By: Dr. Yumeng Chen
“The world is not perfect. Every measurement should come with an error bar.” This is what I learned before I stepped into the fluid dynamics lab as a student many years ago. This statement still echoes now when I work on data assimilation (DA). Because neither observations nor model forecasts are perfect, based on their errors/uncertainties, DA combines both observations and forecasts to provide an estimate of the most likely state of the modelled system. This estimate also comes with an error bar that should give reduced uncertainties compared to both the model forecast and observations.
Arctic sea ice, as an important component of the climate system, regulates solar radiation, provides habitants for marine life, and influences human activities. Like a variety of fields such as numerical weather prediction, marine ecosystems, and land surface modelling, the Arctic sea ice community adopts the DA technique operationally to provide an estimate of the state of the Arctic sea ice for better prediction. The seasonal forecast of the Arctic sea ice is improved by better initialisation of the sea ice concentration (percentage of the Arctic sea ice in each grid cell) and sea ice thickness (Kimmritz et al., 2019). However, initial conditions are not the only source of uncertainty of the Arctic sea ice. Numerical models can also suffer from erroneous parameters. These errors can cause biases in Arctic sea ice prediction especially in long-term simulations.
Fortunately, DA can also provide estimates of unobserved model fields and model parameters. How can DA estimate something that we do not observe? This is based on the relationship between the errors of parameters and observed model forecasts. This means that if we know the error of the model forecast from observations, using this relationship, we can infer and reduce the error in the model parameters. In most operational DA methods, the relationship between parameter and forecast errors is represented by error covariances which are derived from the numerical models. Thus, the performance of parameter estimation using DA depends heavily on the dynamics of the numerical models.
This dependence on model dynamics can cause problems for the parameter estimation. For example, DA could provide wrong estimates when multiple sets of model parameters lead to the same model forecast – if you know, this implies an ill-posed problem. Also, parameter estimation may fail when the parameters are not so sensitive to the observed model fields compared to other sources of uncertainties. To demonstrate the potential issue of parameter estimation when forecast errors of observed model fields do not dominantly come from parameter errors, we can apply DA to estimate parameters in a novel ‘dynamics-only’ sea ice model developed in the scale-aware sea ice project (Chen et al., 2023).
We set up an idealised experiment where a block of sea ice is forced by periodic random storms (Figure 1a). In this idealised setup, no thermodynamics exists, and the wind is the only external forcing. In this dynamics-only sea ice model, depending on the state of the sea ice, the sea ice can deform elastically like a spring, deform permanently like viscous fluid, and break abruptly under its internal dynamics or external forcing which cause damage and fracture of the sea ice. The initial state of the sea ice is not damaged and the main errors of the experiment setup come from the wind of the random storms.
Figure 1: a) An illustration of the experiment setup where the quivers show the wind field, and the sea ice is thicker in the middle of the domain than the boundaries with decreasing the sea ice concentration due to the constant wind forcing; time series of b) air drag coefficient estimation using sea ice velocity observations, c) damage parameter estimation using sea ice velocity observations, and d) damage parameter estimation using sea ice concentration observations.
In such an idealised setup, we can decide the true model state and parameters and assign errors to our chosen parameters. One important model parameter of the sea ice model is the air drag coefficient. This coefficient decides how the wind influences the sea ice velocity. The error in the wind field can be magnified or reduced by this coefficient in the sea ice velocity.
Let us assume that the air drag coefficient is erroneous and all other model parameters are correct. In this case, we can get very accurate estimates of the air drag coefficient using DA when we use sea ice velocity observations (Figure 1b). Another important model parameter of the sea ice model is called damage parameter. This parameter determines the response of sea ice motion to the forces exerted on them when the sea ice is damaged. With low value of the damage parameter, the sea ice can behave like an elastic spring; with high value of the damage parameter, the sea ice is more sluggish like viscous fluid in response to the forces. However, sea ice velocity observations cannot provide a reliable estimation of the damage parameter (Figure 1c) when the damage parameter is the only erroneous model parameter. As sea ice velocity is mostly influenced by the wind field, the sea ice velocity cannot be used to infer the damage parameter reliably. Instead, we see improved parameter estimation with a combination of sea ice concentration (Figure 1d).
Here, our example shows that parameter estimation using DA could be challenging and must be performed carefully. However, DA is still a powerful tool for improving errors in models based on available observations.
References and Further Reading
Kimmritz, M., Counillon, F., Smedsrud, L. H., Bethke, I., Keenlyside, N., Ogawa, F., & Wang, Y. (2019). Impact of ocean and sea ice initialisation on seasonal prediction skill in the Arctic. Journal of Advances in Modeling Earth Systems, 11, 4147–4166. https://doi.org/10.1029/2019MS001825
Chen, Y., Smith, P., Carrassi, A., Pasmans, I., Bertino, L., Bocquet, M., Finn, T. S., Rampal, P., and Dansereau, V.: Multivariate state and parameter estimation with data assimilation on sea-ice models using a Maxwell-Elasto-Brittle rheology, EGUsphere [preprint], https://doi.org/10.5194/egusphere-2023-1809, 2023.