## 1. Introduction

The assimilation of moisture for numerical weather prediction (NWP) purposes has traditionally been treated with less care than the assimilation of the ‘dynamic’ model variables of wind, temperature and pressure. There are several reasons for this:

The forecast models are in general more sensitive to errors in the initial dynamic fields than to errors in the initial humidity fields (Smagorinsky

*et al.*, 1970).The assimilation of moisture is more difficult than assimilation of the dynamic model variables. One reason for this difficulty is that most expressions of moisture do not have Gaussian probability distributions due to the condensation effects near the saturation point and the strict limit at zero humidity.

The strong nonlinear dependency of moisture on temperature via the water vapour saturation pressure with the effect that the distributions of moisture and associated forecast errors tend to be strongly heterogeneous in space and time.

The purpose of the work presented here is to rectify some of these shortcomings in the humidity analysis. Dee and da Silva (2003) discussed the choice of variables for atmospheric moisture analysis. Since most atmospheric models use specific humidity or the mixing ratio to represent the atmospheric moisture content, several modelling groups have chosen to use these model variables also for the analysis. In order to deal with the spatial variability of moisture forecast errors, Rabier *et al.* (1998) introduced empirical models for the moisture background-error standard deviations, depending on the local background temperature and relative humidity fields. Other groups have chosen to use relative humidity as the analysis moisture variable (Hólm *et al.* 2002), thereby solving some of these variability problems. However, with relative humidity as the analysis control variable in a univariate analysis scheme, relative humidity is conserved during the analysis in case there is no influence from relative humidity observations. Thus, temperature observations may in this case have a questionable effect on the atmospheric moisture content as measured by specific humidity or the total water content.

The spatial and temporal variability of moisture forecast errors originates from many processes and from the limitations of the NWP model to describe these processes. An ensemble of model background states may also help to model these uncertainties within the framework of variational data assimilation. Kucukkaraca and Fischer (2006) used ensembles for estimating flow-dependent background-error variances. More generally, ensembles may also be applied for modelling the full background-error covariance in hybrids between variational and ensemble Kalman filter (Evensen, 1994) data assimilation schemes, for example by following the approach of an augmented assimilation control vector, as suggested by Lorenc (2003).

In this study we have followed Dee and da Silva (2003) and we have utilized a pseudo-relative humidity assimilation variable, i.e. the specific humidity assimilation increment normalized by the background saturation specific humidity. With this choice, we avoid the questionable effects of temperature observations on the moisture content at the same time as we obtain improved statistical characteristics of the moisture control variable. In order to reduce the problems at saturation and at zero humidity, a transformation by an additional normalization with a background-error standard deviation that depends on the relative humidity is also utilized. This approach was already suggested by Hólm *et al.* (2002) for a relative-humidity-based analysis control variable. Another approach is to apply *a posteriori* explicit adjustments. This is done also in the HIgh-Resolution Limited-Area Model (HIRLAM) system before running the forecast model and will be used in the two formulations compared in the present paper. A statistical balance constraint where moisture is involved, following the formulation of Berre (2000), is investigated and dicussed here.

This study has been motivated by the increased importance of moist physical processes at increased spatial model resolutions and also by the increased availability of moisture-related observations from weather radars and from satellites. Until recently it has been difficult to show any positive impact at all on NWP from the use of moisture observations; e.g. Bengtsson and Hodges (2005) applied the ECMWF (European Centre for Medium-Range Weather Forecasts) forecasting system in an observing system experiment. With improved moisture assimilation algorithms and with more extensive utilization of remote-sensing data, it has now been possible to prove such a positive impact of moisture observations on NWP, also using the ECMWF forecasting system (Andersson *et al.* 2007).

Section 2 outlines the reformulation of the HIRLAM (Undén *et al.*, 2002) humidity analysis in more detail and in section 3 background-error statistics for the new moisture variable are presented and compared with the background-error statistics for the present moisture control variable. The effects of the new moisture assimilation control variable on simulated and real observation data assimilation and forecast runs are discussed in section 4, followed by a summary with conclusions in section 5.