Chapter 20 · Data assimilation in practice

Overview

Every forecast starts from a state nobody knows. The atmosphere is observed at scattered points, by instruments that disagree, at times that do not line up — and from that a forecast centre must produce a complete, physically consistent initial condition for a model with $10^8$ degrees of freedom. Data assimilation is how.

Chapter 18 and chapter 19 derive the machinery. This chapter asks what happens when you run it for ever: three schemes cycled on the same problem, a check on whether the ensemble they produce is honest, and the effect of observing less often.

The chapter then asks the question that decides observing-system budgets: if every observation were ten times more accurate, how much forecast would that buy? The answer is not “ten times more”. It is a fixed increment of lead time — and the next factor of ten buys the same fixed increment again. Predictability is purchased in units of $\ln$, which is why fifty years of enormous investment in observing systems has moved the useful forecast range by roughly a day per decade rather than transforming it.

The model

A perfect-model twin experiment on Lorenz 63: a nature run stands in for the truth, noisy observations are drawn from it every $\Delta t_{\rm obs}$, and a deliberately wrong background stands in for prior knowledge. Three schemes then compete on the same observations, held identical across all three along with $\mathbf{B}$, $\mathbf{R}$ and the observation times, so that the comparison measures the scheme and not the setup.

The schemes themselves are not derived here. 3D-Var and 4D-Var — the cost function, the adjoint gradient and how to know it is right, the Hessian as the analysis-error covariance, window length, the incremental form — are chapter 18. The EnKF — sampling error and why localisation is compulsory, inflation, deterministic versus stochastic updates, hybrids — is chapter 19. What is left for this chapter is the part those two set aside: a single analysis is a problem in linear algebra, while running one every few time units for ever against a system whose errors double on the same timescale is a different problem, and the one an operational centre actually has.

All three come from chaoslib.assimilate, which is tested against the linear-Gaussian Kalman filter for both the analysis mean and the analysis covariance, and whose 4D-Var gradient is checked against central differences of the same cost function to better than one part in $10^7$.

That gradient test was not decoration. It caught two real interval bugs in the tangent-linear propagator while this chapter was being written — a zero-length window being advanced by one time step, and a propagator covering n_steps*dt instead of the requested $\tau$ — either of which left every 4D-Var gradient about 7% wrong. The symptom was a chapter whose 4D-Var underperformed for no visible reason; nothing in the figures looked amiss.

Exercises

Analytic. Cycling reaches a steady state: the analysis error at one cycle sets the background error at the next, which the next analysis partly removes. Write that as a one-dimensional map — background error grows by $e^{\lambda \Delta t_{\rm obs}}$ over the interval, then the analysis contracts it by the Kalman gain — and find its fixed point. Then show why the fixed point exists only for $\Delta t_{\rm obs}$ below a threshold, and relate that threshold to the divergence seen in section 5.

Computational. Derive $\Delta t = \ln 10/\lambda$ from $t_c = \lambda^{-1}\ln(\delta_c/\delta_0)$, then check the measured slope in section 6 against $1/\lambda_1$ computed independently in chapter 7. They agree to a few percent by two completely different routes — one from the dynamics, one from forecast error curves.

Exploratory. Set the EnKF to $N = 10$ with inflation $1.0$ and hunt for filter divergence: the point where the ensemble becomes so narrow that the analysis stops responding to observations at all. Then find the inflation that recovers it. Is the value that minimises RMSE the same as the value that best calibrates the spread? It is not, and that tension is a real operational problem rather than an artefact of this toy.

A note on what this chapter shows, and what it does not

The three-way RMSE comparison should not be read as a general ranking. This is a three-variable, perfect-model problem in which every component is observed at every analysis time — about the friendliest possible setting for an ensemble method, and one where localisation, the EnKF’s central practical difficulty, cannot bite. The mechanisms transfer; the ordering does not.

The same caution applies to the headline number. The logarithmic law is general, but its constant is $1/\lambda$, and Lorenz 63’s $\lambda$ under the conventional 1 MTU $\approx$ 5 days reading makes it less chaotic per day than the atmosphere it stands in for. The law transfers; the constant does not — which is the argument of chapter 3.

Further reading

• Kalnay, E. (2003). Atmospheric Modeling, Data Assimilation and Predictability. Cambridge University Press — ch. 5 for the algorithms, §6.1 for predictability.
• Evensen, G. (2009). Data Assimilation: The Ensemble Kalman Filter. Springer.
• Bocquet, M. et al. (2023). A guide to ensemble Kalman methods with implementation in Python. arXiv:2305.00087.
• Bauer, P., Thorpe, A. and Brunet, G. (2015). The quiet revolution of numerical weather prediction. Nature, 525, 47–55 — the historical record of the “day per decade” improvement.
• Palmer, T. and Hagedorn, R., eds. (2006). Predictability of Weather and Climate, ch. 8 [citation needed: pages].