How do observations change the predictions of a numerical model? This hands-on experiment uses a simple chaotic system to show how data assimilation updates a forecast. No background in meteorology or mathematics is required.
What Is Data Assimilation?
Weather forecasts use numerical models based on physical laws to predict how the atmosphere will evolve. However, the estimated state at the start of a forecast (the initial conditions) contains errors, and the forecast may diverge from what actually happens.
Data assimilation combines model forecasts with observations to estimate better initial conditions. Because observations are uncertain too, assimilation does not simply force a forecast to pass through every observed point. It accounts for the uncertainties in both sources of information. Data assimilation is an essential part of modern weather prediction.
The Lorenz-63 Model and Chaos
The Lorenz-63 model is a mathematical system with three variables, x, y, and z, derived from a simplified representation of fluid convection. It does not directly predict real-world air temperature or wind, but it illustrates important properties of atmospheric predictability.
Chaos means that even when a system follows deterministic equations, tiny differences in initial conditions can grow into major differences over time. Lorenz-63 trajectories trace a butterfly-shaped attractor with two lobes. The trajectory switches irregularly between the lobes, and a small change in initial conditions can change which lobe it visits next.
Here, the initial forecast and the reference “truth” start from different conditions. We use synthetic observations to update the forecast initial conditions. Watch how the trajectory changes, especially the timing of transitions between the two lobes.
Before You Start: How to Read the Plots
The simulation uses line colors, line styles, points, and shading to distinguish the different quantities.
Line and symbol examples
About the initial view: Before assimilation starts, only the observation markers are visible. The forecast and shaded deviations appear progressively after you press “Start Assimilation.” Current and previous analysis trajectories appear after the first optimization update. The black truth curve remains hidden until you select “Show Truth.” The shaded deviations can still appear while the truth line is hidden.
Try Data Assimilation
Press “Start Assimilation” in the simulation below to see the forecast evolve and observe how the model initial conditions are adjusted using observations.
How to Use the Simulation
- Start assimilation: Watch the initial forecast evolve forward in time, followed by a backward replay of the adjoint calculation used to obtain the information needed for optimization.
- Next iteration: Update the initial conditions once to reduce the cost function. Compare the teal solid and dashed trajectories to see what changed from the previous iteration.
- Show truth: Display the black reference trajectory and compare it with the forecast and analyses. Pay attention to the translucent shading between the curves.
- Cost function: Open this tab to see how the objective changes with each iteration and to compare the root-mean-square errors (RMSE) relative to the truth and observations.
- Change the settings: Adjust the initial conditions or observation errors, or click a time-series plot to add an observation. The new measurement is generated at the clicked time from the truth plus random observational error.
What to look for: Even when an analysis fits the observations better, it may not be closer to the truth at every time. Try changing the number or timing of observations, their uncertainty, and the initial conditions.
Key Terms
- Truth: The reference trajectory designated as the correct state in this synthetic experiment. The exact true state of the real atmosphere is not generally known.
- Forecast: A trajectory produced by the numerical model. Here, it includes the initial forecast computed before assimilation.
- Analysis: An estimate obtained after incorporating observations. In this experiment, the analysis trajectory is integrated from the updated initial conditions.
- Observation: A measurement of a system’s state. Here, observations are synthetic values produced by adding Gaussian errors to the truth.
- Four-dimensional variational assimilation (4D-Var): A method that adjusts the initial conditions to minimize a cost function using observations over a time window.
- Cost function: A weighted measure that accounts for both disagreement with observations and departure from the original initial estimate. Reducing it does not guarantee a smaller error relative to the truth.
- Iteration: One update of the initial conditions during the optimization process.
- Adjoint variable: A quantity used to compute how changes in the initial conditions affect the cost function.
About This Experiment
This educational simulation applies strong-constraint four-dimensional variational data assimilation (4D-Var) to the Lorenz-63 system. By default, it uses synthetic observations of all three variables x, y, and z at t = 1 and t = 2. The observation values include Gaussian noise.
The model, initial conditions, and observation setup are chosen for learning purposes; this is not a real atmospheric forecast. The references below provide scientific background, but the simulation is not a quantitative reproduction of a published figure.
References and Further Reading
- Lorenz, E. N. (1963): Deterministic Nonperiodic Flow. Journal of the Atmospheric Sciences, 20, 130–141. Read the paper.
- Evensen, G., F. C. Vossepoel, and P. J. van Leeuwen (2022): 3Dvar and SC-4DVar for the Lorenz 63 Model. In Data Assimilation Fundamentals, Springer. Read the open-access chapter.
- European Centre for Medium-Range Weather Forecasts (ECMWF): Data assimilation. ECMWF overview.
This simulation is designed to illustrate the basic ideas of data assimilation. Operational weather prediction uses many more variables and observations, together with much more detailed physical models.