# Reducing Forecast Errors of a Regional Climate Model Using Adaptive Filters
## 1 Introduction  
Climate forecasts aim at providing the most probable evolution of the climate in a future time window. Numerical methods are applied to model and solve the physics laws that govern atmospheric circulation [1]. An atmospheric model is a tool to predict the evolution of climate conditions for a certain region [2; 3]. The Eta Regional Climate Model (RCM) has provided seasonal forecasts over South America since 2002 [4]. The Eta model [5; 6] has been employed by the Brazilian National Institute for Space Research (INPE) since 1996 [7] for high-resolution numerical weather forecasts. The Eta model employs as prognostic variables: air temperature, zonal and meridional wind components, specific humidity, surface pressure, turbulent kinetic energy, soil humidity and temperature, and liquid or ice water in clouds [8; 9]. Climate forecasts using any numerical method/model (such as the Eta) will generally differ from the observed climatic conditions. To reduce forecast errors, one normally carries out statistical bias correction of climate model outputs by using methods based on distribution mapping, power transformation, local intensity, and linear scaling [10; 11; 12; 13]. These methods adjust the statistics of the climate forecast such as mean, variance, and distribution matching; other methods proceed by attempting to adjust wet days probability [14]. In addition, these studies show that approaches based on quantile mapping with gamma distribution result in better performance on RCM bias correction. In [15], the authors compared the quantile mapping (QM) with gamma distribution adjustment against power transformation, considering the Eta RCM precipitation. They concluded that the former method is more appropriate for daily precipitation bias correction. In the literature, we found bias correction being performedin the frequency domain, usually by employing Discrete Fourier Transform. This framework treats both univariate [16; 17] and multivariate [18] cases. The frequency-domain bias correction comprises several timescales simultaneously, whereas the QM techniques are restricted to specific timescales as shown in [19; 20; 21]. Recently, in [22], the authors used discrete wavelet transforms to correct systematic bias in spectral attributes from raw General Circulation Model simulations results, more specifically, on global mean sea level and the Arctic sea-ice extent. This work aims to investigate the possibility of improving the quality of climate forecasts by filtering the predicted climate variables using adaptive filters. The adaptive filters are applied to the Eta seasonal forecasts. In this work, we consider the design of filters for postprocessing of the prognostic variables in order to reduce forecast errors. The filters are designed for different spatial regions using adaptive algorithms and are also inherently capable of processing different timescales simultaneously. The spatial regions are defined through a clustering algorithm. Therefore, in our proposal, temporal variability is treated by adaptive filtering and spatial variability is treated by clustering. Our work innovates by employing adaptive filtering techniques for RCM bias correction. The NCEP (National Centers for Environmental Prediction) provides a dataset aiming to be a bona fide portrait of the observed climatic conditions. This dataset is known as NCEP reanalysis, from the term NCEP/CFSR (NCEP/_Climate Forecast System Reanalysis_). It is constructed from a combination of forecasts and observed meteorological data, which are then processed and fused into a regular grid to produce the NCEP reanalysis [23]. These data can be used to verify the forecast skill of meteorological models. The filter coefficients are adjusted and adapted, comparing the Eta model forecasts against the NCEP reanalysis data. This can result in filtered prognoses having both a temporal behavior and statistical characteristics that are more similar to the observed conditions than the original forecasts.