Reduction of uncertainties through Data Model Integration (DMI)

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Data Model Integration (DMI) combines numerical simulation models with observations to improve estimates of model parameters, system states and forecasts. Its two main applications are (1) Model calibration, in which uncertain model parameters are estimated from observations, and (2) sequential data assimilation, in which model states are updated as new observations become available. Data assimilation can reduce forecast uncertainty, particularly over short prediction horizons, and is important for early warning of hazardous events. Unlike purely data-driven techniques discussed in Data analysis techniques for the coastal zone, DMI also uses a description of the physical processes. This can improve predictions beyond the conditions covered by observations, although their reliability remains dependent on the model and its assumptions.


Introduction

Data Model Integration is applied in many fields, including hydrodynamics, meteorology and hydrology. Examples include improving simulations of currents, water levels and waves, updating weather and flood forecasts, and identifying inconsistencies in observations.

DMI combines two sources of information: prior knowledge represented by a model and additional observations. Both are uncertain. When their errors can be characterized statistically, the two sources can be combined to improve estimates. The principal approaches are:

  1. Model calibration (parameter estimation): adjusting uncertain parameters to improve agreement with observations.
  2. Sequential data assimilation: updating estimates of the evolving system state as observations become available, thereby improving initial conditions for forecasts.

Error criteria and Goodness of Fit

To evaluate a model, the quantities to be compared with observations must first be identified. Agreement is commonly expressed by a cost function (CF), which measures discrepancies between modelled and observed values. Least-squares cost functions are widely used. Model calibration generally seeks to minimize a cost function; goodness-of-fit (GoF) measures provide alternative descriptions of model performance. The relationship between CF and GoF depends on their definitions.

Role and reduction of uncertainties

Uncertainties in models and data

Numerical models are simplified representations of real processes, such as currents, waves, sediment transport, rainfall-runoff and morphological change. Approximations in their formulation, uncertain parameters and imperfect forcing data introduce errors. Uncertainty may arise from:

  1. model structure and numerical approximations,
  2. model parameters,
  3. external forcing and boundary conditions,
  4. initial conditions and unresolved processes.

A further source of uncertainty is the difference between the quantities represented by the model and those actually measured, including differences in spatial and temporal scales.

Observations are also uncertain, owing to:

  1. equipment accuracy,
  2. instrument drift,
  3. equipment fouling or malfunctioning,
  4. temporal and spatial sampling frequency,
  5. data processing and interpretation,
  6. spatial and temporal representativeness.

Consequently, discrepancies between model results and observations are unavoidable. Neither source necessarily represents the true system state exactly.

Stochastic models

A deterministic model can be embedded in a stochastic formulation by representing model errors statistically. Observation errors can be described in a similar way. Data assimilation combines these descriptions with the model equations and observations to estimate the system state and its uncertainty.

Formulation of uncertainty

Effective DMI requires plausible estimates of model and observation uncertainties, including their spatial and temporal correlations. The characteristic scales of these errors should be consistent with the processes represented by the model. Physical knowledge is therefore important when specifying the uncertainty model. Poorly specified uncertainties can lead to misleading estimates, even when the assimilation algorithm is mathematically sound.

Reduction of uncertainty through DMI

When model and observation uncertainties are adequately described, their estimates can be combined using weights that reflect their relative reliability. Consider a model estimate [math]\xi_M[/math] with standard deviation [math]\sigma_M[/math] and an observed estimate [math]\xi_O[/math] with standard deviation [math]\sigma_O[/math], both referring to the same variable, location and time. For unbiased, independent errors, the minimum-variance linear estimate is

[math]\hat\xi = \Large\frac{\xi_M \, {\sigma_O}^2 + \xi_O \, {\sigma_M}^2}{ {\sigma_O}^2 + {\sigma_M}^2} \, , \qquad (1)[/math]

with a spread [math]\hat\sigma[/math] that satisfies

[math]{\hat\sigma}^{-2} = {\sigma_O}^{-2} + {\sigma_M}^{-2}[/math].

These expressions assume unbiased and statistically independent model and observation errors with correctly estimated variances. Under these assumptions, the variance of the combined estimate is smaller than either individual variance. Correlated errors, systematic biases and incorrect variance estimates require a more general treatment and can invalidate the expected reduction in actual uncertainty.

This example illustrates the basic principle of DMI. In numerical models with many grid points, interacting state variables and nonlinear dynamics, the same principle is extended using spatial and temporal error covariances and methods that account for the model equations.

Calibration of models

Objective

Model calibration estimates uncertain parameters or control variables by comparing model predictions with observations. The model structure is normally kept fixed, although discrepancies may also arise from deficiencies in the model equations, forcing or boundary conditions. Parameters are adjusted to improve agreement according to a specified criterion, taking measurement uncertainties into account where possible. Calibration is distinct from model validation, in which performance is assessed using independent data without readjusting the calibrated parameters.

Approach

Calibration is generally formulated as an optimization problem: parameters are chosen to minimize a cost function or maximize a goodness-of-fit measure. The criterion specifies which aspects of the observations the model should reproduce. Measurement uncertainties may be included through statistical weights; more reliable observations then generally receive greater weight, provided their errors and correlations are appropriately represented.

Statistical interpretation

Calibration can be interpreted statistically as well as deterministically. Instead of prescribing an arbitrary cost function, one can derive it from an assumed probability distribution of the observation errors. For independent normally distributed errors, maximum likelihood estimation yields a weighted least-squares criterion, with weights related to error variances. Correlated Gaussian errors require a covariance-weighted criterion. Statistical methods can also estimate parameter uncertainty, subject to the model and error assumptions.

Calibration techniques

When model predictions depend linearly on the parameters, analytical methods such as those described in Linear regression analysis of coastal data may be used. Nonlinear models generally require numerical optimization. Gradient-based methods, including conjugate-gradient and quasi-Newton methods, can be efficient when derivatives of the cost function are available. For large dynamical models, adjoint methods can calculate gradients with respect to many parameters at a cost that does not increase proportionally with the number of parameters (see Appendix A)[1][2]. Their development and maintenance may nevertheless be demanding. Gradient-free methods, such as DUD (Doesn't Use Derivatives), provide alternatives, particularly when the number of parameters and model evaluation cost are manageable.[3]

Sequential data assimilation in dynamic (time-stepping) models

Objective

Even a well-calibrated model can produce inaccurate forecasts because of uncertain initial conditions, forcing, parameters and model structure, or changes in system behaviour. Sequential data assimilation incorporates new observations during a simulation to improve the estimated state from which subsequent forecasts are made.

Approach

Sequential data assimilation combines the model forecast with new observations using statistical descriptions of their uncertainties. The model provides an estimate of the evolving state; observations are used to correct that estimate, with the size and spatial extent of the correction determined by error covariances. The updated state is then used to continue the simulation and produce a new forecast. This cycle is repeated as observations become available.

Simple techniques

The simplest method is direct data insertion: at observation times, selected model values are replaced by measurements. Such abrupt changes may be inconsistent with the surrounding flow field. In hydrodynamic models they can generate artificial gravity waves (gravity noise) that propagate through the domain and temporarily distort the solution. Optimal interpolation instead applies a weighted correction based on the difference between observations and model predictions and on prescribed error covariances. Under suitable statistical assumptions this gives a minimum-variance linear update, but it does not necessarily preserve dynamical balance. Initialization or balance-preserving procedures may be needed to limit spurious oscillations.

Kalman filtering

The Kalman filter (Kalman, 1960[4]) updates a model forecast using observations and their estimated error covariances. Unlike optimal interpolation with fixed background-error statistics, it propagates the estimated state-error covariance through the model in time. The classical filter gives the minimum-variance linear estimate for linear systems under its statistical assumptions (see Appendix B). For nonlinear models, the Extended Kalman Filter (EKF) uses local linearization, while the Ensemble Kalman Filter (EnKF) estimates forecast-error covariances from an ensemble of model simulations (Evensen, 1994[5]). Reduced-rank filters approximate the covariance using a limited number of dominant directions, reducing computational demands (Verlaan et al. 1997[6]). Applications include tidal and storm-surge forecasting and river-flow prediction. [7][8][9][10][11] Different current DMI methods are further compared in the paragraph below.

Current data assimilation methods

Two widely used families of data assimilation methods are variational and ensemble methods (Moore et al. 2019[12]; Evensen et al. 2022[13]). Both combine model estimates and observations according to their uncertainties, extending the principle of Eq. (1). In spatial models, a central difficulty is estimating how errors at different locations and in different variables are related. These correlations determine how observations at a limited number of locations can improve estimates elsewhere.

Variational methods estimate the model state by minimizing a cost function that measures disagreement with observations and departure from a prior estimate, taking account of their uncertainties (Carrassi et al. 2019[14]). Three-dimensional variational assimilation (3D-Var) usually updates the state at an analysis time. Four-dimensional variational assimilation (4D-Var) fits observations distributed over a time window by adjusting the initial state and propagating the model through that window; adjoint methods are often used to compute gradients. In conventional formulations, the uncertainty of the prior estimate and its spatial correlations (the background-error covariance) are prescribed beforehand rather than computed by the model.

Ensemble methods, notably the Ensemble Kalman Filter, propagate multiple model realizations with different initial conditions, forcing or parameters. The ensemble spread and covariation provide estimates of forecast uncertainty and of relationships between errors at different locations. No adjoint model is required, but repeated model simulations are computationally expensive. With a limited number of runs, the ensemble can suggest relations between errors at distant locations that are not real; these are usually suppressed (localization), and the ensemble spread is often enlarged to prevent it from becoming too small (inflation). Hybrid methods combine elements of variational and ensemble approaches.

Particle filters propagate an ensemble of possible states, or particles, and assign each a weight according to the likelihood of the observations[15]. Particles with low weights may be removed and those with high weights replicated through resampling. Unlike standard Kalman-filter methods, particle filters do not require a Gaussian representation of the state distribution. However, in large systems almost all weight tends to end up on a few particles, so that a very large number of runs is needed.



Appendix A. The adjoint method for efficient gradient computation

Model calibration generally requires finding a set of parameters that minimizes a cost function measuring the differences between model results and observations. For a model with many uncertain parameters, calculating the gradient of the cost function by separately perturbing each parameter can be computationally expensive.

Consider a numerical model represented by

[math]\mathbf{F}(\mathbf{x},\mathbf{p})=0, \qquad (A1)[/math]

where [math]\mathbf x[/math] represents the model state variables (for example, water levels and velocities at all grid points and relevant time steps) and [math]\mathbf p[/math] is the vector of uncertain model parameters. The cost function [math]J(\mathbf{x},\mathbf{p})[/math] measures the discrepancy between model results and observations (for example the sum of squared discrepancies).

Because the model state depends on the parameters, the derivative of the cost function is

[math]\dfrac{dJ}{d\mathbf{p}}=\dfrac{\partial J}{\partial\mathbf{p}}+\dfrac{\partial J}{\partial\mathbf{x}}\dfrac{d\mathbf{x}}{d\mathbf{p}}. \qquad (A2)[/math]

Differentiating (A1) gives [math]\quad \dfrac{\partial \mathbf{F}}{\partial \mathbf{x}}\dfrac{d\mathbf{x}}{d\mathbf{p}} = −\dfrac{\partial \mathbf{F}}{\partial \mathbf{p}} \, . [/math]

Direct evaluation requires calculating how the model state changes with every parameter. This can be avoided by introducing an auxiliary vector [math]\boldsymbol{\lambda}[/math], called the adjoint variable, which satisfies the adjoint equation

[math]\left(\dfrac{\partial\mathbf{F}}{\partial\mathbf{x}}\right)^T\boldsymbol{\lambda}=\left(\dfrac{\partial J}{\partial\mathbf{x}}\right)^T. \qquad (A3)[/math]

The gradient can then be calculated from

[math]\dfrac{dJ}{d\mathbf{p}}=\dfrac{\partial J}{\partial\mathbf{p}}-\boldsymbol{\lambda}^T\dfrac{\partial\mathbf{F}}{\partial\mathbf{p}}. \qquad (A4)[/math]

The important advantage is that the adjoint equation needs to be solved only once for a scalar cost function, irrespective of the number of uncertain parameters[16]. After a forward model calculation and an adjoint calculation, the gradient with respect to all parameters can be obtained without separately repeating the full model simulation for each parameter.

For time-dependent models, the adjoint calculation generally proceeds backward in time and requires information from the forward simulation. Implementing and storing this information can be demanding, but the method is particularly attractive when a large number of parameters must be calibrated against a single cost function.

Appendix B. Kalman filtering as sequential Bayesian updating

The Kalman filter implements the Bayesian principle that prior information is updated when new observations become available. In a numerical model, the prior information consists of the predicted system state and its uncertainty. The observations provide additional information, also subject to uncertainty.

For a linear model, the predicted state at time step [math]k[/math] is

[math]\mathbf{x}_k^-=\mathbf{M}_k\mathbf{x}_{k-1}^+, \qquad (B1)[/math]

where [math]\mathbf{M}_k[/math] is the [math]m \times m[/math] model operator and the superscripts [math]-[/math] and [math]+[/math] denote the state before and after assimilation, respectively, and where [math]\mathbf{x}_k[/math] is the [math]m[/math]-dimensional state vector of the dependent variables at all the grid points in the considered domain at time [math]t_k[/math].

The uncertainty covariance matrix is propagated according to

[math]\mathbf{P}_k^-=\mathbf{M}_k\mathbf{P}_{k-1}^+\mathbf{M}_k^T+\mathbf{Q}_k, \qquad (B2)[/math]

where [math]\mathbf{Q}_k[/math] represents uncertainty introduced by the model dynamics.

The covariance matrix [math]\mathbf{P}_{k-1}^+[/math] describes the uncertainties and error correlations of the estimated state after the preceding assimilation step. It is obtained recursively from the previous forecast and assimilation cycle using Eqs. (B2) and (B5). An initial covariance matrix must be prescribed, based on prior knowledge of model uncertainties, historical model errors or ensemble simulations. In conventional Kalman filtering, the covariance matrix is propagated explicitly; in Ensemble Kalman filtering, it is estimated from an ensemble of model simulations.

The [math]d[/math]-dimensional observation vector [math]\mathbf{y}_k[/math] is related to the [math]m[/math]-dimensional model state vector through the [math]d \times m[/math] observation matrix [math]\mathbf{H}_k[/math]:

[math]\mathbf{y}_k=\mathbf{H}_k\mathbf{x}_k+\mathbf{e}_k,[/math]

where [math]\mathbf{e}_k[/math] represents observation errors with covariance matrix [math]\mathbf{R}_k[/math].

The difference [math]\mathbf{y}_k-\mathbf{H}_k\mathbf{x}_k^-[/math] between the observations and the predicted observations is called the innovation. According to Bayes' theorem, the probability distribution of the model state after assimilation is proportional to the product of the prior state distribution and the likelihood of the observations. For Gaussian uncertainties, this leads to the following explicit update equations for the estimated state and its covariance.

[math]\mathbf{x}_k^+=\mathbf{x}_k^-+\mathbf{K}_k(\mathbf{y}_k-\mathbf{H}_k\mathbf{x}_k^-) \, . \qquad (B3)[/math]

The matrix [math]\mathbf{K}_k[/math], called the Kalman gain, determines how strongly the observations influence the corrected state:

[math]\mathbf{K}_k=\mathbf{P}_k^-\mathbf{H}_k^T(\mathbf{H}_k\mathbf{P}_k^-\mathbf{H}_k^T+\mathbf{R}_k)^{-1}, \qquad (B4)[/math]

where [math]\mathbf{R}_k[/math] is the observation-error covariance matrix. The updated uncertainty is

[math]\mathbf{P}_k^+=(\mathbf{I}-\mathbf{K}_k\mathbf{H}_k)\mathbf{P}_k^-. \qquad (B5)[/math]

For a single state variable observed directly, the Kalman gain reduces to

[math]K=\dfrac{\sigma_M^2}{\sigma_M^2+\sigma_O^2}. \qquad (B6)[/math]

Substitution into Eq. (B3) gives precisely the uncertainty-weighted estimate of Eq. (1) in the main text. Thus, the Kalman filter extends the same principle to systems with many interacting variables and observations, while continuously updating their uncertainties.

For linear systems with Gaussian, unbiased and independent model and observation errors, these equations give the Bayesian posterior mean and covariance. The forecast and update steps are repeated whenever new observations become available. Nonlinear models require extensions or approximations, such as the Extended Kalman Filter or Ensemble Kalman Filter[17].


Related articles

Data analysis techniques for the coastal zone
Modelling coastal hydrodynamics

Further reading

Weather forecasts
De Mey-Frémaux, P., Ayoub, N., Barth, A., Brewin, R., Charria, G., Campuzano, F., Ciavatta, S., Cirano, M., Edwards, C.A., Federico, I., Gao, S., Garcia Hermosa, I., Garcia Sotillo, M., Hewitt, H., Hole, L.R., Holt, J., King, R., Kourafalou, V., Lu, Y., Mourre, B., Pascual, A., Staneva, J., Stanev, E.V., Wang, H. and Zhu, X. 2019. Model-observations synergy in the coastal ocean. Frontiers in Marine Science 6: 436. https://doi.org/10.3389/fmars.2019.00436

References

  1. ↑ Lardner, R.W., Al-Rabeh, A.H., Gunay, N. 1993. Optimal estimation of parameters for a two-dimensional model of the Arabian Gulf. Journal of Geophysical Research. 98(C10): 18,229-18,242
  2. ↑ Panchang, V.G., O'Brien, J.J. 1990. On the determination of hydraulic model parameters using the adjoint state formulation. In Davies A.M. (ed.), Modelling Marine Systems I, Chapter 2: 5-18. Boca Raton, Florida, CRC Press, Inc.
  3. ↑ Ralston, M.L. and R.I. Jennrich, 1978. Dud, a derivative-free algorithm for nonlinear least squares. Technometrics 20: 7-14
  4. ↑ Kalman R. E. 1960. A new approach to linear filtering and prediction problems. Journal of Basic Engineering 83(1): 35-45.
  5. ↑ Evensen, G. 1994. Sequential data assimilation with a non-linear quasi-geostrophic model using Monte Carlo methods to forecast error statistics, J. Geophys. Res. 99(C5): 10,143-10,162
  6. ↑ Verlaan, M., Heemink, A.W., 1997. Tidal flow forecasting using reduced-rank square root filters. Stochastic Hydrology and Hydraulics 11(5): 349-368.
  7. ↑ Gelb, A. 1974. Applied Optimal Estimation. Cambridge, Massachusetts: The MIT Press.
  8. ↑ Heemink, A.W., Kloosterhuis, H. 1990. Data assimilation for non-linear tidal models. International Journal for Numerical Methods in Fluids 11(12): 1097-1112.
  9. ↑ Heemink, A.W., Bolding, K., Verlaan, M. 1997. Storm surge forecasting using Kalman Filtering. Journal Meteorological Society Japan 75(1B): 305-318.
  10. ↑ Verlaan, M., Zijderveld, A., de Vries, H. and Kroos, J. 2005. Operational storm surge forecasting in the Netherlands: developments in the last decade. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 363(1831): 1441–1453
  11. ↑ van Leeuwen, P.J., Künsch, H.R., Nerger, L., Potthast, R. and Reich, S. 2019. Particle filters for high-dimensional geoscience applications: A review. Quarterly Journal of the Royal Meteorological Society 145(723): 2335-2365
  12. ↑ Moore, A.M., Martin, M.J., Akella, S., Arango, H.G., Balmaseda, M., Bertino, L., Ciavatta, S., Cornuelle, B., Cummings, J., Frolov, S., Lermusiaux, P., Oddo, P., Oke, P.R., Storto, A., Teruzzi, A., Vidard, A. and Weaver, A.T. 2019. Synthesis of ocean observations using data assimilation for operational, real-time and reanalysis systems: A more complete picture of the state of the ocean. Frontiers in Marine Science 6, 90
  13. ↑ Evensen, G., Vossepoel, F.C. and van Leeuwen, P.J. 2022. Data Assimilation Fundamentals. Springer, 245 p.
  14. ↑ Carrassi, A., Bocquet, M., Bertino, L. and Evensen,G. 2018. Data assimilation in the geosciences: An overview of methods, issues, and perspectives. WIREs Climate Change 9(5), e535
  15. ↑ Weerts, A. and G. Y. El Serafy, 2006. Particle Filtering and Ensemble Kalman Filtering for runoff nowcasting using conceptual rainfall runoff models, Water Resources Research 42(9), W09403, doi:10.1029/2005WR004093
  16. ↑ Chavent, G. 1980. Identification of distributed parameter systems: about the output least square method, its implementation, and identifiability. In Iserman R. (ed), Proc. 5th IFAC Symposium on Identification and System Parameter Estimation. 1: 85-97. New York: Pergamon Press.
  17. ↑ Evensen, G. 2003.The Ensemble Kalman Filter: theoretical formulation and practical implementation. Ocean Dynamics 53: 343-367


The main authors of this article are Gerritsen, Herman, Boogaard, Henk van den and Job Dronkers
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