissions is increased for the UK, France, Italy, eastern Europe and the Balkans, depending on the configuration of prior uncertainty. Further increasing the number of stations or the sampling frequency improves the uncertainty reduction (up to 40 to 70 %) in high emitting regions, but the performance of the inversion remains limited over low-emitting regions, even assuming a dense observation network covering the whole of Europe. This study also shows that both the theoretical uncertainty reduction (and resulting posterior uncertainty) from the inversion and the posterior estimate of emissions itself, for a given prior and “true” estimate of the emissions, are highly sensitive to the choice between two configurations of the prior uncertainty derived from the general estimate by inventory compilers or computations on existing inventories. In particular, when the configuration of the prior uncertainty statistics in the inversion system does not match the difference between these prior and true estimates, the posterior estimate of emissions deviates significantly from the truth. This highlights the difficulty of filtering the targeted signal in the model–data misfit for this specific inversion framework, the need to strongly rely on the prior uncertainty characterization for this and, consequently, the need for improved estimates of the uncertainties in current emission inventories for real applications with actual data. We apply the posterior uncertainty in annual emissions to the problem of detecting a trend of FFCO2, showing that increasing the monitoring period (e.g., more than 20 years) is more efficient than reducing uncertainty in annual emissions by adding stations. The coarse spatial resolution of the atmospheric transport model used in this OSSE (typical of models used for global inversions of natural CO2 fluxes) leads to large representation errors (related to the inability of the transport model to capture the spatial variability of the actual fluxes and mixing ratios at subgrid scales), which is a key limitation of our OSSE setup to improve the accuracy of the monitoring of FFCO2 emissions in European regions. Using a high-resolution transport model should improve the potential to retrieve FFCO2 emissions, and this needs to be investigated.

Abstract A meaningful solution to an inversion problem should be composed of the preferred inversion model and its uncertainty and resolution estimates. The model uncertainty estimate describes an equivalent model domain in which each model generates responses which fit the observed data to within a threshold value. The model resolution matrix measures to what extent the unknown true solution maps into the preferred solution. However, most current geophysical electromagnetic (also gravity, magnetic and seismic) inversion studies only offer the preferred inversion model and ignore model uncertainty and resolution estimates, which makes the reliability of the preferred inversion model questionable. This may be caused by the fact that the computation and analysis of an inversion model depend on multiple factors, such as the misfit or objective function, the accuracy of the forward solvers, data coverage and noise, values of trade-off parameters, the initial model, the reference model and the model constraints. Depending on the particular method selected, large computational costs ensue. In this review, we first try to cover linearised model analysis tools such as the sensitivity matrix, the model resolution matrix and the model covariance matrix also providing a partially nonlinear description of the equivalent model domain based on pseudo-hyperellipsoids. Linearised model analysis tools can offer quantitative measures. In particular, the model resolution and covariance matrices measure how far the preferred inversion model is from the true model and how uncertainty in the measurements maps into model uncertainty. We also cover nonlinear model analysis tools including changes to the preferred inversion model (nonlinear sensitivity tests), modifications of the data set (using bootstrap re-sampling and generalised cross-validation), modifications of data uncertainty, variations of model constraints (including changes to the trade-off parameter, reference model and matrix regularisation operator), the edgehog method, most-squares inversion and global searching algorithms. These nonlinear model analysis tools try to explore larger parts of the model domain than linearised model analysis and, hence, may assemble a more comprehensive equivalent model domain. Then, to overcome the bottleneck of computational cost in model analysis, we present several practical algorithms to accelerate the computation. Here, we emphasise linearised model analysis, as efficient computation of nonlinear model uncertainty and resolution estimates is mainly determined by fast forward and inversion solvers. In the last part of our review, we present applications of model analysis to models computed from individual and joint inversions of electromagnetic data; we also describe optimal survey design and inversion grid design as important applications of model analysis. The currently available model uncertainty and resolution analyses are mainly for 1D and 2D problems due to the limitations in computational cost. With significant enhancements of computing power, 3D model analyses are expected to be increasingly used and to help analyse and establish confidence in 3D inversion models.

We have developed a Bayesian method for Dix inversion and illustrated it with examples from the North Sea. The method is a constrained Dix inversion in which the uncertainty of the estimated interval velocities is an integral part of the solution. The method combines available geologic prior knowledge with the information in the picked rms velocities so that the prior model stabilizes and constrains the inversion. The definition of layers or intervals is flexible. One possibility is the classical layering whereby the top and bottom of the intervals are defined directly from the times of the picked rms velocities, but the intervals can be defined arbitrarily and independent of the picked velocities. Even a pseudocontinuous velocity profile with dense time sampling can be predicted with corresponding uncertainty. The Dix inversion is conventionally formulated as a 1D inversion problem in the locations with picked rms velocities. Under the assumption that Dix inversion is adequate, we have defined a method for spatial prediction at any position in a 3D model. This involves spatial smoothing and interpolation, and provides a method for building a velocity cube with the corresponding uncertainty on any grid. Explicit analytic expressions were found for both the optimal solution and the uncertainty. In general, the uncertainty of the estimated interval velocities increases with depth and with decreasing layer thickness. The uncertainty of the spatial prediction decreases with increasing spatial correlation. Because the solution is of an explicit analytic form, the Bayesian Dix inversion is computationally fast and does not require stochastic simulation.

The subject of this dissertation is use of computer models as data analysis tools in several different geoscience settings, including integrated surface water/groundwater modeling, tephra fallout modeling, geophysical inversion, and hydrothermal groundwater modeling. The dissertation is organized into three chapters, which correspond to three individual publication manuscripts. In the first chapter, a linear framework is developed to identify and estimate the potential predictive consequences of using a simple computer model as a data analysis tool. The framework is applied to a complex integrated surface-water/groundwater numerical model with thousands of parameters. Several types of predictions are evaluated, including particle travel time and surface-water/groundwater exchange volume. The analysis suggests that model simplifications have the potential to corrupt many types of predictions. The implementation of the inversion, including how the objective function is formulated, what minimum of the objective function value is acceptable, and how expert knowledge is enforced on parameters, can greatly influence the manifestation of model simplification. Depending on the prediction, failure to specifically address each of these important issues during inversion is shown to degrade the reliability of some predictions. In some instances, inversion is shown to increase, rather than decrease, the uncertainty of a prediction, which defeats the purpose of using a model as a data analysis tool. In the second chapter, an efficient inversion and uncertainty quantification approach is applied to a computer model of volcanic tephra transport and deposition. The computer model simulates many physical processes related to tephra transport and fallout. The utility of the approach is demonstrated for two eruption events. In both cases, the importance of uncertainty quantification is highlighted by exposing the variability in the conditioning provided by the observations used for inversion. The worth of different types of tephra data to reduce parameter uncertainty is evaluated, as is the importance of different observation error models. The analyses reveal the importance using tephra granulometry data for inversion, which results in reduced uncertainty for most eruption parameters. In the third chapter, geophysical inversion is combined with hydrothermal modeling to evaluate the enthalpy of an undeveloped geothermal resource in a pull-apart basin located in southeastern Armenia. A high-dimensional gravity inversion is used to define the depth to the contact between the lower-density valley fill sediments and the higher-density surrounding host rock. The inverted basin depth distribution was used to define the hydrostratigraphy for the coupled groundwater-flow and heat-transport model that simulates the circulation of hydrothermal fluids in the system. Evaluation of several different geothermal system configurations indicates that the most likely system configuration is a low-enthalpy, liquid-dominated geothermal system.

Abstract Knowledge of the thermal conditions in the lithosphere is based on theoretical models of heat transfer constrained by geological and geophysical data. The present dissertation focuses on the uncertainties of calculated temperature and heat flow density results and on how they depend on the uncertainties of thermal properties of rocks, as well as on the relevant boundary conditions. Due to the high number of involved variables of typical models, the random simulation technique was chosen as the applied tool in the analysis. Further, the random simulation technique was applied in inverse Monte Carlo solutions of geothermal models. In addition to modelling technique development, new measurements on thermal conductivity and diffusivity of middle and lower crustal rocks in elevated pressure and temperature were carried out. In the uncertainty analysis it was found that a temperature uncertainty of 50 K at the Moho level, which is at a 50 km's depth in the layered model, is produced by an uncertainty of only 0.5 W m-1 K-1 in thermal conductivity values or 0.2 orders of magnitude uncertainty in heat production rate (mW m-3). Similar uncertainties are obtained in Moho temperature, given that the lower boundary condition varies by ± 115 K in temperature (nominal value 1373 K) or ± 1.7 mW m-2 in mantle heat-flow density (nominal value 13.2 mW m-2). Temperature and pressure dependencies of thermal conductivity are minor in comparison to the previous effects. The inversion results indicated that the Monte Carlo technique is a powerful tool in geothermal modelling. When only surface heat-flow density data are used as a fitting object, temperatures at the depth of 200 km can be inverted with an uncertainty of 120 - 170 K. When petrological temperature-depth (pressure) data on kimberlite-hosted mantle xenoliths were used also as a fitting object, the uncertainty was reduced to 60 - 130 K. The inversion does not remove the ambiguity of the models completely, but it reduces significantly the uncertainty of the temperature results.

L'inversion de forme d'onde complète (FWI) est une méthode d'inversion non-linéaire qui a pour but l'obtention de modèles précis des propriétés physiques du sous-sol terrestre. Ces modèles, véritables cartes de propriétés physiques, sont indispensables pour l'exploration et l'étude des structures internes de la Terre.Généralement formulée sous la forme d'un schéma d'optimisation par la méthode des moindres carrés, la FWI compare des enregistrements sismiques observés en surface, avec des données synthétiques calculées à partir d'un modèle numérique de sous-sol. Alors qu'une infinité de modèles peut potentiellement expliquer les observations, la FWI, du fait de sa formulation, ne permet d'obtenir qu'un seul modèle du sous-sol fortement conditionné par le choix de modèle de départ. À cette ambiguïté s'ajoute la difficulté d'estimer l'incertitude de la solution, à cause du coût de calcul prohibitif de la FWI. La non-unicité de la solution et le manque de moyens d'estimation d'incertitude rend l'exploitation des modèles de FWI compliquée.Dans cette thèse, nous proposons une méthode non conventionnelle et abordable, intégrant l’estimation d’incertitude au coeur de la solution de FWI. Notre méthode combine la FWI conventionnelle et l’assimilation de données par méthodes d’ensemble. De ce fait, elle tire avantage de la vitesse de convergence de la FWI conventionnelle, ainsi que des capacités d'estimation d'incertitude du Filtre de Kálmán d'Ensemble dit "Transform" (ETKF). Cette combinaison est permise par les fondements théoriques communs aux problèmes d'optimisation en FWI conventionnelle et au filtrage bayésien de l'ETKF. Nous utilisons ce schéma, l’ETKF-FWI, afin de transposer le problème de FWI dans le cadre de l'inférence Bayésienne locale. Au lieu d’une unique solution, l’ETKF-FWI retourne un ensemble de modèles qui permet à la fois de calculer la meilleure solution au sens des moindres carrés, mais aussi l'information d’incertitude et de résolution associée à chaque paramètre. Cette estimation d’incertitude est rendue possible par l’approximation de bas-rang de la matrice de covariance a posteriori, calculée à partir de l’ensemble. Les valeurs de variance permettent d’évaluer le degré de variabilité de la solution au sein de l’ensemble. La résolution est quant à elle, donnée par les termes hors diagonaux de la matrice de corrélation, qui est préférée à la matrice de covariance pour sa nature adimensionnelle.L'application de l'ETKF-FWI à deux cas d'études (un test synthétique et une application sur données de terrain) nous permet d'évaluer la faisabilité, ainsi que les limites de notre technique. Malgré le coût de calcul important lié à la représentation d’ensemble, cette stratégie permet une implémentation complètement parallèle, la rendant avantageuse au regard des solutions existant dans la littérature.Ces tests nous permettent d’évaluer l’influence de la taille de l’ensemble sur l’estimation de la variance, en caractérisant le biais de sous-échantillonnage associé aux petits ensembles. Bien que ce biais soit classiquement corrigé grâce aux méthodes d’inflation d’ensemble, celles-ci ne semblent pas adaptées à l’ETKF-FWI, limitant l’estimation d’incertitude à des évaluations qualitatives. De plus, la complexité de l’application sur données de terrain impacte la création de l’ensemble initial, ce qui influence directement les capacités de l’ETKF-FWI à produire une estimation quantitative de l’incertitude.Nous terminons par l’application de l'ETKF-FWI à une inversion de plusieurs paramètres physique (vitesse des ondes P et densité), considéré comme un défi majeur en FWI conventionnelle. Ce test nous permet d’évaluer qualitativement les liens de corrélation et d'ambiguïté entre vitesse et densité, ainsi que leurs incertitude et résolution respectives. De plus, le modèle moyen issu de l’ETKF-FWI semble être de qualité supérieure, ce qui laisse supposer d’un possible effet de préconditionnement fourni par la covariance
Full Waveform Inversion (FWI) is an ill-posed non-linear inverse problem, aiming at recovering detailed pictures of subsurface physical properties, which are crucial to explore and understand Earth structures.Classically formulated as a least-squares optimization scheme, FWI yields a single subsurface model amongst an infinite possibility of solutions. With the general lack of systematic and scalable uncertainty estimation, this formulation makes interpretation of FWI's outcomes complex.In this thesis, we propose an unconventional, scalable way of tackling the lack of uncertainty estimation in FWI, thanks to data assimilation ensemble methods. We develop a scheme combining both classical FWI and the Ensemble Transform Kalman Filter, that we call ETKF-FWI, and which is successfully applied on two 2-D test cases. This scheme takes advantage of the theoretical common-ground between least-squares optimization problems and Bayesian filtering. We use it to recast FWI in a local Bayesian inference framework, thanks to the ensemble representation. The ETKF-FWI provides high-resolution subsurface tomographic models and yields a low-rank approximation of the posterior covariance, holding the uncertainty and resolution information of the proposed solution. We show how the ETKF-FWI can be applied to qualitatively evaluate uncertainty and resolution of the solution. Instead of providing a single solution, the filter yields an ensemble of models, from which statistical information can be inferred.Uncertainty is evaluated from the ensemble's variance, which relates to the diversity of solution amongst the ensemble members for each parameter. We show that lines of the correlation matrix are ideal to evaluate qualitatively parameters resolution, thanks to their adimentionality. While the methodology is computationally intensive, it has the benefit of being fully scalable. Its applicability is demonstrated on a synthetic benchmark. This preliminary test allows us to assess the sensitivity of the ensemble representation to the common undersampling bias encountered in ensemble data assimilation. While undersampling does not affect the image reconstruction in any way, it results in variance underestimation, which makes the whole exercise of quantitative uncertainty assessment complicated. Ensemble inflation has been used to mitigate this bias, but does not seems to be a practical solution.A field data experiment is also discussed in this thesis. It makes it possible to test the sensitivity of the ETKF-FWI to complex noise structure and realistic physics. As it stands, the complexity of the problem reduces flexibility in the ensemble generation, and hence on the uncertainty estimate. Despite these limitations, results are consistent with the synthetic benchmark, and we are able to provide a qualitative uncertainty assessment. The field data case also allows us to evaluate the possibilities to use the ETKF-FWI on multiparameter inversion, which is still regarded as a challenging topic in FWI. The ETKF-FWI multiparameter inversion yields improved models compared with conventional ones. More importantly, it makes it possible to assess the uncertainty associated with parameters cross-talks

L'objectif de cette thèse est de résoudre un problème d'inversion sous incertitudes de fonctions coûteuses à évaluer dans le cadre du paramétrage du contrôle d'un système de dépollution de véhicules.L'effet de ces incertitudes est pris en compte au travers de l'espérance de la grandeur d'intérêt. Une difficulté réside dans le fait que l'incertitude est en partie due à une entrée fonctionnelle connue à travers d'un échantillon donné. Nous proposons deux approches basées sur une approximation du code coûteux par processus gaussiens et une réduction de dimension de la variable fonctionnelle par une méthode de Karhunen-Loève.La première approche consiste à appliquer une méthode d'inversion de type SUR (Stepwise Uncertainty Reduction) sur l'espérance de la grandeur d'intérêt. En chaque point d'évaluation dans l'espace de contrôle, l'espérance est estimée par une méthode de quantification fonctionnelle gloutonne qui fournit une représentation discrète de la variable fonctionnelle et une estimation séquentielle efficace à partir de l'échantillon donné de la variable fonctionnelle.La deuxième approche consiste à appliquer la méthode SUR directement sur la grandeur d'intérêt dans l'espace joint des variables de contrôle et des variables incertaines. Une stratégie d'enrichissement du plan d'expériences dédiée à l'inversion sous incertitudes fonctionnelles et exploitant les propriétés des processus gaussiens est proposée.Ces deux approches sont comparées sur des fonctions jouets et sont appliquées à un cas industriel de post-traitement des gaz d'échappement d'un véhicule. La problématique est de déterminer les réglages du contrôle du système permettant le respect des normes de dépollution en présence d'incertitudes, sur le cycle de conduite
This thesis deals with the inversion problem under uncertainty of expensive-to-evaluate functions in the context of the tuning of the control unit of a vehicule depollution system.The effect of these uncertainties is taken into account through the expectation of the quantity of interest. The problem lies in the fact that the uncertainty is partly due to a functional variable only known through a given sample. We propose two approaches to solve the inversion problem, both methods are based on Gaussian Process modelling for expensive-to-evaluate functions and a dimension reduction of the functional variable by the Karhunen-Loève expansion.The first methodology consists in applying a Stepwise Uncertainty Reduction (SUR) method on the expectation of the quantity of interest. At each evaluation point in the control space, the expectation is estimated by a greedy functional quantification method that provides a discrete representation of the functional variable and an effective sequential estimate from the given sample.The second approach consists in applying the SUR method directly to the quantity of interest in the joint space. Devoted to inversion under functional uncertainties, a strategy for enriching the experimental design exploiting the properties of Gaussian processes is proposed.These two approaches are compared on toy analytical examples and are applied to an industrial application for an exhaust gas post-treatment system of a vehicle. The objective is to identify the set of control parameters that leads to meet the pollutant emission norms under uncertainties on the driving cycle

Abstract This work is devoted to the theory of joint interpretation of multimethod geophysical data and its application to the solution of real geophysical inverse problems. The targets of such joint interpretation can be geological bodies with an established dependence between various physical properties that cause anomalies in several geophysical fields (geophysical multiresponse). The establishing of the relationship connecting the various physical properties is therefore a necessary first step in any joint interpretation procedure. Bodies for which the established relationship between physical properties is violated (single-response bodies) can be targets of separate interpretations. The probabilistic (Bayesian) approach provides the necessary formalism for addressing the problem of the joint inversion of multimethod geophysical data, which can be non-linear and have a non-unique solution. Analysis of the lower limit of resolution of the non-linear problem of joint inversion using the definition of e-entropy demonstrates that joint inversion of multimethod geophysical data can reduce non-uniqueness in real geophysical inverse problems. The question can be formulated as a multiobjective optimisation problem (MOP), enabling the numerical methods of this theory to be employed for the purpose of geophysical data inversion and for developing computer algorithms capable of solving highly non-linear problems. An example of such a problem is magnetotelluric impedance tensor inversion with the aim of obtaining a 3-D resistivity distribution. An additional area of application for multiobjective optimisation can be the combination of various types of uncertain information (probabilistic and non-probabilistic) in a common inversion scheme applicable to geophysical inverse problems. It is demonstrated how the relationship between seismic velocity and density can be used to construct an algorithm for the joint interpretation of gravity and seismic wide-angle reflection and refraction data. The relationship between the elastic and electrical properties of rocks, which is a necessary condition for the joint inversion of data obtained by seismic and electromagnetic methods, can be established for solid- liquid rock mixtures using theoretical modelling of the elastic and electrical properties of rocks with a fractal microstructure and from analyses of petrophysical data and borehole log data.

La caractérisation des tremblements de terre est un domaine de recherche primordial en sismologie, où l'objectif final est de fournir des estimations précises d'attributs de la source sismique. Dans ce domaine, certaines questions émergent, par exemple : quand un tremblement de terre s’est-il produit? quelle était sa taille? ou quelle était son évolution dans le temps et l'espace? On pourrait se poser d'autres questions plus complexes comme: pourquoi le tremblement s'est produit? quand sera le prochain dans une certaine région? Afin de répondre aux premières questions, une représentation physique du phénomène est nécessaire. La construction de ce modèle est l'objectif scientifique de ce travail doctoral qui est réalisé dans le cadre de la modélisation cinématique. Pour effectuer cette caractérisation, les modèles cinématiques de la source sismique sont un des outils utilisés par les sismologues. Il s’agit de comprendre la source sismique comme une dislocation en propagation sur la géométrie d’une faille active. Les modèles de sources cinématiques sont une représentation physique de l’histoire temporelle et spatiale d’une telle rupture en propagation. Cette modélisation est dite approche cinématique car les histoires de la rupture inférées par ce type de technique sont obtenues sans tenir compte des forces qui causent l'origine du séisme.Dans cette thèse, je présente une nouvelle méthode d'inversion cinématique capable d'assimiler, hiérarchiquement en temps, les traces de données à travers des fenêtres de temps évolutives. Cette formulation relie la fonction de taux de glissement et les sismogrammes observés, en préservant la positivité de cette fonction et la causalité quand on parcourt l'espace de modèles. Cette approche, profite de la structure creuse de l’histoire spatio-temporelle de la rupture sismique ainsi que de la causalité entre la rupture et chaque enregistrement différé par l'opérateur. Cet opérateur de propagation des ondes connu, est différent pour chaque station. Cette formulation progressive, à la fois sur l’espace de données et sur l’espace de modèle, requiert des hypothèses modérées sur les fonctions de taux de glissement attendues, ainsi que des stratégies de préconditionnement sur le gradient local estimé pour chaque paramètre du taux de glissement. Ces hypothèses sont basées sur de simples modèles physiques de rupture attendus. Les applications réussies de cette méthode aux cas synthétiques (Source Inversion Validation Exercise project) et aux données réelles du séisme de Kumamoto 2016 (Mw=7.0), ont permis d’illustrer les avantages de cette approche alternative d’une inversion cinématique linéaire de la source sismique.L’objectif sous-jacent de cette nouvelle formulation sera la quantification des incertitudes d’un tel modèle. Afin de mettre en évidence les propriétés clés prises en compte dans cette approche linéaire, dans ce travail, j'explore l'application de la stratégie bayésienne connue comme Hamiltonian Monte Carlo (HMC). Cette méthode semble être l’une des possibles stratégies qui peut être appliquée à ce problème linéaire sur-paramétré. Les résultats montrent qu’elle est compatible avec la stratégie linéaire dans le domaine temporel présentée ici. Grâce à une estimation efficace du gradient local de la fonction coût, on peut explorer rapidement l'espace de grande dimension des solutions possibles, tandis que la linéarité est préservée. Dans ce travail, j'explore la performance de la stratégie HMC traitant des cas synthétiques simples, afin de permettre une meilleure compréhension de tous les concepts et ajustements nécessaires pour une exploration correcte de l'espace de modèles probables. Les résultats de cette investigation préliminaire sont encourageants et ouvrent une nouvelle façon d'aborder le problème de la modélisation de la reconstruction cinématique de la source sismique, ainsi, que de l’évaluation des incertitudes associées
The earthquake characterization is a fundamental research field in seismology, which final goal is to provide accurate estimations of earthquake attributes. In this study field, various questions may rise such as the following ones: when and where did an earthquake happen? How large was it? What is its evolution in space and time? In addition, more challenging questions can be addressed such as the following ones: why did it occur? What is the next one in a given area? In order to progress in the first list of questions, a physical description, or model, of the event is necessary. The investigation of such model (or image) is the scientific topic I investigate during my PhD in the framework of kinematic source models. Understanding the seismic source as a propagating dislocation that occurs across a given geometry of an active fault, the kinematic source models are the physical representations of the time and space history of such rupture propagation. Such physical representation is said to be a kinematic approach because the inferred rupture histories are obtained without taking into account the forces that might cause the origin of the dislocation.In this PhD dissertation, I present a new hierarchical time kinematic source inversion method able to assimilate data traces through evolutive time windows. A linear time-domain formulation relates the slip-rate function and seismograms, preserving the positivity of this function and the causality when spanning the model space: taking benefit of the time-space sparsity of the rupture model evolution is as essential as considering the causality between rupture and each record delayed by the known propagator operator different for each station. This progressive approach, both on the data space and on the model space, does require mild assumptions on prior slip-rate functions or preconditioning strategies on the slip-rate local gradient estimations. These assumptions are based on simple physical expected rupture models. Successful applications of this method to a well-known benchmark (Source Inversion Validation Exercise 1) and to the recorded data of the 2016 Kumamoto mainshock (Mw=7.0) illustrate the advantages of this alternative approach of a linear kinematic source inversion.The underlying target of this new formulation will be the future uncertainty quantification of such model reconstruction. In order to achieve this goal, as well as to highlight key properties considered in this linear time-domain approach, I explore the Hamiltonian Monte Carlo (HMC) stochastic Bayesian framework, which appears to be one of the possible and very promising strategies that can be applied to this stabilized over-parametrized optimization of a linear forward problem to assess the uncertainties on kinematic source inversions. The HMC technique shows to be compatible with the linear time-domain strategy here presented. This technique, thanks to an efficient estimation of the local gradient of the misfit function, appears to be able to rapidly explore the high-dimensional space of probable solutions, while the linearity between unknowns and observables is preserved. In this work, I investigate the performance of the HMC strategy dealing with simple synthetic cases with almost perfect illumination, in order to provide a better understanding of all the concepts and required tunning to achieve a correct exploration of the model space. The results from this preliminary investigation are promising and open a new way of tackling the kinematic source reconstruction problem and the assessment of the associated uncertainties

The oil and gas industry need for high performing and low cost multiphase meters is ever more justified given the rapid depletion of conventional oil reserves. This has led oil companies to develop smaller/marginal fields and reservoirs in remote locations and deep offshore, thereby placing great demands for compact and more cost effective soluti8ons of on-line continuous multiphase flow measurement. The pattern recognition approach for clamp-on multiphase measurement employed in this research study provides one means for meeting this need. Cont/d.

We study a scalable approach to information fusion for large sensor networks. The algorithm, field inversion by consensus and compressed sensing (FICCS), is a distributed method for detection, localization, and estimation of a propagating field generated by an unknown number of point sources. The approach combines results in the areas of distributed average consensus and compressed sensing to form low dimensional linear projections of all sensor readings throughout the network, allowing each node to reconstruct a global estimate of the field. Compressed sensing is applied to continuous source localization by quantizing the potential locations of sources, transforming the model of sensor observations to a finite discretized linear model. We study the effects of structured modeling errors induced by spatial quantization and the robustness of ℓ1 penalty methods for field inversion. We develop a perturbations method to analyze the effects of spatial quantization error in compressed sensing and provide a model-robust version of noise-aware basis pursuit with an upperbound on the sparse reconstruction error. Numerical simulations illustrate system design considerations by measuring the performance of decentralized field reconstruction, detection performance of point phenomena, comparing trade-offs of quantization parameters, and studying various sparse estimators. The method is extended to time-varying systems using a recursive sparse estimator that incorporates priors into ℓ1 penalized least squares. This thesis presents the advantages of inter-sensor measurement mixing as a means of efficiently spreading information throughout a network, while identifying sparse estimation as an enabling technology for scalable distributed field reconstruction systems.

Récemment, plusieurs tentatives ont été faites afin de quantifier les émissions de CO2 à l'échelle de la ville et ainsi d’améliorer les inventaires existants. Les mesures de concentration de CO2 et autres gaz peuvent ainsi être utilisées dans une approche dite “descendante” afin de contraindre les inventaires d’émission qui sont traditionnellement construits via une approche dite “ascendante” à partir d’une quantification des activités.Dans le cadre de cette thèse, nous avons évalué le potentiel d’une nouvelle technique de surveillance du CO2, référencée sous le nom de sous le nom de GreenLite™ (Green Imaging Tomography Experiment). Le système a été déployé pendant un an dans la ville de Paris. Il permet une mesure en continu de la concentration le long de 30 segments horizontaux, proches de la surface. Il a donc une couverture spatiale beaucoup plus large que l'échantillonnage in situ traditionnel et apporte une information dont la représentativité spatiale est plus cohérente avec celle des modèles de transport atmosphérique à résolution à l'échelle kilométrique utilisés pour l'inversion atmosphérique à l'échelle de la ville.J’ai développé un outil de modélisation complet centré sur le modèle à haute résolution WRF avec un couplage (WRF-Chem), et en utilisant des inventaires d'émissions anthropique de CO2, des estimations des flux par la végétation et des conditions aux limites fournies par une simulation à grande échelle. Ce modèle permet d’interpréter les mesures.Le chapitre 1 est une large introduction au sujet tandis que les chapitres 2-4 sont construits autour de trois articles publiés dans la littérature scientifique.Le chapitre 2 évalue si le modèle WRF à une résolution spatiale de 3 km peut reproduire les champs météorologiques dans la région IdF mieux que ne le fait le modèle Européen CEPMMT à 16 km de résolution. Les comparaisons entre les analyses des deux modèles sont faites avec un focus sur trois variables atmosphériques (température de l'air, vent et hauteur de la couche limite) qui sont les plus pertinentes en ce qui concerne le transport du CO2 atmosphérique dans un environnement urbain. Les résultats ont permis de sélection une version du modèle et une option de nudging qui permet la meilleure adéquation entre les simulations numériques et les observations de terrain, et ces options sont utilisés dans la suite du travail.Le chapitre 3 vise à comprendre les variations temporelles et spatiales des concentrations de CO2 dans Paris et ses environs pendant la période de mesure du système GreenLITE™ (Sep 2015 à dec 2016). Les données permettent de démontrer qu’un schéma de canopée urbaine (BEP) est en bien meilleur accord avec la réalité, par rapport à l'autre (UCM), en particulier pendant l’hiver. Pendant cette période, le mélange vertical est réduit ce qui peut conduire à des accumulations du CO2 dans les basses couches de l’atmosphère, qui sont difficiles à modéliser. Cependant, les mesures GreenLITE™ montrent aussi un bruit important et des indications de biais, ce qui limite leur potentiel d’interprétation. De plus, les inadéquations entre modèles et observations dans ce chapitre soulignent clairement la difficulté de modélisation du CO2 dans les zones urbaines en raison des grandes incertitudes tant dans le transport atmosphérique que dans l'inventaire des émissions.Le chapitre 4 vise à étudier en détail les sources d'erreurs principales qui conduisent aux différences entre mesures et résultats de simulation en ce qui concerne le CO2 atmosphérique au-dessus de Paris. Ces sources d’erreur incluent les hypothèses sur les distributions des émissions anthropiques, le transport atmosphérique en particulier le mélange vertical, les flux de CO2 biogéniques, et les conditions aux limites du domaine de simulation
Cities play an important role in tackling climate change as they account for more than 70% of global anthropogenic CO2 emissions. In recent years, several efforts have attempted to quantify city-scale CO2 emissions and establish a high spatially and temporally resolved inventory for supporting urban emission mitigation strategies. The so-called "top-down" inverse estimation of CO2 emissions constrained by independent atmospheric observations could serve to evaluate the consistency of traditional "bottom-up" inventories. A novel CO2 monitoring technique, known as the Greenhouse gas Laser Imaging Tomography Experiment (GreenLITE™) trace gas measurement system, was deployed in central Paris for a 1-year monitoring of near-surface atmospheric CO2 concentrations along 30 horizontal chords. This system has a much wider spatial coverage than traditional in situ sampling and was expected to be more consistent with the spatial representativeness of the kilometer-scale resolution atmospheric transport models used for the city-scale atmospheric inversion.The primary objective of this thesis is to assess the potential contribution of this GreenLITE™ system, in addition to two urban and four peri-urban in situ CO2 measurement stations, for a better understanding of the spatiotemporal variations of CO2 concentrations within Paris and its vicinity. For this objective, I have developed a full modeling framework around the high-resolution Weather Research and Forecasting model (WRF) and its coupling with Chemistry (WRF-Chem), using CO2 emission inventories, estimates of the vegetation fluxes and boundary conditions provided by a large-scale simulation.Chapter 1 is a broad introduction to the subject while chapter 2-4 are built around three separate and publishable papers.Chapter 2 aims at evaluating whether the WRF model running at a 3-km horizontal resolution, with its various configurations, can reproduce the meteorological fields over the IdF region better than the 16-km resolution ECMWF global operational forecasts. The comparisons between WRF and ECMWF forecasts with respect to observations are carried out with a focus on three atmospheric variables (air temperature, wind and PBL height). The results of the sensitivity tests of different physics schemes and nudging options obtained in this chapter are used in subsequent research for the selection of appropriate WRF-Chem model setup in support of atmospheric CO2 transport modeling.Chapter 3 aims at understanding the spatiotemporal variations of CO2 concentrations within Paris and its vicinity during the 1-year GreenLITE™ operating period from September 2015 to December 2016. The analyses are based on CO2 data provided by GreenLITE™ together with six in situ stations and the 1 km-resolution WRF-Chem model coupled with two urban canopy schemes (Urban Canopy Model - UCM; Building Effect Parameterization - BEP). The GreenLITE™ data provide clear information that favors BEP over UCM in the description of vertical mixing and CO2 concentrations during the winter. However, there are indications of measurement noise in summer that limit the usefulness of the data. Furthermore, the model-observation mismatches clearly stress the difficulty of CO2 modeling within urban areas due to the large uncertainties both in the atmospheric transport and the emission inventory.Chapter 4 aims at investigating in detail the critical sources of errors that lead to the model-observation mismatches in the atmospheric CO2 modeling over Paris. These sources of misfit include uncertainties in the assumed distribution of anthropogenic emission, errors in the atmospheric transport, in biogenic CO2 fluxes and in CO2 boundary conditions at the edges of the atmospheric transport model domain. The lessons and insights from this chapter provide requirements and recommendations for the assimilation of CO2 measurements into the atmospheric inversion, when aiming at the quantification of CO2 emissions for the Paris region

Adaptive control designs using neural networks (NNs) based on dynamic inversion are investigated for aerospace vehicles which are operated at highly nonlinear dynamic regimes. NNs play a key role as the principal element of adaptation to approximately cancel the effect of inversion error, which subsequently improves robustness to parametric uncertainty and unmodeled dynamics in nonlinear regimes. An adaptive control scheme previously named composite model reference adaptive control is further developed so that it can be applied to multi-input multi-output output feedback dynamic inversion. It can have adaptive elements in both the dynamic compensator (linear controller) part and/or in the conventional adaptive controller part, also utilizing state estimation information for NN adaptation. This methodology has more flexibility and thus hopefully greater potential than conventional adaptive designs for adaptive flight control in highly nonlinear flight regimes. The stability of the control system is proved through Lyapunov theorems, and validated with simulations. The control designs in this thesis also include the use of pseudo-control hedging techniques which are introduced to prevent the NNs from attempting to adapt to various actuation nonlinearities such as actuator position and rate saturations. Control allocation is introduced for the case of redundant control effectors including thrust vectoring nozzles. A thorough comparison study of conventional and NN-based adaptive designs for a system under a limit cycle, wing-rock, is included in this research, and the NN-based adaptive control designs demonstrate their performances for two highly maneuverable aerial vehicles, NASA F-15 ACTIVE and FQM-117B unmanned aerial vehicle (UAV), operated under various nonlinearities and uncertainties.

Macroeconomic summary Several factors contributed to an increase in projected inflation on the forecast horizon, keeping it above the target rate. These included inflation in December that surpassed expectations (5.62%), indexation to higher inflation rates for various baskets in the consumer price index (CPI), a significant real increase in the legal minimum wage, persistent external and domestic inflationary supply shocks, and heightened exchange rate pressures. The CPI for foods was affected by the persistence of external and domestic supply shocks and was the most significant contributor to unexpectedly high inflation in the fourth quarter. Price adjustments for fuels and certain utilities can explain the acceleration in inflation for regulated items, which was more significant than anticipated. Prices in the CPI for goods excluding food and regulated items also rose more than expected. This was partly due to a smaller effect on prices from the national government’s VAT-free day than anticipated by the technical staff and more persistent external pressures, including via peso depreciation. By contrast, the CPI for services excluding food and regulated items accelerated less than expected, partly reflecting strong competition in the communications sector. This was the only major CPI basket for which prices increased below the target inflation rate. The technical staff revised its inflation forecast upward in response to certain external shocks (prices, costs, and depreciation) and domestic shocks (e.g., on meat products) that were stronger and more persistent than anticipated in the previous report. Observed inflation and a real increase in the legal minimum wage also exceeded expectations, which would boost inflation by affecting price indexation, labor costs, and inflation expectations. The technical staff now expects year-end headline inflation of 4.3% in 2022 and 3.4% in 2023; core inflation is projected to be 4.5% and 3.6%, respectively. These forecasts consider the lapse of certain price relief measures associated with the COVID-19 health emergency, which would contribute to temporarily keeping inflation above the target on the forecast horizon. It is important to note that these estimates continue to contain a significant degree of uncertainty, mainly related to the development of external and domestic supply shocks and their ultimate effects on prices. Other contributing factors include high price volatility and measurement uncertainty related to the extension of Colombia’s health emergency and tax relief measures (such as the VAT-free days) associated with the Social Investment Law (Ley de Inversión Social). The as-yet uncertain magnitude of the effects of a recent real increase in the legal minimum wage (that was high by historical standards) and high observed and expected inflation, are additional factors weighing on the overall uncertainty of the estimates in this report. The size of excess productive capacity remaining in the economy and the degree to which it is closing are also uncertain, as the evolution of the pandemic continues to represent a significant forecast risk. margin, could be less dynamic than expected. And the normalization of monetary policy in the United States could come more quickly than projected in this report, which could negatively affect international financing costs. Finally, there remains a significant degree of uncertainty related to the duration of supply chocks and the degree to which macroeconomic and political conditions could negatively affect the recovery in investment. The technical staff revised its GDP growth projection for 2022 from 4.7% to 4.3% (Graph 1.3). This revision accounts for the likelihood that a larger portion of the recent positive dynamic in private consumption would be transitory than previously expected. This estimate also contemplates less dynamic investment behavior than forecast in the previous report amid less favorable financial conditions and a highly uncertain investment environment. Third-quarter GDP growth (12.9%), which was similar to projections from the October report, and the fourth-quarter growth forecast (8.7%) reflect a positive consumption trend, which has been revised upward. This dynamic has been driven by both public and private spending. Investment growth, meanwhile, has been weaker than forecast. Available fourth-quarter data suggest that consumption spending for the period would have exceeded estimates from October, thanks to three consecutive months that included VAT-free days, a relatively low COVID-19 caseload, and mobility indicators similar to their pre-pandemic levels. By contrast, the most recently available figures on new housing developments and machinery and equipment imports suggest that investment, while continuing to rise, is growing at a slower rate than anticipated in the previous report. The trade deficit is expected to have widened, as imports would have grown at a high level and outpaced exports. Given the above, the technical staff now expects fourth-quarter economic growth of 8.7%, with overall growth for 2021 of 9.9%. Several factors should continue to contribute to output recovery in 2022, though some of these may be less significant than previously forecast. International financial conditions are expected to be less favorable, though external demand should continue to recover and terms of trade continue to increase amid higher projected oil prices. Lower unemployment rates and subsequent positive effects on household income, despite increased inflation, would also boost output recovery, as would progress in the national vaccination campaign. The technical staff expects that the conditions that have favored recent high levels of consumption would be, in large part, transitory. Consumption spending is expected to grow at a slower rate in 2022. Gross fixed capital formation (GFCF) would continue to recover, approaching its pre-pandemic level, though at a slower rate than anticipated in the previous report. This would be due to lower observed GFCF levels and the potential impact of political and fiscal uncertainty. Meanwhile, the policy interest rate would be less expansionary as the process of monetary policy normalization continues. Given the above, growth in 2022 is forecast to decelerate to 4.3% (previously 4.7%). In 2023, that figure (3.1%) is projected to converge to levels closer to the potential growth rate. In this case, excess productive capacity would be expected to tighten at a similar rate as projected in the previous report. The trade deficit would tighten more than previously projected on the forecast horizon, due to expectations of an improved export dynamic and moderation in imports. The growth forecast for 2022 considers a low basis of comparison from the first half of 2021. However, there remain significant downside risks to this forecast. The current projection does not, for example, account for any additional effects on economic activity resulting from further waves of COVID-19. High private consumption levels, which have already surpassed pre-pandemic levels by a large margin, could be less dynamic than expected. And the normalization of monetary policy in the United States could come more quickly than projected in this report, which could negatively affect international financing costs. Finally, there remains a significant degree of uncertainty related to the duration of supply chocks and the degree to which macroeconomic and political conditions could negatively affect the recovery in investment. External demand for Colombian goods and services should continue to recover amid significant global inflation pressures, high oil prices, and less favorable international financial conditions than those estimated in October. Economic activity among Colombia’s major trade partners recovered in 2021 amid countries reopening and ample international liquidity. However, that growth has been somewhat restricted by global supply chain disruptions and new outbreaks of COVID-19. The technical staff has revised its growth forecast for Colombia’s main trade partners from 6.3% to 6.9% for 2021, and from 3.4% to 3.3% for 2022; trade partner economies are expected to grow 2.6% in 2023. Colombia’s annual terms of trade increased in 2021, largely on higher oil, coffee, and coal prices. This improvement came despite increased prices for goods and services imports. The expected oil price trajectory has been revised upward, partly to supply restrictions and lagging investment in the sector that would offset reduced growth forecasts in some major economies. Elevated freight and raw materials costs and supply chain disruptions continue to affect global goods production, and have led to increases in global prices. Coupled with the recovery in global demand, this has put upward pressure on external inflation. Several emerging market economies have continued to normalize monetary policy in this context. Meanwhile, in the United States, the Federal Reserve has anticipated an end to its asset buying program. U.S. inflation in December (7.0%) was again surprisingly high and market average inflation forecasts for 2022 have increased. The Fed is expected to increase its policy rate during the first quarter of 2022, with quarterly increases anticipated over the rest of the year. For its part, Colombia’s sovereign risk premium has increased and is forecast to remain on a higher path, to levels above the 15-year-average, on the forecast horizon. This would be partly due to the effects of a less expansionary monetary policy in the United States and the accumulation of macroeconomic imbalances in Colombia. Given the above, international financial conditions are projected to be less favorable than anticipated in the October report. The increase in Colombia’s external financing costs could be more significant if upward pressures on inflation in the United States persist and monetary policy is normalized more quickly than contemplated in this report. As detailed in Section 2.3, uncertainty surrounding international financial conditions continues to be unusually high. Along with other considerations, recent concerns over the potential effects of new COVID-19 variants, the persistence of global supply chain disruptions, energy crises in certain countries, growing geopolitical tensions, and a more significant deceleration in China are all factors underlying this uncertainty. The changing macroeconomic environment toward greater inflation and unanchoring risks on inflation expectations imply a reduction in the space available for monetary policy stimulus. Recovery in domestic demand and a reduction in excess productive capacity have come in line with the technical staff’s expectations from the October report. Some upside risks to inflation have materialized, while medium-term inflation expectations have increased and are above the 3% target. Monetary policy remains expansionary. Significant global inflationary pressures and the unexpected increase in the CPI in December point to more persistent effects from recent supply shocks. Core inflation is trending upward, but remains below the 3% target. Headline and core inflation projections have increased on the forecast horizon and are above the target rate through the end of 2023. Meanwhile, the expected dynamism of domestic demand would be in line with low levels of excess productive capacity. An accumulation of macroeconomic imbalances in Colombia and the increased likelihood of a faster normalization of monetary policy in the United States would put upward pressure on sovereign risk perceptions in a more persistent manner, with implications for the exchange rate and the natural rate of interest. Persistent disruptions to international supply chains, a high real increase in the legal minimum wage, and the indexation of various baskets in the CPI to higher inflation rates could affect price expectations and push inflation above the target more persistently. These factors suggest that the space to maintain monetary stimulus has continued to diminish, though monetary policy remains expansionary. 1.2 Monetary policy decision Banco de la República’s board of directors (BDBR) in its meetings in December 2021 and January 2022 voted to continue normalizing monetary policy. The BDBR voted by a majority in these two meetings to increase the benchmark interest rate by 50 and 100 basis points, respectively, bringing the policy rate to 4.0%.