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The Community Inversion Framework v a unified system for atmospheric inversion studies

 Computation modes in the CIF

The present version of the CIF includes three main categories of inversion methods: (1) analytical, i.e., algebraic solution of the unbiased Gaussian Bayesian problem; (2) ensemble methods with the ensemble square root filter (EnSRF); and (3) variational with two examples of minimizing Actual Window Minimizer Licenses key (M1QN3 and CONGRAD). Other types of data assimilation methods (e.g., direct sampling of probability density functions through Monte Carlo approaches) are also used by the community. The choice of implementing the three aforementioned methods first is motivated Actual Window Minimizer Licenses key their dominant use and because these three use very different approaches for solving the Bayesian inversion problem, i.e., with/without random sampling of probability distributions and with/without the use of the adjoint of the observation operator. The adjoint of the observation operator, noted ℋ*, is built following the mathematical definition of the adjoint; heuristically, it operates backwards compared to the observation operator (e.g., Errico, ) in the sense that it determines the sensitivity to inputs (e.g., fluxes) given an incremental perturbation to outputs (e.g., concentrations). In addition to the mentioned data assimilation methods, the CIF also includes the possibility to run forward simulations and to test the adjoint and the tangent linear of the observation operator for given inversion configurations. In the following we call all inversion methods and other types of computation in the CIF “computation modes”. With these computation modes implemented in a flexible and general manner, it is anticipated that other inversion methods could be easily added to the CIF in the future (see Sect. ).

 Data assimilation methods

Analytical inversions

Analytical inversions compute the algebraic solution of the Gaussian Bayesian problem when it adobe acrobat pro dc crack ita windows 10 - Crack Key For U linear, and they are used extensively at all scales (e.g.,  Stohl Actual Window Minimizer Licenses key Turner and Jacob, ; Kopacz et al., ; Bousquet et al., ; Wang et al., ; Palmer et al., ). When the observation operator is linear, ℋ equals its Jacobian matrix H; conversely, its adjoint ℋ* is the transpose of the Jacobian HT. In that case, xa and A can be explicitly written as matrix products. There are two equivalent formulations of the matrix products for the solution of the problem (e.g.,  Tarantola and Valette, ):

or

with K the Kalman gain matrix as .

Analytical inversions can also be used on slightly non-linear problems, by linearizing the observation operator around a given reference point using the tangent linear of the observation operator. It formulates as follows:

with δx a small deviation from xb within a domain where the linear assumption is valid, the tangent linear of ℋ at xb and the Jacobian matrix of ℋ at xb.

Then Eq. (3) can be easily adapted by replacing (yoHxb) by (yo−ℋ(xb)) and H by .

The computation of an analytical Bitdefender Total Security 2018 Build 23.0.8.17 Crack - Free Activators faces two main computational limitations. First, the matrix H representing the observation operator ℋ must be built explicitly. This can be done either column by column, in the so-called response function method, or row by row, in the so-called footprint method. The two approaches require dim(𝒳), the dimension of the target space, and dim(𝒴), the dimension of the observation space, independent simulations. In the response function method, each column is built by computing with ℬχ being the canonical basis of the target space. For a given increment δxi, the corresponding column gives the sensitivity of observations to changes in the corresponding component of the target space. In the footprint method, each row is built by computing with ℬ𝒴 the canonical basis of the observation space. For a given perturbation of δyi of a component of the observation vector, the corresponding row of H gives the sensitivity of the inputs to that perturbation.

Depending on the number of available observations or the size of the target vector, one of the two is preferred to limit the number of observation operator computations to be carried out explicitly. When the dimension of the target vector is relatively small, the response function is generally preferred; conversely, when the observation vector is small, the footprint approach is preferred. The type of transport model used to compute the matrix H also plays a role in the choice of the approach; for Eulerian models, the response function approach is preferred for multiple reasons: (i) their adjoint is often much more costly than their forward, (ii) the adjoint may not be available for some models or is difficult to generate, and (iii) the computation time of the forward is constant no matter how numerous the observations are; for Lagrangian models, the footprint approach is preferred as they often compute backward transport simulations for each observation, allowing a straightforward computation of the adjoint (Seibert and Frank, ). In both cases, the explicit construction of the matrix H requires numerous independent simulations, which can be an insurmountable computational challenge.

The second obstacle consists of the fact that the computation of the Kalman gain matrix in Eq. (3) (left) requires inverting a matrix of the dimension of the observation space, dim(𝒴), while for the other formulation (Eq. 3, right) the matrix is of dimension dim(𝒳), which is the dimension of the target space. If the dimensions of both the observation and the target spaces are very high, as in many inversion applications, the explicit computation of Eq. (3) with matrix products and inverses is not computationally feasible. For this reason, smart adaptations of the inversion framework (including approximations and numerical solvers) are often necessary to tackle problems even when they are linear; in the following, we choose to elaborate on some of the most frequent approaches used in the atmospheric inversion community: the variational approach and one ensemble method, the Ensemble Square Root Filter (EnSRF). Less frequently, intermediate adaptations of the analytical inversion also include sequential applications (e.g., Michalak, ; Bruhwiler et al., ; Brunner et al., ), which are a compromise between tackling the above-mentioned computational obstacles while maintaining the simplicity of the analytical inversion; however, such sequential analytical inversions are limited to specific linear and simple cases.

Ensemble methods

Ensemble methods are commonly used to tackle high-dimensional problems and to approximately characterize the optimal solution. In ensemble methods, such as ensemble Kalman filters (EnKFs) or smoothers (e.g.,  Whitaker and Hamill, ; Peters et al., ; Zupanski et al., ; Zupanski, ; Feng et al., ; Chatterjee et al., ), the issue of high dimensions in the system of Eq. (3) is avoided using the two following Actual Window Minimizer Licenses key procedures:

  • For very dense observations, such as datasets from new-generation high-resolution satellites, the sequential assimilation of observations may not be sufficient, or at least methods may be needed to make the observation errors between sequential assimilation windows independent, e.g., by applying a whitening transformation to the observations to form a new set with uncorrelated errors as suggested by Tippett et al. (). The challenge is exacerbated for long-lived species such as CO2, for which assimilation windows must be long enough to maintain the propagation of information on the fluxes over long distances through transport; propagating a covariance matrix from assimilation windows to assimilation windows as accurate as possible could in principle limit the later issue, as suggested in Kang et al. (, ), but this could still prove hard to apply in very high resolution problems.

  • The posterior distribution at a given step of the filter is then characterized explicitly by applying Eq. (1) on each member of the ensemble; the new intermediate posterior distribution is then sampled and propagated to the next data assimilation window.

In the atmospheric inversion community, another ensemble method is widely used, based on the CarbonTracker system (Peters et al., ), which uses an ensemble square root filter (EnSRF; Whitaker and Hamill, ). In that approach, the observations are split using running data assimilation windows as for other ensemble methods, but instead of directly characterizing the posterior distribution from the ensemble, the statistics of the ensemble are used to solve the analytical equation, Eq. (3), approximately. Thus, the EnSRF method is less general than EnKFs methods, as it relies on the Gaussian assumption, as well as limited non-linearity in the inversion problem, but it proves very efficient at computing an approximated solution of the inversion problem. Matrix products in Eq. (3) involving the target vector covariance matrix B (HBHT and BHT) are approximated by reducing the space of uncertainties to a low-rank representation; this is done in practice by using a Monte Carlo ensemble of possible target vectors sampling the distribution 𝒩(xb,B); with such an approximation, matrix products can be written as follows:

where N is the size of the ensemble.

From there, Eq. (1) is solved analytically by replacing HBHT and BHT by their respective approximations.

By using random sampling, ensemble methods are able to approximate large dimensional matrices at a reduced cost Actual Window Minimizer Licenses key using the adjoint of the observation operator (see variational inversion below) that can be challenging to implement. Small ensembles Actual Window Minimizer Licenses key cause the posterior ensemble to collapse; i.e., the posterior distribution is dominated by one or a very small number of members, which does not allow for a reliable assessment of the posterior uncertainties (Morzfeld et al., ); moreover, small ensembles introduce spuriousness in the posterior uncertainties, with unrealistic correlations being artificially generated. In the EnSRF, small ensembles rather cause a misrepresentation of uncertainty structures, which limits the accuracy of the computed solution and may require fixes as described in, for example, Bocquet (). In any case, the level of approximation necessary for this approach to work is strongly different for each problem, which requires preliminary studies before consistent application. In particular, the so-called localization of the ensemble can be used to improve the consistency of the inversion outputs (e.g., Zupanski et al., ; Babenhauserheide et al., ).

In the current version, only the EnSRF approach is implemented in the CIF. One should note that the EnSRF, as a direct approximation of the analytical solution, can be very sensitive to non-linearity in the observation operator (e.g., Tolk et al., ). It can generally cope only with slight non-linearity over the assimilation window; thus, the assimilation window length has to be chosen appropriately, contrary to other ensemble methods which are usually not sensitive to non-linearity.

Variational inversions

Variational inversions use the fact that finding the mode of the posterior Gaussian distribution in Eq. (2) is equivalent to finding the minimum xa of the cost function J:

In variational inversions, the minimum of the cost function in Eq. (6) is numerically estimated iteratively using quasi-Newtonian algorithms based on the gradient of the cost function:

Quasi-Newtonian methods are a group of algorithms designed to compute the minimum of a function iteratively. It should be noted that in high-dimensional problems it can take a very large number of iterations to reach the minimum of the cost function J, forcing the user to stop the algorithm before convergence, thus reaching only an approximation of xa; in addition, iterative algorithms can reach local minima without ever reaching the global minimum, making it essential to thoroughly verify variational inversion results; this can happen in non-linear cases but also due to numerical artefacts in linear cases (some points in the cost function have gradients so close to zero that the algorithm sees them as convergence points, whereas the unique global minimum is somewhere else). In the community, examples of quasi-Newtonian algorithms commonly used are the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm (Zheng et al., ; Bousserez et al., ), M1QN3 (Gilbert and Lemaréchal, ) and the CONGRAD algorithm (applicable only to linear or linearized problems; Fisher, ; Chevallier et al., ) based on the Lanczos method, which iterates to find the eigenvalues and eigenvectors of the Hessian matrix, which is then used (in a single step) to calculate the analysis vector, xa. In general, quasi-Newtonian methods require an initial regularization, or “pre-conditioning” of x, the vector to be optimized, for better efficiency. In atmospheric inversions, such a regularization is generally made by optimizing instead of x; we denote 𝔄 as the regularization space: χ∈𝔄. This transformation translates in Eqs. (6) and (7) as follows:

Solving Eqs. (6) and  (7) in the target vector space or Eq. (8) in the regularization space is mathematically fully equivalent, but the solution in the regularization space is often reached in fewer iterations. Moreover, in the regularization space, one can force the algorithm to focus on the main modes of the target vector space by filtering the smallest eigenvalues of the matrix B. This reduces the dimension of χ and accelerates further the rate of convergence, although the solution of the reduced problem is only an approximation of the solution of the full problem. In the following we thus prefer calling the “regularization space” the “reduction space”. The link between the two can be written as follows:

withActual Window Minimizer Licenses key and Λ being the matrices of the eigenvector and the matrix of the corresponding eigenvalues of the matrix B respectively. Q spyhunter 5 crack free download Λ are the reduced matrices of eigenvalues and eigenvectors with a given number of dominant eigenvalues.

Overall, variational inversions are a numerical approximation to the solution of the inversion problem: they involve the gradient of the cost function in Eq. (7) and require us to run forward and adjoint simulations iteratively (e.g., Meirink et al., ; Bergamaschi et al., ; Houweling et al., , ; Fortems-Cheiney et al., ; Chevallier et al., , ; Thompson and Stohl, ; Monteil and Scholze, ; Wang et al., ).

The variational formulation does not require calculation of complex matrix products and inversions, contrary to the analytical inversion, and is thus not limited by vector dimensions. Still, the inverses of the uncertainty matrices B and R need to be computed, potentially prohibiting the use of very large and/or complex general matrices; this challenge is often overcome by reducing B and R to manageable combinations of simple matrices (e.g., Kronecker products of simple shape covariance matrices; see Sect. ).

When the observation operator is linear, the posterior uncertainty matrix A is equal to the inverse of the Hessian matrix at the minimum of the cost function. In most cases the Hessian cannot be computed explicitly because of memory limitations, which is a major drawback of variational inversions. But some variational algorithms such as CONGRAD provide a coarse approximation of the Hessian: in the case of CONGRAD based on the Lanczos method, leading eigenvectors of the Hessian can be computed, together with their eigenvalues (Fisher, ). The approximation of the posterior uncertainty matrix A in the case of CONGRAD reads as follows:

with being the dominant eigenvectors of the Hessian matrix at the point xa, and being the matrix of the dominant eigenvalues of the Hessian matrix. Please note that the dominant eigenvalues of the Hessian matrix correspond to components with low posterior uncertainties in A.

Another approach to quantify the posterior uncertainty matrix A, valid for both linear and non-linear cases, is to carry out a Monte Carlo ensemble of independent inversions with sampled prior vectors from the prior distribution 𝒩(xb,B)(e.g., Liu et al., ). An ensemble of posterior vectors are inferred and used to compute the posterior matrix as follows:

with N being the size of the Monte Carlo ensemble, the posterior vector corresponding to the prior of the Monte Carlo ensemble and the average over sampled posterior vectors.

 Auxiliary computation modes

Forward simulations

Forward simulations simply use the observation operator to compute simulated observation equivalents. It reads as

This mode is used to make quick comparisons between observations and simulations to check for inconsistencies before running a full inversion. It is also used by the analytical inversion mode to build response functions.

Test of the adjoint

The test of the adjoint is a crucial diagnostic for any inversion system making use of the adjoint of the observation operator. Such a test is typically required after making any edits to the code (to the forward observation operator or its adjoint) before running an inversion. Coding an adjoint is prone to errors and even small errors can have significant impacts on the computation of the gradient of the cost function in Eq. (7). Thus, one needs to make sure that the adjoint rigorously corresponds to the forward. This test consists of checking the definition of the mathematical adjoint of the observation operator. It writes as follows for a given target vector x and incremental target perturbation δx:

where dx(δx) is the linearization of the observation operator ℋ at the point x for a given increment δx; it is computed with the tangent linear model, which is the numerical adaptation of dx(δx). Then,

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