# Reducing Forecast Errors of a Regional Climate Model Using Adaptive Filters
## 2 Seasonal Climate Forecasts  
Weather and climate forecasts differ basically on the forecast period, and the integration length employed in the numeric solution. Weather forecasts extend for about ten (10) days. In contrast, climate forecasts encompass the analysis of a longer time scale (months or years) of meteorological variables that are treated statistically. Both forecasts are interested in temperature, geopotential height, wind, specific humidity, surface pressure, precipitation, clouds, among other variables. The climate forecasts considered in this work are within the region inside the dashed rectangle in Figure 1. This region is located within the latitudes 6\({}^{\circ}\) S and 30\({}^{\circ}\) S and the longitudes 33\({}^{\circ}\) W and 83\({}^{\circ}\) W, covering most of Brazil. The forecast dataset is produced by the Eta climate model using a spatial resolution of 40 km. Regional climate forecast requires initial conditions in the model domain and lateral boundary conditions along its contour. The initial conditions in the region must describe the observed atmospheric state so that they may be a reliable scenario for the atmospheric variables to evolve. Meanwhile, the contour conditions (which are necessary for regional climate models) describe the state of the atmosphere in the region's border from a coarser model dataset. These conditions drive the regional model domain, which in turn provides a smaller scale climate forecasts that is dynamically downscaling [2,3]. The Eta Regional Climate Model outputs used in this study are derived from ten-year seasonal reforecasts [28], which have been shown to add value over the driver coarse global model forecasts, especially during the rainy seasons. The evaluation was based on the temporal correlation between forecasts and observations of precipitation seasonal anomaly. In addition, the regional forecasts reproduce the average upper and lower level winds in different seasons of the year. However, regional climate forecasts are not completely accurate and differ from the observed climate. The differences have several origins and tend to increase as the integration time (the forecast time range) advances. There are approximations such as those inherent to the physics of the model or the initial conditions errors (inside the region and along the borders). All these aspects contribute to produce climate forecast inaccuracies, i.e., errors. While the initial conditions may affect short- and medium-term prognoses (days to weeks), the lateral boundary conditions along the borders affect the long-term (months and years) prognosis of regional climate [3,29]. For example, the prognostics from Eta-40-km are used as contour conditions for even higher resolution models such as the Eta-15-km, affecting numerical simulation results. Figure 1: Limits of the region considered for adjusting the Eta-40-km climate forecast using as reference the NCEP reanalysis. Eta outputs prognoses with a six-hour time interval. They may be envisioned as 3-dimension data that evolve with time since each prognostic variable is spatially sampled using latitude, longitude, and altitude coordinates. Therefore, a temporal sequence of the predicted values of a given climatic variable in a region of interest can be understood as a discrete volumetric signal. There is a 3D matrix (a tensor) for each climate variable that evolves depending on the time index \(n\). Let \(\mathbf{X}(n)\) denote this tensor. It contains the value of the variable in each volumetric cell at time \(n=1,2,\ldots,L\)--\(L\) being the prognosis interval. The elements in the tensor \(\mathbf{X}(n)\) are indexed by their spatial positions; consequently, the i-th element in \(\mathbf{X}(n)\) is the value of the variable \(x\) at the coordinates \((lat_{i},long_{i},alt_{i})\) at time \(n\). Consequently, \(x_{i}(n)\) represents the time-series (signal) containing the values of the prognosis for the variable \(x\) at the \(i\)-th cell over time.