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[EQUATION] This cost function is slightly more general than the one in Eq. ( 30 ) since we do not require independence betweeen [MATH] and [MATH] By taking the gradient with respect to [MATH] we obtain the following closed system of nonlinear equations for [MATH] |
[EQUATION] In the case of a linear model, the solution is the standard Kalman filter update equation for the mean. However, the presence of the nonlinear function [MATH] makes the solution more elaborate. All of ES, ESMDA and IES are developed to solve this system of equations. |
4.3 Derivation of ES update equations To derive the ES solution, we will use a Taylor expansion and approximation that allows us to obtain a closed form solution for each realization of [MATH] , i.e., |
[EQUATION] where we have defined [EQUATION] Thus, we approximate the nonlinear function [MATH] with its linearization in Eq. ( 34 ) around [MATH] and in addition evaluate the gradient in Eq. ( 33 ) at [MATH] We now have the gradient of the model, [MATH] defined in Eq. ( 35 ), which differ for each realization. We wish ... |
[EQUATION] By solving for [MATH] and using the matrix identity [EQUATION] (which can be derived from the Woodbury identity), where we substitute [MATH] for [MATH] , and [MATH] for [MATH] , we obtain the solution for [MATH] as |
[EQUATION] 4.4 Linear regression for [MATH] Evensen 2018 used a Taylor expansion of [MATH] around the ensemble mean [MATH] and could then replace the individual gradients in Eq. ( 35 ) evaluated at [MATH] with the gradient [MATH] evaluated at the ensemble mean [MATH] He then showed that [MATH] and [MATH] |
Evensen 2018 , Eqs. 29 and 30) , and he replaced the analytical gradients with ensemble covariances. We will here use an interpretation based on linear regression (see also Reynolds et al. 2006 Chen and Oliver 2013 |
where we start by defining [EQUATION] Thus, we view [MATH] as the sensitivity matrix in a linear regression between [MATH] and [MATH] as |
[EQUATION] When we introduce the ensemble-anomaly matrices [EQUATION] we can write Eq. ( 40 ) as [EQUATION] or equivalently [EQUATION] |
where the approximation is the use of a finite ensemble size. From Eq. ( 39 ) we have [EQUATION] where the superscript denotes pseudo inverse. The unbiasedness of |
[MATH] can be shown, i.e., [EQUATION] Also, using Eq. ( 39 ), we can write [EQUATION] and when we introduce the ensemble representation from Eq. ( 45 ), we obtain |
[EQUATION] The projection [MATH] is just the orthogonal projection onto the range of [MATH] We will now consider three cases: 4.4.1 Linear model and measurement operator |
For a linear model and measurement operator we can write, e.g., [MATH] , and Eq. ( 48 ) becomes [EQUATION] Hence, Eq. ( 49 ) is consistent with the definition of the covariance matrix [MATH] for linear models and all combinations of [MATH] and [MATH] |
4.4.2 Nonlinear model and [MATH] In the case with a nonlinear model and [MATH] the rank of [MATH] is [MATH] , and the projection [MATH] with [MATH] being a vector with all elements equal to one |
(see the Appendix in Sakov et al. 2012 . This result is seen from the fact that [MATH] has only one singular value equal to zero corresponding to the right singular vector [MATH] . The projection [MATH] is then just the subtraction of the ensemble mean. Since we have already removed the ensemble mean from [MATH] , we c... |
[EQUATION] and as in the linear case, Eq. ( 48 ) exactly corresponds to the definition of the ensemble covariance [MATH] This nonlinear case with [MATH] is the most considered case for history matching, data assimilation with nonlinear measurement operators, and iterative smoothers used for sequential data assimilation... |
4.4.3 Nonlinear model and [MATH] In the case of a nonlinear model and [MATH] , which applies for the example considered in Section , the expression in Eq. ( 48 ) is not equal to [MATH] and we must include the projection and redefine the sample covariance as |
[EQUATION] i.e., we compute the covariance of the predicted measurement anomalies projected onto the range of [MATH] This case also applies for nonlinear models in the limit of infinite ensemble size. |
Thus, we must use the definition Eq. ( 51 ) to evaluate [MATH] to ensure consistency in the derivation of the update equation. 4.5 ES algorithm |
We can replace the gradient [MATH] in the update Eq. ( 38 ), using Eqs. ( 40 ) and ( 47 to obtain [EQUATION] The solution of this equation is identical to the solution of Eq. ( 36 ) with [MATH] defined by Eq. ( 39 ). However, if we replace |
[MATH] with [MATH] in Eq. ( 52 ), the solutions of Eqs. ( 36 ) and ( 52 ) will differ in the nonlinear case. By representing the covariances using a finite sample size and sample covariance matrices, we get the update equation for the finite ensemble of [MATH] realizations as |
[EQUATION] where [MATH] is defined in ( 51 ). If we omit the projection in Eq. ( 51 ), then the solution computed by Eqs. ( 52 ) (using sample covariances) and ( 53 ) will differ in the case of a nonlinear model and [MATH] , also when [MATH] becomes infinitely large. Thus, our update will be biased. |
To compute the ES update, we start by sampling the Gaussian prior variables for the parameters [MATH] , the model errors [MATH] , and the measurement perturbations [MATH] |
[EQUATION] The vector [MATH] contains all stochastic model errors over the time interval of the model integration, and the errors can also have correlations in time. We obtain the model prediction from the model written on the form |
[EQUATION] where the model operator depends nonlinearly on the model-error term. Next, we can compute the sample covariances [MATH] , and [MATH] , from the ensembles of [MATH] [MATH] , and [MATH] , and |
[MATH] from the definition in Eq. ( 51 ). Eq. ( 53 ) defines the final update equations for [MATH] and [MATH] which becomes [EQUATION] |
We then rerun the model using the updates [MATH] and [MATH] to get [EQUATION] Alternatively, we can also compute the update of the predicted measurements from |
[EQUATION] and in the case of a linear model the result would be identical to that obtained by integrating the model in Eq. ( 60 ). |
Iterative smoothers in the presence of model errors The critical approximations used in the derivation of ES are, firstly, the linearization in Eq. ( 34 ) of the model about [MATH] meaning that large updates will have large errors, and secondly, that an averaged statistical ensemble gradient replaces the exact analytic... |
The minimization problem in Eq. ( 10 ) can be solved using iterative methods like IES, ESMDA, and IEnKF. The iterative ensemble smoother (IES) by Chen and Oliver 2012 2013 minimizes the ensemble of cost functions by direct minimization using an approximate ensemble gradient. Alternatively, the Ensemble Smoother with Mu... |
Sakov et al. 2012 ); Bocquet and Sakov 2014 derived the iterative EnKF (IEnKF) and iterative Ensemble Kalman Smoother (IEnKS) to better handle nonlinearities in the dynamical model and the observation operator. The focus was on state estimation where the model state at the time [MATH] is updated using measurements of t... |
In the following, we will present variants of ESMDA and IES that take more general model errors into account as is required when solving the weak constraint history-matching problem. |
5.1 ESMDA As explained in Evensen 2018 , ESMDA solves the standard ES update equations using a predefined number of recursive steps. In each step, the measurement error covariance and associated measurement perturbations are inflated to reduce the impact of the measurements. With correctly chosen inflation factors and ... |
From the previous discussion, it is clear that, in the presence of model errors, we need to recursively update both the parameters and the model errors. It is easiest to derive ESMDA by using a tempering of the likelihood function Neal 1996 which leads to a recursive minimization of a sequence of [MATH] cost functions,... |
[EQUATION] where we evaluate [MATH] at the [MATH] th iterate [MATH] , and we must have [EQUATION] Similarly to the derivation of the ES in the previous section, we obtain the recursive update equations for ESMDA given by Eqs. ( 69 ) and ( 70 ) in the algorithm below. As in ES, the update direction is computed based on ... |
To compute the ESMDA solution, we start by sampling the initial ensembles from Eqs. ( 54 ) and ( 55 ) to initialize the recursion in ESMDA |
[EQUATION] Then the model is integrated according to Eq. ( 57 ) to obtain the prior ensemble prediction for the first ESMDA step, |
[EQUATION] and we compute recursively the following for each iteration [MATH] We construct the sample covariances [MATH] , and [MATH] , from the ensembles of [MATH] [MATH] , and [MATH] and [MATH] from the definition in Eq. ( 51 ), and we sample the measurement perturbations |
[EQUATION] used to generate the perturbed measurements [EQUATION] We then compute the updates [EQUATION] and rerun the model to obtain the updated prediction |
[EQUATION] for step [MATH] . We repeat this procedure until [MATH] , which results in the ESMDA solution for [MATH] [MATH] and [MATH] |
5.2 IES In IES we use a gradient based minimization method, and we need to evaluate the first and second order derivatives of the cost function in Eq. ( 32 ) with respect to [MATH] The gradient of the cost function in Eq. ( 32 ) is already derived above as Eq. ( 33 |
[EQUATION] An approximation to the Hessian of the cost function is obtained by operating again by [MATH] on the gradient in Eq. ( 72 ) to obtain |
[EQUATION] where we have neglected the second derivatives or Hessian of the vector function [MATH] , i.e., [MATH] We can then write a Gauss-Newton iteration |
[EQUATION] where we define [MATH] as the gradient normalized by the approximate Hessian as follows [EQUATION] and we define [EQUATION] |
as the gradient of the model, evaluated at iteration [MATH] and for ensemble member [MATH] The Eqs. ( 74 ) and ( 75 ) defines the Ensemble Randomized Likelihood method Kitanidis 1995 Oliver et al. 1996 |
Since we are not computing the analytical gradient of the model we will need to aproximate the ensemble of gradients with an averaged gradient like [MATH] from Eq. ( 45 ), or we can evaluate the gradient at the ensemble average for the local iterate |
[MATH] as was explained by Evensen 2018 . The same model gradient is now used for all realizations, and this leads to a different solution than the solution of the originally posed problem. |
Eq. ( 75 ) is exactly Eq. (2) in Chen and Oliver 2013 . Also, Chen and Oliver 2013 suggested using the state covariance in the Hessian evaluated at the local iterate to simplify further computations, since changing the Hessian does not change the gradient and thus the final converged solution (although it changes the s... |
Thus, we can rewrite Eq. ( 75 ) with the averaged model gradient and introduce the state covariance for the local iterate in the Hessian, to obtain |
[EQUATION] Then using the corollaries [EQUATION] which are derived from the Woodbury identity, we can write Eq. ( 77 ) as [EQUATION] |
Now, from Eqs. ( 40 ) and ( 47 ) we can write Eq. ( 80 ) as [EQUATION] In the original algorithm the expression [MATH] was replaced with the covariance [MATH] . But, as we have seen, this will break the consistency between Eqs. ( 77 ) and ( 81 ) in the nonlinear case with [MATH] , and we need to use the definition in E... |
The numerical solution method for this equation is discussed in more detail by Chen and Oliver 2013 . It is clear that it is the introduction of low-rank ensemble representations of the covariances that makes it possible to compute the update steps [MATH] , and the computation requires the use of singular-value decompo... |
Examples To verify the new algorithms, we will use the scalar example from Evensen 2018 The example resembles the use of conditioning methods in history matching, i.e., there is a parameter [MATH] that serves as an input to a forward model to predict [MATH] . We assume an initial state [MATH] and a prediction [MATH] , ... |
[EQUATION] Here [MATH] is a parameter that determines the nonlinearity of the model. In the current example, we have used [MATH] for the linear cases and [MATH] for the nonlinear cases. Clearly, in this case the model error is additive to make the linear case completely linear. If we have a product of [MATH] and [MATH]... |
The model error [MATH] is a random variable sampled from [MATH] with [MATH] in the case with no model errors and [MATH] in the case including model errors. |
In all the four cases we sample the prior ensemble for [MATH] from a Gaussian distribution with mean [MATH] and variance [MATH] and we sample the perturbed observations of [MATH] from a Gaussian error distribution with mean [MATH] and variance [MATH] Thus, in the current example, [MATH] represents the initial state or ... |
In this example, we use a sufficiently large number of samples, i.e., [MATH] , to generate accurate estimates of the probability density functions and this allows us to work directly with the pdfs and to examine the converged solutions of the methods. |
6.1 Results from the linear case In Figs. and we show the results from the linear cases without and with model errors. In Fig. we plot the joint pdfs for the prior and the updated solutions, and in Fig. we plot the corresponding marginal pdfs. |
The joint pdfs illustrate the effect of including stochastic model errors. Without model errors, there is a unique mapping from [MATH] to [MATH] , and the pdf is zero except along the curve (or line in the linear case) defined by the model function [MATH] . The prior joint pdf has a maximum value located at [MATH] whil... |
[MATH] for all the methods. When we introduce the model errors, we notice that we obtain a stronger update in [MATH] and weaker update in [MATH] , than in the case without model errors. Still, we observe that all the smoother methods give a result that is identical to the Bayesian update. We can better visualize these ... |
In the case without model errors, we see that the prediction pdf for [MATH] and the measurement pdf have the same variance and only differ in the value of the means. The measurement is at [MATH] while the mean prediction is located at [MATH] . The update from ES, ESMDA, and IES, exactly matches the Bayesian update in t... |
When we include model errors, the effect is that the prediction gets a higher variance, although the mean is the same (in this particular case). Due to the higher variance, we give more weight to the measurement in the update and the update for [MATH] is stronger than in the case without model errors. On the other hand... |
So, how can the update for [MATH] be shifted towards the observation in this case? Afterall, we compute [MATH] as a prediction from [MATH] . Here the inclusion of the model errors in the inversion plays a vital role. We simultaneously update the ensemble for [MATH] and the estimate of the model errors [MATH] . In Fig. ... |
This example illustrates how model errors impact the updates of [MATH] and [MATH] as well as how we also need to include the model errors as a parameter in the estimation and then use it in the prediction to obtain the correct estimate of [MATH] . Finally, we also demonstrate that in the linear case with and without mo... |
6.2 Results from the nonlinear case In Figs. and we show the results from the nonlinear cases with and without model errors, where Fig. plots the joint pdfs for the prior and the updated solutions, and in Fig. we plot the corresponding marginal pdfs. |
From the joint pdfs, we notice that the various smoother methods give different results both with and without the inclusion of model errors, although the general shape and locations of the pdfs are reasonably consistent with the theoretical solution as given by Bayes theorem. |
We get a clearer picture from the marginal pdfs in Fig. As for the linear case, we get a weaker update of [MATH] and a stronger update of [MATH] . We also notice that the introduction of model errors is handled well by the iterative methods, and the results are somewhat better and more consistent with the theoretical s... |
The dashed green line in the plots for [MATH] in Fig. is the direct ES update of [MATH] using the predicted ensemble for [MATH] and the measurement. It is clear that the update of [MATH] followed by an integration of the model to obtain [MATH] gives a better result than a direct update of [MATH] . Furthermore, the addi... |
The impact of using Eq. ( 51 ) for evaluating [MATH] in the update schemes is illustrated in Fig. where we show results including and excluding the projection. The impact is most pronounced when using ES and ESMDA with few update steps where the use of [MATH] instead of [MATH] severely impacts the computation of the lo... |
Including model errors in history matching The need for including model errors in iterative ensemble smoothers became apparent while working with the paper Evensen and Eikrem 2018 , which considered the conditioning of reservoir models on production-rate data. Typically, in history matching, we assign errors to the rat... |
The errors in rate data are considered as the dominant model errors in a reservoir simulation model when we exclude errors in the model parameters that we estimate during the history matching. Also, Evensen and Eikrem 2018 pointed out that there are strong time correlations in the errors in rate data due to the rate al... |
The functional form [MATH] can represent the prediction of the produced rates (that we observe) from a reservoir simulation model. Note that using ensemble methods, we do not need to explicitly construct the functional form [MATH] since we represent the gradients using ensemble covariances. We only need access to a num... |
If we associate the dominant model errors with the rates used to force the simulation model, then the size of the vector of model errors [MATH] is equal to the number of rate data used to force the model. A typical number of data for a well can be 12 data points per year, i.e., if we force the model using monthly reser... |
The prior error statistics for the rate data used to force the model should be the same as is used for the rate data in the update step. Thus, we can simulate a prior ensemble of time series with mean zero, a specified variogram in time, and a specified variance, to represent the model errors. These time series are the... |
[MATH] is then defined by the ensemble [MATH] The conventional procedure of deriving the production rates from rate-allocation tables, which we only update in connection with separator tests, often several years apart, means that the errors in the rate data will be nearly perfectly correlated in between each separator ... |
The expected impact of including model errors is first a larger and more realistic spread of the prior ensemble. Second, we will obtain a weaker and more correct update of the reservoir parameters in conditioning step. Furthermore, the posterior ensemble will give a more accurate and consistent prediction at the end of... |
Summary In this paper, we have given consistent formulations of iterative ensemble smoothers when we include model errors. We demonstrate the consistency by showing that the ensemble smoothers all converge to the Bayesian posterior in the linear case. The main result is that the model errors need to be treated as anoth... |
We demonstrate that the inclusion of model errors leads to a weaker update of the input parameters to the model, but a stronger update of the measured model prediction. Vice verse, the negligence of model errors that should be present, will lead to a too substantial update of the model input parameters with an associat... |
Thus, the results open for a more consistent solution of the history-matching problem, given that significant model errors are neglected in all previous history-matching applications with iterative ensemble smoothers. |
We also briefly discuss an inconsistency of the linearizations in the analysis scheme that appear for nonlinear operators and when the state dimension is less than the ensemble size minus one, and we show that we must include an additional projection of the predicted measurements for consistency in the derivation of th... |
Acknowledgement This work was supported by the Research Council of Norway and the companies Aker–BP, DEA, ENI, Shell, Petrobras, Equinor, and VNG, through the Petromaks–2 project DIGIRES. Also, the work has benefited from the interaction and collaborations with the Nordforsk Nordic center of excellence in data assimila... |
# Source: arxiv 1806.00315 # Title: Expansion of Presburger arithmetic with the exchange property # Sections: all # Downloaded: 2026-03-03T02:37:29.844368+00:00 |
Expansion of Presburger arithmetic with the Exchange Property Abstract Let [MATH] be a model of Presburger arithmetic. Let [MATH] be an expansion of the language of Presburger [MATH] . In this paper we prove that the [MATH] -theory of [MATH] is [MATH] -minimal iff it has the exchange property and any bounded definable ... |
Introduction Let [MATH] be a model of Presburger arithmetic i.e., [MATH] is a group elementary equivalent to [MATH] (as ordered group i.e. in the language [MATH] - where [MATH] is the congruence relation). We fix [MATH] an expansion of [MATH] and an interpretation of the elements of [MATH] in [MATH] . We will consider ... |
Definition 1.1 We say that [MATH] is [MATH] -minimal if for all [MATH] definable, [MATH] is [MATH] -definable. [MATH] is [MATH] -minimal if for all [MATH] (as [MATH] -structures), [MATH] is [MATH] -minimal. |
Remark. All definable sets are definable with parameters. It is known that [MATH] -minimal theories have many good properties. For instance in |
, R. Cluckers proves a cell decomposition theorem for models of such theory. It follows that it admits a good notion of dimension and the exchange property. |
Definition 1.2 Let [MATH] be a [MATH] -structure. Let [MATH] . Then [MATH] is the set of [MATH] such that there is [MATH] [MATH] -formula and [MATH] such that [MATH] |
Definition 1.3 A theory [MATH] satisfies EP (exchange property) if for all [MATH] , for all [MATH] , for all [MATH] [EQUATION] If [MATH] , then the theory [MATH] satisfies EP if and only if it is [MATH] -minimal. This follows easily from results of C. Michaux and R. Villemaire |
. In their paper the authors prove that (1) [MATH] is [MATH] -minimal iff [MATH] is [MATH] -minimal and (2) that [MATH] is [MATH] -minimal iff there is no definable expanding set in [MATH] |
Definition 1.4 Let [MATH] definable. We say that [MATH] is expanding if for all [MATH] there are infinitely many [MATH] such that [MATH] |
It is then not difficult to prove that the exchange property implies that there is no definable expanding set: Proposition 1.5 Let [MATH] . If [MATH] satisfies EP then for all [MATH] , for all [MATH] definable, [MATH] is not expanding. |
Proof. Let [MATH] be a model of [MATH] and [MATH] definable. Assume that [MATH] is expanding. Then by compactness there is [MATH] elementary expansion of [MATH] and [MATH] such that [MATH] [MATH] for all [MATH] . Let [MATH] be the successor of [MATH] in [MATH] . By compactness again there is an elementary extension [MA... |
So [MATH] while [MATH] . This contradicts the assumption that [MATH] has the exchange property. In general it is not true that EP implies [MATH] -minimality. For instance consider [MATH] be a nonstandard model of Presburger and the expansion of [MATH] by a unary predicate interpreted in [MATH] by a proper convex subgro... |
Definition 1.6 [MATH] satisfies DC (definable completeness) if any definable unary set has a supremum in [MATH] Remark. DC is a first-order property. So it is also a property of the theory of [MATH] |
For the rest of this paper, we assume that [MATH] and that [MATH] satisfies EP (so by Proposition 1.5 , there is no definable expanding set). We fix [MATH] [MATH] -definable subset of [MATH] . Under these hypotheses, we shall prove that [MATH] is [MATH] -definable (Proposition 2.24 ). Then, the main theorem follows: |
Theorem 1.7 [MATH] is [MATH] -minimal iff [MATH] satisfies EP and DC. Proof. One direction is done in : EP follows for instance from cell-decomposition and DC is an immediate consequence of [MATH] -minimality. The other direction will be proved in Proposition 2.24 |
The proof of Proposition 2.24 has two main steps. First, we prove a non-standard version of Michaux-Villemaire . More precisely following their strategy we prove that if [MATH] is definable then [MATH] is a finite union of points and of cosets of [MATH] (for some integer [MATH] ). Then by DC, we can extend this propert... |
Remark. It is already known that the above theorem fails for generalisation of [MATH] -minimality. For coset-minimal groups (in the sense of |
; note that [MATH] -minimal groups are coset-minimal), there is an example in of coset-minimal group which does not have the exchange property. |
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