Haonan Ren – h.ren@pgr.reading.ac.uk
Data assimilation is a mathematical method to combine forecasts with observations, in order to improve the accuracy of the original forecast. Normally data assimilation methods are performed with the perfect-model assumption (weak-constraint). However, there are different sources that can produce model error, such as missing description of the dynamic system and numerical discretisation. Therefore, in recent years, the model error has been more often considered in the data assimilation process (strong-constraint settings).
There are several data assimilation methods applied in various fields. My PhD project mainly focuses on the ensemble/Monte Carlo formulation of the Kalman Filter-based methods, more specifically, the ensemble Kalman Smoother (EnKS). Different from the filter, a smoother updates the state of the system using observations from the past, present and possibly the future. The smoother does not only improve the forecast at the observation time, instead, it updates the whole simulation period.
The main purpose of my research is to investigate the performance of the data assimilation methods with auto-correlated model error. We want to know what will happen if we propose a misspecified auto-correlation in the model error, for both state update and parameter estimation. We start our project with a very simple linear auto-regressive model. As for the auto-correlation in model error, we propose an exponential decaying decorrelation. Naturally, the system has a exponential decaying parameter ωr, and the parameter we use in the forecast and data assimilation is ωg which can be different from the real one.
A simple example can illuminate the decorrelation issue. In Figure 1, we show results of a smoothing process for a simple one-dimensional system over a time window of 20 nature time steps. We use an ensemble Kalman Smoother with two different observation densities in time. The memories in the nature model and the forecasts models do not coincide. We can see that with ωr = 0.0, when the actual model error is a white-in-time random variable, the evolution of the true state of the system behaves rather erratically with the present model settings. If we do not know the memory and assume the model error is a bias in the data assimilation process (ωg → ∞), the estimation made by the data assimilation method is not even close to the truth, even with very dense observations in the simulation period, as shown in the left two subplots in Figure 1. On the other hand, if the model error in the true model evolution behaves like a bias, and we assume that the model error is white in time in the data assimilation process, the results are quite different with different observation frequencies. As shown in two subplots on the right in Figure 1, with very frequent observations, we can see a fairly good performance of the data assimilation process, but with a single observation, the estimation is still not accurate.

In order to evaluate the performance of the EnKS, we need to compare the root-mean-square error (RMSE) with the ensemble spread of the posterior. The best performance of the EnKS is when ratio of RMSE over the spread is equal to 1.0. The results are shown in Figure 2. As we can see, the Kalman Smoother works well when ωg = ωr for all the cases, with the ratio of RMSE over the spread equal to 1.0. With relatively high observational frequency, 5 observations or more in the simulation window, the RMSE is larger than the spread when ωg > ωr, and vice versa. In a further investigation, the mismatch between the two timescales ωr and ωg barely has any impact on the RMSE. The ratio is dominated by the ensemble spread.

Then, we move to estimating the parameter encoded in the auto-correlation of the model error. We estimate the exponential decaying parameter by the state augmentation using the EnKS, and the results are shown in Figure 3. Instead of the exponential parameter, ωg, we use the log scale of the memory timescale to avoid negative memory estimates. The initial log-timescale values are drawn from a normal distribution: ln ωgi ∈ N (ln ωg, 1.0). Hence we assume that the prior distribution of the memory time scale is lognormal distributed. According to Figure 3, with an increasing number of windows we obtain better estimates. Also, the convergence is faster with more observations. And in some cases the solution did not converge to the correct value. This is not surprising given the highly nonlinear character of the parameter estimation problem, especially with only one observation per window. When we observe every time step the convergence is much faster, and the variance in the estimate decreases, as shown in the lower two subplots. In this case we always found fast convergence with different first guess and true timescale combinations, demonstrating that more observations bring us closer to the truth, and hence make the parameter estimation problem more linear.

As the results of the experiments show, the influence of an incorrect decorrelation timescale in the model error can be significant. We found that when the observation density is high, state augmentation is sufficient to obtain converging results. The new element is that online estimation is possible beyond a relatively simple bias estimate of the model error.
As a next step we will explore the influence of incorrectly specified model errors in nonlinear systems, and a more complex auto-correlation in the model error.







A true revolutionary in the field of theoretical physics and abstract algebra, Amelie Emmy Noether was a German-born inspiration thanks to her perseverance and passion for research. Instead of teaching French and English to schoolgirls, Emmy pursued the study of mathematics at the University of Erlangen. She then taught under a man’s name and without pay because she was a women. During her exploration of the mathematics behind Einstein’s general relativity alongside renowned scientists like Hilbert and Klein, she discovered the fundamentals of conserved quantities such as energy and momentum under symmetric invariance of their respective quantities: time and homogeneity of space. She built the bridge between conservation and symmetry in nature, and although Noether’s Theorem is fundamental to our understanding of nature’s conservation laws, Emmy has received undeservedly small recognition throughout the last century.
Claudine Hermann is a French physicist and Emeritus Professor at the École Polytechnique in Paris. Her work, on physics of solids (mainly on photo-emission of polarized electrons and near-field optics), led to her becoming the first female professor at this prestigious school. Aside from her work in Physics, Claudine studied and wrote about female scientists’ situation in Europe and the influence of both parents’ works on their daughter’s professional choices. Claudine wishes to give girls “other examples than the unreachable Marie Curie”. She is the founder of the Women and Sciences association and represented it at the European Commission to promote gender equality in Science and to help women accessing scientific knowledge. Claudine is also the president of the European Platform of Women Scientists which represents hundreds of associations and more than 12,000 female scientists.
For most people being handpicked to be one of three students to integrate West Virginia’s graduate schools would probably be the most notable life achievements. However for Katherine Johnson’s this was just the start of a remarkable list of accomplishments. In 1952 Johnson joined the all-black West Area Computing section at NACA (to become NASA in 1958). Acting as a computer, Johnson analysed flight test data, provided maths for engineering lectures and worked on the trajectory for America’s first human space flight.
Women however were not allowed on such ships, thus Marie Tharp was stationed in the lab, checking and plotting the data. Her drawings showed the presence of the North Atlantic Ridge, with a deep V-shaped notch that ran the length of the mountain range, indicating the presence of a rift valley, where magma emerges to form new crust. At this time the theory of plate tectonics was seen as ridiculous. Her supervisor initially dismissed her results as ‘girl talk’ and forced her to redo them. The same results were found. Her work led to the acceptance of the theory of plate tectonics and continental drift.
Ada Lovelace was a 19th century Mathematician popularly referred to as the “first computer programmer”. She was the translator of “Sketch of the Analytical Engine, with Notes from the Translator”, (said “notes” tripling the length of the document and comprising its most striking insights) one of the documents critical to the development of modern computer programming. She was one of the few people to understand and even fewer who were able to develop for the machine. That she had such incredible insight into a machine which didn’t even exist yet, but which would go on to become so ubiquitous is amazing!
As a student, being an RMetS member can lead to conversations that could develop your career and bring unexpected opportunities. This has been greatly enhanced with the RMetS mentoring scheme.
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