## Numerical Weather Prediction (NWP) Basics & Glossary

Air is a fluid that obeys the laws of fluid dynamics and thermodynamics. The governing equations for these laws are coupled nonlinear partial differential equations that cannot be solved analytically. Instead, the equations are approximated by a discrete form, which converts them into algebra. One approach to achieve this applies the equations at discrete locations called grid points or grid cells within some 3-D domain. The domain can be a limited area (such as over N. America), or can be global. The horizontal grid size is ∆x, which can vary depending on location.

The particular choice of algebraic approximations, along with the grid set up, are called the dynamics and numerics. The computer code that solves these equations is called a model. To make a forecast, the model starts with initial conditions (ICs, provided by observations or from a larger-domain forecast model), and steps forward in time. At each time step, the weather at any one grid cell is affected by the weather at all surrounding grid cells. So all grid cells must step forward into the future in lockstep (by amount ∆t) until they reach the desired forecast horizon. (Some portions of the model take many “baby” time steps for each parent time step, but they still progress together.) If the forecast is for a limited area domain, then both initial and lateral-boundary conditions (IBCs) must be provided from outside information.

Many atmospheric processes are too numerous, individually too small, or not well enough understood to describe exactly. Examples include individual snowflakes, raindrops, cloud drops, turbulent eddies, bands of radiation, leaves on trees, layers of soil, etc. Instead, the cumulative (resolvable-scale) effects of these subgrid processes are approximated or parameterized, and are collectively known as the model physics. Each combination of dynamics, numerics, and physics can be a different Numerical Weather Prediction (NWP) model. Different models are often better than others at different locations, for different weather variables (wind, temperature, precipitation, etc.) and in different seasons. Smaller values of ∆x and ∆t generally give better weather forecasts (except for mountainous terrain), but take longer to run on the computer.

Often compromises are made in the numerics and physics to make a timely forecast; namely, to finish the forecast before the actual weather happens. Another method to achieve timeliness is to use nested grids, where a time-consuming finer-resolution small-domain is nested inside a less-computationally-expensive coarser-resolution larger domain. The large domain captures the larger-scale weather systems, and this info is fed (often with two-way coupling between the grids) during each time step to the finer grid that can better resolve mountains, valleys, coastlines, etc. over the region of interest.

No forecast is perfect. To make matters worse, the nonlinear equations describing the atmosphere are sensitively dependent on initial conditions, and on the model parameters, as was famously shown by Ed Lorenz in his 1963 papers on nonlinear dynamics and chaos. Sadly, since our initial conditions are never perfect, and since the models contain imperfect approximations and parameterizations, we are guaranteed that the forecast will generally deviate further and further from reality as the model steps forward in time. Thus, forecast skill (accuracy relative to some baseline reference) gets worst for longer forecast horizons.

Different NWP models get worse is different ways. So instead of running just one model on the computer each day, we can run an ensemble of many models each day. Each gives a slightly different forecast, but taken together, this is a way to partially compensate for the random errors associated with non-linear dynamics and chaos. But by averaging all of the ensemble members together, the resulting ensemble average forecast is usually the best deterministic forecast. Also, the spread between ensemble members gives a measure of uncertainty.

Ensemble prediction systems have been run by large government NWP centers for decades (Buizza et al. 2005). At UBC, we have been making daily real-time ensemble forecasts since 1996. The ensembles can be created with multiple models (dynamical cores), multiple ICs, multiple realizations of the physics, multiple compiler options, etc. Usually greater numbers of ensemble members give better ensemble forecasts (Candille 2009, Satterfield et al. 2016).

Even the ensemble forecast is not perfect. Systematic errors can be reduced after the fact via statistical post-processing. For example, if the forecast was 2°C too cold during the past week at some location, then add a 2°C bias correction to the raw NWP forecast for that location. Various methods are used to estimate and correct the bias, including linear regression, Kalman-filtering, and various forms of machine learning including artificial neural networks. Normally, each ensemble member is individually bias-corrected before they are combined into the ensemble.

Even with all these corrections, the forecast is not perfect. Post-processing can also include calculation of the error. The forecasted values are compared to values from some reference such as weather-station observations; satellite and radar observations; or an analysis (a merger of observations with a previous NWP forecast) or re-analysis. The results are verification statistics. Many different verification metrics exist – we focus on the metrics preferred by our clients. Finally, additional post-processing includes publishing the forecasts on the internet for our clients, producing weather maps and meteograms (plots of a weather variable vs. time), and archiving the forecasts into databases.