*Uncertainty* is an unavoidable element in numerical weather prediction (NWP). It makes its appearance, however, even before engaging on the task of predicting (i.e. determining the future state of the atmosphere): it arises from the very attempt to determine the present state of it.

At any given moment, the atmospheric variables (wind, pressure, density, humidity) take a given value in each location of the atmosphere; let us call these values the *truth*. We have ways to try to estimate this truth, but these ways are imperfect. Perhaps the first one that comes to mind is the use of *observations*. Say we take a thermometer and measure the temperature at a given location. This *direct* measurement is not error-free: instruments have calibration errors, users make mistakes, etc. Moreover, taking direct measurements is often difficult, so for some areas (think of the Southern Ocean for example) we rely on *remote* sensing. Typical examples are satellites, which observe atmospheric variables from space. This is not a direct observation, however, and as intuition may suggest, the errors satellites make are way more complicated than those made by a thermometer. Figure 1 is an schematic of the different components in the Earth-observing system.

**Figure 1.** Components of the Earth-observing system (source: World Meteorological Organization).

The second way to estimate the true state of the atmosphere is through *models*. These are mathematical representations of the atmosphere based upon physical laws drawn from fluid dynamics and thermodynamics, amongst other fields. The atmosphere is a myriad of processes at different scales (in both space and time). There are some processes that we know better than others: while we are certain about some mechanisms, there are some which are poorly known and still an area of intense research, and their non-optimal representation introduces error in the models. Furthermore, the equations that result from these models cannot be solved analytically; instead they are solved via numerical methods in a massive computer. To do this, the atmosphere has to be discretised into a three-dimensional grid (see Figure 2). We can see it as a collection of boxes spanning the whole atmosphere. As one may imagine, the smaller the boxes the more precise the solution is. If the boxes are too big there may be processes that cannot even be represented explicitly! Of course, the smaller the boxes are – i.e. the highest resolution of the model – the more computational resources are needed (and this need *does not* grow linearly).

* Figure 2. Equations in the atmosphere are solved numerically using discretisations of the atmosphere, both in space and tim*e.

So determining the most likely current state of the atmosphere is not a trivial task. One needs precise models, accurate observations with a decent coverage, and fast supercomputers. But on top of this, producing this so-called *analysis* (that is what we call the *best estimate* of the truth) is a careful blending of models and observations. This blending uses statistical techniques to consider the relative precision of both. Furthermore, these techniques allow to use observations to inform (or sometimes correct) the values of unobserved variables. This is done by taking into consideration statistical correlations amongst variables. This is one of the main problems studied in Data Assimilation. In the University of Reading, we have the largest academic group in the world dedicated to this problem, the Data Assimilation Research Centre (DARC).