Use of a nonlinear pseudo-relative humidity variable in a multivariate formulation of moisture analysis

Authors


Abstract

We present a reformulation of the humidity part of the HIRLAM (HIgh-Resolution Limited-Area Model) variational data assimilation. The purpose is to rectify some of the shortcomings of the present formulation which uses specific humidity, q, as an assimilation control variable with homogeneous and static covariances. One problem is that specific humidity forecast errors tend to have a non-Gaussian probability distribution, in particular near saturation and near zero humidity. In addition, the variance of the distribution tends to change in space and time due to the dependency of the water vapour saturation pressure on temperature. A modified pseudo-relative humidity variable has been adapted to the statistical balance background constraint, including the associated moisture balance formulation. Background-error statistics for the new moisture control variable and the moisture-related balances were derived, taking differences between forecasts valid at the same time as a proxy for background forecast errors. The background-error statistics were compared with the corresponding statistics for specific humidity as the moisture assimilation control variable. In connection with the nonlinearity of the change of the variable, it was noted that specified background-error standard deviations were chosen to be substantially reduced for nearly dry and saturated states, which can raise difficulties. The impact of the new moisture assimilation control variable is illustrated with simulated observation experiments as well as data assimilation experiments using real observations, for one summer month and one winter month in a 4D-Var assimilation cycle using two outer loop iterations in the 4D-Var minimization. The impact of the new formulation on forecast verification scores is small and essentially neutral, while using the second outer loop in the old formulation has a small positive impact. Copyright © 2011 Royal Meteorological Society

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.

2. A new formulation of the humidity analysis

2.1. A multivariate moisture balance formulation

The background-error formulation for the HIRLAM variational data assimilation (Gustafsson et al., 2001; Lindskog et al., 2001) is based on a multivariate statistically determined balance constraint (Berre, 2000) with specific humidity as the assimilation control variable for moisture:

equation image(1)

where a, b and c are scale-dependent 3D linear regression matrices.

The total assimilation increment of moisture δqtotal consists of two parts, one balanced part that is obtained through regression from the other assimilation control variables and one unbalanced part δqub used as the assimilation control variable. Both parts are expressed in terms of specific humidity. The predictors used to derive the balanced specific humidity increment are the vorticity increment, via a balanced linearized geopotential increment δPb, an unbalanced divergence increment δDub, an unbalanced temperature increment δTub, and an unbalanced surface pressure increment (δlnps)ub. One strength of this formulation is the coupling between the flow field increment and the moisture increment, important for example in low-level convergence areas with moistening due to the convergence.

Following Dee and Da Silva (2003), we have now introduced the pseudo-relative humidity increment (δRH) as an alternative moisture control variable:

equation image(2)

where qs(Tb) is the saturation specific humidity with respect to the background temperature Tb.

With pseudo-relative humidity as the assimilation control variable, a similar multivariate moisture statistical balance relation was applied:

equation image(3)

where δRHtotal is the total pseudo-relative humidity increment and δRHub is the unbalanced part of the pseudo-relative humidity increment.

The statistical balance relations expressed in Eqs (1) and (3) were obtained via separate regression derivations which are illustrated and discussed in the next section.

2.2. A renormalized pseudo-relative humidity control variable

Starting with the moisture balance relation, Eq. (3), we may select the unbalanced specific humidity increment, normalized by the corresponding background saturation specific humidity, as the humidity assimilation control variable. This control variable may also be referred to as an unbalanced pseudo-relative humidity control variable, which should have statistical properties similar to those of a relative humidity control variable. We could also expect the probability density function (PDF) of pseudo-relative humidity forecast errors to deviate from the Gaussian one due to

  • (i)the possible dependency of the error standard deviations on the forecast (background) relative humidity, and
  • (ii)the asymmetries that occur close to saturation and close to zero humidity.

In order to obtain a moisture control variable with a more Gaussian behaviour, we followed Hólm et al. (2002), by additionally normalizing the unbalanced pseudo-relative humidity background-error by a standard deviation that is a function of the background relative humidity increased by half of its background-error δRH, thus

equation image(4)

Note that this normalization of the background-error by these standard deviations is nonlinear, because these standard deviations vary with the background-error value δRH. This implies in particular that, when implementing this background-error model in the analysis, the analysis problem becomes nonlinear. In order to keep quadratic minimization over all inner loop minimization iterations, we carried out a relinearization on the level of outer loop minimization iterations. In terms of implementation in the analysis scheme, the control variable transforms are applied to analysis increment variables. As the analysis problem becomes nonlinear, this raises a specific issue in terms of error variance specification. The choice which was made in the present study was to use the nonlinear form σ(RHb + 0.5δRH) with δRH = 0 for the normalization in the first outer loop iteration, and with δRH = RHfgRHb in further outer loop iterations. RHfg denotes the first-guess relative humidity, i.e. the result of the previous outer loop iteration. As will be further discussed in section 3, this choice implies that specified background-error standard deviations are chosen so that they are substantially reduced for nearly dry and saturated states, which can raise a risk that observed information tends to be neglected for these states. The realism of this choice and of the associated change of standard deviations may thus deserve further study. In the present study, we have restricted ourselves to illustrate the effect of outer loop minimization iterations on assimilation increments in cases with nearly saturated model background states (see below).

The statistical properties of the new moisture control variable are illustrated and discussed in the next section.

3. Estimation of background-error covariances

Before applying the new moisture assimilation control variable it was necessary to derive the corresponding background-error statistics. The NMC method (Parrish and Derber, 1992) was applied for this purpose, taking differences between +36 h and +12 h forecasts as a proxy for +6 h forecast errors. Forecast data were taken from the operational HIRLAM forecast runs at the Swedish Meteorological and Hydrological Institute (SMHI), based on the reference HIRLAM forecast model and 6-hourly four-dimensional variational data assimilation (4D-Var) cycles with a 6 h assimilation window. The forecast domain covered Europe and the northern North Atlantic with a grid resolution of 22 km and with 40 vertical levels. Background-error statistics were derived for one period, May–August 2008, representing the summer season, and for another period, November 2008–February 2009, representing the winter season. We have concentrated on the summer season results and we show statistics for the winter season only when significant differences between the two seasons were found. Background-error statistics were derived, including cross-covariances between different assimilation control variables used to construct balance operators as well as spatial covariances of the assimilation control variables. Here, we show only background-error statistics for moisture, together with a comparison between the statistics for the old and the new moisture assimilation control variables. In particular, we give the statistics for model level 20 (≈550 hPa) and for model level 30 (≈850 hPa).

The frequency distributions for pseudo-relative humidity forecast differences at model level 30, separated into classes depending on the background relative humidity only, are illustrated for the summer period in Figure 1(a). It can be noticed that the widths of the frequency distributions differ depending on the background relative humidity. Further, asymmetries exist where the background relative humidity is near zero and near saturation. It is also important to note that the negative tail of the PDF at saturation and the positive tail of the PDF at zero background relative humidity are quite similar to the corresponding tails at 50% background relative humidity. Actually, this result could be a motivation for treating q as a Gaussian variable together with an explicit removal of negative and supersaturating humidities, as done in many operational NWP centres. However, another approach has been considered in the present article. The separation into classes depending on the background relative humidity increased by half of the difference improves the symmetry of the frequency distributions (Figure 1(b)). Note that this is done at the expense of changes in the ‘physical side’ of the tails, which has implications on specified standard deviations, as discussed later.

Figure 1.

Frequency distributions of forecast differences for pseudo-relative humidity at model level 30 for the summer season and at three values (0.025, 0.525 and 0.975): (a) for the ‘background relative humidity’ (one of the forecasts forming the difference), and (b) for the ‘background relative humidity increased by half of the difference’ (the average of the two forecasts forming the difference).

The standard deviations of pseudo-relative humidity forecast differences as a function of ξ = RHb + 0.5δRH were approximated by a fourth-degree polynomial Pk(ξ) over 20 relative humidity bins of size 0.05. Vertical averaging was also applied at each level in order to reduce the effect of sampling errors (the averaging was carried out over ± 8 vertical levels). Figure 2 shows results for the summer season and model levels 20 and 30. The original standard deviations of pseudo-relative humidity forecast differences as functions of the background relative humidity RHb have been included in the figures for comparison purposes. We may refer to these as the linear background-error standard deviations, and these are roughly constant with respect to the background relative humidity. This was no longer the case with the nonlinear standard deviations being functions of ξ = RHb + 0.5δRH. The decreasing background-error standard deviations toward saturation and zero background relative humidities are clearly discernible. For instance, at level 30 the standard deviation near 0% is divided by a factor of 6.5 (it is equal to 2.5 in the nonlinear case, instead of 16% in the linear case). When applied in data assimilation, this has the effect of reduced increments for such background relative humidities and the degree of realism of this aspect may deserve further investigation. The decreased background-error standard deviations at saturation and at zero background relative humidity is the ‘price we pay’ in order to get a more symmetrical probability distribution and this may be a weakness in our approach. In order to avoid too small total increments, for example in a case where the background field is saturated while observations indicate a drier atmosphere, relinearizations of the variable transform in an outer minimization loop are needed (see above), since during the first outer loop minimization iteration we do not know the assimilation increment. There is certainly a risk with our approach of ‘getting stuck’ in the near-saturation or near-zero relative humidity regimes during data assimilation.

Figure 2.

Standard deviations σb of pseudo-relative humidity forecast differences as functions of RHb (symbol ×) and of RHb + 0.5δRH (symbol + and full lines) for (a) level 20 and (b) level 30 for the summer season. Symbols + indicate empirical (raw) standard deviation values in bins of width 0.05 and full lines denote the fitted polynomials.

Another visible change in the nonlinear standard deviation profile is the increase of standard deviations for relative humidity values between 40% and 80%. For instance, for a relative humidity of 65% at level 20, the standard deviation is increased by a factor of 1.4 (it is nearly equal to 33 and 23% in the nonlinear and linear profiles, respectively). This increase in standard deviations is likely to correspond, for instance, to nearly dry background states affected by large errors (which are supposed to be known in the nonlinear profile). The realism of this standard deviation increase may be questioned similarly with respect to the first minimization in particular, since the specified standard deviation can be seen as artificially increased for these background states, without knowing what the actual background error value is. In other words, it should be noted that, in terms of implementation in the minimization, while background-error standard deviations will be underestimated for nearly dry and saturated states, they will be overestimated for intermediate states.

It should be pointed out that the standard deviations of pseudo-relative humidity presented in Figure 2 are estimated for the total pseudo-relative humidity forecast differences, while the assimilation control variable is an ‘unbalanced’ pseudo-relative humidity increment. We have made the choice to use the standard deviation polynomials based on the total pseudo-relative humidity forecast differences. The inconsistency is relatively small in summer, since the fraction of total pseudo-relative humidity variance explained by the balanced part on average is on the order of 33% in the troposphere. This approximation implies that, in the implementation of the RH formulation, specified standard deviations of humidity have been inflated implicitly by a factor of around 1.25 in the troposphere (relative to the formulation based on q), and the influence of this change may merit further investigation in the future.

The standard deviations of pseudo-relative humidity forecast differences have a dependency on relative humidity for the winter season (not shown) similar to that for the summer season, although these standard deviations are systematically larger for the winter season than for the summer season. The larger standard deviations during winter indicate that factors other than the local temperature and relative humidity influence the moisture background-error standard deviations, and these factors need to be taken into account by, for example, modelling based on ensembles.

The balance operators of the HIRLAM statistical balance background-error constraint act in spectral space and with participation of all vertical levels for a particular wave number. This is made possible through the assumption of horizontal homogeneity with respect to spatial correlations. A nice effect is that balance relations like geostrophy become scale-dependent. As we will show, this is also important for (statistical) balance relations where moisture is involved.

Vertical cross-covariances between the moisture control variable and two other assimilation control variables, the balanced geopotential derived from vorticity and the unbalanced temperature, are presented in Figures 34 for the summer season as averages over all horizontal scales. To simplify the comparison between the cross-covariances, those involving the new moisture control variable have been rescaled at each level i by equation image (≈ qs(Tb) according to Eq. (2)) to have the same physical units as the cross-covariances involving the old moisture control variable. The corresponding fractions of humidity variance, explained by the other assimilation control variables, are presented as functions of vertical level in Figure 5.

Figure 3.

Vertical cross-covariances (a) between specific humidity and vorticity-balanced geopotential and (b) between scaled pseudo-relative humidity and vorticity-balanced geopotential, both for the summer season and averaged for all horizontal scales. The scaling factor for pseudo-relative humidity, the standard deviation for specific humidity divided by the standard deviation for pseudo-relative humidity, is selected such that the covariances in both panel will have the same physical units (10−3kg kg−1J kg−1).

Figure 4.

As Figure 3, but (a) between specific humidity and unbalanced temperature and (b) between scaled pseudo-relative humidity and unbalanced temperature. The covariances in both panels have the same physical units (10−3 kg kg−1 K).

Figure 5.

Spectral averages of the percentage of explained humidity variance as a function of height for the summer season for (a) the old moisture variable and (b) the new moisture variable: total (full line), balanced geopotential (dashed line), unbalanced divergence (symbol +) and unbalanced temperature and surface pressure (symbol ×).

The main signal from the cross-covariances between humidity and vorticity-balanced geopotential is negative at low levels and positive at upper levels (Figure 3) for both the old and the new moisture variables. Note that the main difference between the old and the new moisture variables is the stronger penetration to higher vertical levels of the negative covariances between the vorticity-balanced geopotential and moisture for the new moisture control variable and that this also corresponds to a higher percentage of the variance of humidity explained by the balanced geopotential (Figure 5(b)) than implied by the old variable (Figure 5(a)) at these higher levels. The covariance patterns in Figure 3 may be interpreted as a coupling between vorticity (and surface pressure via vorticity) forecast errors in cyclone developments and low-level moistening due to frictional inflow. Additionally, the cross-covariance between specific humidity and unbalanced divergence (not shown) indicates a link between a low-level moistening and convergence at lower levels, while moistening at upper tropospheric levels is associated with divergence at upper levels.

Significant values of cross-covariance of humidity with unbalanced temperature are mainly negative for both the old (Figure 4(a)) and the new (Figure 4(b)) moisture variables. For both humidity variables, the link is strongest along the diagonal of the vertical covariance matrix. This indicates that a more local process is responsible for this link and it is natural to interpret this as due to condensation coupled with latent heating. The smaller amount of moisture variance explained by unbalanced temperature, when using pseudo-relative humidity instead of specific humidity, may possibly be explained by the latitude weighting in the calculation of the horizontally averaged covariances. Using pseudo-relative humidity gives larger weights to higher latitudes, while the coupling between moisture and unbalanced temperature may be stronger at lower latitudes.

The most significant differences between the winter and summer seasons occur for the coupling between humidity and unbalanced temperature. This can most easily be recognized by comparing the spectral averages of fractions of explained moisture variance (Figure 5 for summer and Figure 6 for winter). For the winter season, there is a much stronger coupling (positive correlation) between the unbalanced temperature and the pseudo-relative humidity in the planetary boundary layer than in the summer season, while during the summer season the most significant coupling between pseudo-relative humidity and unbalanced temperature occurs at higher vertical levels. This could possibly be interpreted as meaning that latent heating due to convection is dominant during the warm season, while boundary-layer processes are more important during the cold season. Since the winter season coupling between unbalanced temperature and pseudo-relative humidity increases all the way down to the lowest model level, it is likely that this coupling is linked to evaporation from both land and sea surfaces. One reason may be that the potential evaporation over land and the evaporation over sea is partly a function of the water vapour saturation pressure, which has a strong nonlinear dependency on temperature. This dependency is directly taken care of when the pseudo-relative humidity variable is applied as the assimilation control variable. Another possibly linked interpretation is that the use of pseudo-relative humidity gives larger weight to higher latitudes. These interpretations may merit further investigation in the future.

Figure 6.

As Figure 5, but for the winter season.

Vertical profiles of standard deviations for the two different moisture assimilation control variables are presented in Figure 7 together with profiles of the parts of these standard deviations explained by the other assimilation control variables. The values here represent the standard deviations for the differences between +36 h and +12 h forecasts valid at the same time, and for application in data assimilation these values are rescaled with a factor of 0.9 to represent standard deviations of +6 h background forecast errors. This rescaling factor was obtained by tuning with innovation (observation minus background) data. A more elaborate rescaling algorithm could be considered for the unbalanced pseudo-relative humidity by, for example, taking the vertical variation of the explained (balanced) moisture variance into account. Due to the normalization with the saturation water vapour pressure, the standard deviations of the pseudo-relative humidity take their maximum values at higher levels in the vertical than the corresponding standard deviations for the specific humidity. The standard deviations of both moisture control variables take values close to zero for the model levels 1–7. The reason for this is that achieving accuracy for moisture at stratospheric levels is too poor, and this forced us to artificially set the input moisture forecast fields to zero (or, in fact, to small random numbers to avoid ill-conditioning of the vertical covariance matrices) at these levels.

Figure 7.

Vertical profiles of standard deviations for (a) specific humidity (g kg−1) and (b) pseudo-relative humidity, both for the summer season: total standard deviation (full line), unbalanced part (dashed line), explained by balanced geopotential (symbol +), by unbalanced divergence (symbol ×), and by unbalanced temperature and surface pressure (symbol º).

Horizontal correlation spectra (not shown), derived by assuming homogeneity and isotropy with respect to these horizontal correlations in physical space, are very similar for the two moisture control variables with a maximum correlation density around wave number 10 to 20, and with a shift towards smaller horizontal scales at lower model levels. Vertical correlations are very similar for the two moisture control variables, with a shift towards narrower vertical correlations for smaller horizontal scales (not shown). The vertical correlations (as well as the horizontal spectral densities) behave more noisily at the smallest horizontal scales.

Background-error statistics were also derived from differences between +36 h and +12 h forecasts taken from the archives of regular HIRLAM forecast runs, also based on the HIRLAM forecast model applied together with HIRLAM 4D-Var on a larger horizontal domain and with 60 vertical levels. A total of 112 pairs of forecasts were selected from the period from 1 September to 31 December 2008. The resulting statistics were very similar to the winter season statistics derived from operational SMHI HIRLAM forecasts; in particular the strong coupling between unbalanced temperature and pseudo-relative humidity in the boundary layer was almost identical in the two statistical datasets. This gives us some confidence that the derived statistics are not accidental or, for example, based on undersampling.

4. Data assimilation experiments

4.1. Model and experimental set-up

Simulated observation assimilation experiments and full-scale real observation assimilation experiments, using the HIRLAM model (Undén et al., 2002) were performed over an area covering Northern Europe and the northern North Atlantic to study the effects of the new assimilation control variable. Three-dimensional variational assimilation (3D-Var; Gustafsson et al., 2001; Lindskog et al., 2001) was used for the simulated observation experiments, and 4D-Var (Huang et al., 2002; Gustafsson, 2006) was used for the full-scale experiment with real observations. The HIRLAM 4D-Var applies a multi-incremental minimization (Veersé and Thépaut, 1998) and includes the simplified physical parametrization schemes of Janisková et al. (1999). Conventional observations (radiosonde, PILOT wind, SYNOP, SHIP, DRIBU and aircraft observations) and satellite radiance data from the Advanced Microwave Sounding Unit (AMSU-A) over open sea surface areas were assimilated in a 6 h assimilation cycle. For the lateral boundary conditions, forecasts from the ECMWF were used. The model grid mesh consisted of 306 × 306 horizontal grid points at 22 km grid spacing and 40 levels. Background-error variances of q are specified as static and homogeneous in the reference HIRLAM system, although a flow-dependent variance specification could be considered for q, following Rabier et al. (1998). For radiosonde moisture observations, an empirical regression relation is used to obtain the standard deviations of relative humidity observation errors, with a dependency on the background temperature Tb:

equation image

For temperatures below 240 K, σo takes the value 0.18 and for temperatures above 320 K it takes the value 0.06. The observation-error standard deviation for specific humidity is then derived by the tangent-linear relationship between specific and relative humidity (Lindskog et al., 2001). An a posteriori correction of negative and supersaturated humidities is applied in both formulations before running the forecast model.

The experiments were based on the current HIRLAM reference model physics. The physical parametrizations used were the CBR (Cuxart et al., 2000) turbulence scheme, the Kain–Fritsch convection scheme (Kain, 2004), the Rasch–Kristjánsson cloud water scheme (Rasch and Kristjánsson, 1998), the Savijärvi (1990) radiation scheme and the ISBA (Interface, Soil, Biosphere and Atmosphere) surface scheme (Noilhan and Mahfouf, 1996).

4.2. Simulated observation assimilation experiments

The new moisture balance operator and the handling of the new moisture assimilation control variable have been introduced into the HIRLAM variational data assimilation software. In order to illustrate the effects of the new moisture assimilation, simulated observation experiments were carried out and compared with equivalent experiments using the reference assimilation based on specific humidity as the assimilation control variable and the moisture balance formulation of Berre (2000). The simulated observation assimilation experiments were carried out with a single iteration in the outer minimization loop.

Simulated observation experiments were first carried out with a 6 h forecast from 0000 UTC on 3 December 1999 as the background model state. The assimilation background-error statistics for 3D-Var with the reference system background-error constraint are completely independent of the background model state, while a dependency on the background state is introduced via the new assimilation control variable, i.e. a dependency on the background temperature through the normalization with background water vapour saturation pressure and a dependency on the background relative humidity via the further normalization with a background-error standard deviation that is a function of the background relative humidity. For this reason, the background relative humidity and temperature fields at 850 hPa are presented in Figure 8.

Figure 8.

Background fields at 0600 UTC on 3 December 1999 for (a) 850 hPa relative humidity (contour interval 25%) and (b) 850 hPa temperature (contour interval 2°C) used for the simulated observation experiments. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

The assimilation increments of specific humidity at 850 hPa due to simulated 850 hPa specific humidity observation increments δq = +10 g kg−1 at five locations are shown in Figure 9 for both formulations of the moisture assimilation control variable. With the reference assimilation control variable, the specific humidity assimilation increments are independent of horizontal position, while with the new moisture control variable, the assimilation increments depend on the background temperature and the spatially varying background-error standard deviations. With decreasing background temperatures (57°N, 03°W and, in particular, 70°N, 05°E), we will have decreasing water vapour saturation pressures, used for the normalization of the moisture assimilation control variables, and thus implicitly defining decreasing background-error standard deviations with regard to specific humidity. The influence of the background relative humidity on the assimilation increment with the new moisture control variable can be observed in the vicinity of the observation increments at 48°N, 30°W (background relative humidity ≈ 95%) and 45°N, 10°W (background relative humidity ≈ 50%). The smaller moisture increment at 95% background relative humidity, as compared to the increment at 50% background relative humidity, certainly is reasonable for a positive moisture increment but more questionable for a negative moisture increment, for which several iterations in the outer loop minimization are needed in order to increase the magnitude of the assimilation increment. Notice also the elliptical assimilation increment at 51°N, 30°W, stretched along the gradient in the relative humidity background field. The strong reduction of the assimilation increment is a direct effect of the symmetrization transform, but this may also be a disadvantage in the case, for example, when the forecast model has a tendency to establish excessively dry background forecast fields. The flow dependency of the assimilation increment is visible, although this is not a proof of actuality as such.

Figure 9.

The assimilation increments of specific humidity (contour interval 1 g kg−1) at 850 hPa due to simulated 850 hPa specific humidity observation increments of δq = 10 g kg−1 at 48°N, 30°W; 51°N, 30°W; 45°N, 10°W; 57°N, 03°W; and 70°N, 05°E. The assumed standard deviation of the observation error is σo = 1 g kg−1. (a) shows the reference moisture assimilation variable, and (b) the new moisture assimilation variable. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

Due to the statistically derived balance relations, the new moisture assimilation variable also enables a flow-dependent impact on moisture from, for example, surface pressure observations. This is illustrated in Figure 10. A negative surface pressure increment (–5 hPa) was simulated that also resulted in convergence and moistening due to increased low-level inflow. With the new moisture control variable (Figure 10(c)), a stronger moistening, in terms of specific humidity, occurs in the warm air compared with the situation in the cold air. When using homogeneous background-error standard deviations for specific humidity, the specific humidity increment is spatially symmetric and independent of the background field (Figure 10(b)). Spatially close to the simulated observation, a frontal system is present in the background temperature field (Figure 10(a)). This is the main cause of the flow-dependent moisture increment structure in this case. However, due to the relatively weak amplitude of the coupling between mositure and the other control variables, the resulting moisture assimilation increment remains relatively small (it is probably smaller than the humidity observation error for instance).

Figure 10.

Vertical cross-sections along 30°W between 40°N and 70°N. (a) The temperature background field valid at 0600 UTC on 3 December 1999, with contour interval 5°C, and (b, c) specific humidity assimilation increments due to a simulated surface pressure observation increment of –5 hPa at 56°N, 30°W using (b) specific humidity as assimilation control variable, and (c) the new assimilation control variable. The assumed standard deviation of the observation error is σo = 0.5 hPa, while the standard deviation of the full background error σb is of the order 1.2 hPa. In (b) and (c), the contour interval is 0.1 g kg−1.

Because of the large amount of available satellite data related to moisture, a second experiment was carried out using a simulated satellite radiance observation for one humidity-sensitive channel of the Spinning Enhanced Visible and InfraRed Imager (SEVIRI; Schmetz et al., 2002) on board the Meteosat-8 satellite. The characteristics of satellite observations differ from most conventional observations, since they usually contain information about a more or less broad atmospheric layer. In this example, a channel was selected which is spectrally located around 7.3 μm within a strong water vapour absorption band. Generally, the vertical impact of such an observation is determined by the vertical sensitivities of radiation in this spectral range with respect to the model control variables (Jacobians), i.e. temperature and humidity. These are provided by the Radiative Transfer Model for TOVS (RTTOV, version 8; Eyre, 1991; Saunders et al., 1999), which is the observation operator for satellite radiances in the HIRLAM assimilation system. The exact values and shape of the temperature and moisture Jacobians for this channel depend, as does the background brightness temperature, on the background profiles. The Jacobians for the chosen channel usually show significant values in the troposphere with the highest amplitudes between 500 and 600 hPa on average (e.g. Stengel et al., 2009). Beside the Jacobians, the vertical impact of such an observation is further determined by the vertical profile of the background-error standard deviations and a smoothing by the vertical background-error correlations.

The simulated radiance observation was used in an assimilation experiment for 0000 UTC on 3 June 2006. The simulated observation was chosen to create a brightness temperature innovation (observation minus background) of –2 K with respect to the background and the observation is placed at 50°N, 15°E between a very dry region with relative humidity values down to 5% and a very humid region with values over 95%. The standard deviation of the observation error was set to 2 K. The experiment was carried out with the old assimilation control variable and repeated with the new control variable. The corresponding assimilation increments for specific humidity are shown in Figure 11. We found a clearly decreased amplitude of the specific humidity assimilation increment for the new moisture variable compared to the old. The maximum value was almost halved, which is likely to be related to relatively low background temperatures. Furthermore, the shape of the increments differed significantly and was no longer symmetrical for the new formulation as it is for the old. This may be explained by the fact that the background-error standard deviations expressed in specific humidity were lower on the drier and colder side of the increment. In the upper part, the increment for the new moisture variable seems to follow the shape of the contours of the strong humidity gradient within which this observation is located.

Figure 11.

Vertical cross-sections along 15°E between 45°N and 55°N of (a) background relative humidity (%), and (b, c) specific humidity assimilation increments (units of 0.01 g kg−1) with (b) specific humidity as the assimilation control variable and (c) the new assimilation control variable. Assimilation cycle done at 0000 UTC on 3 June 2006 with one single simulated SEVIRI brightness temperature innovation of –2 K located at 50°N, 15°E.

4.3. Real observation assimilation experiments

The following sets of summer (June 2005)/winter (January 2007) data assimilation and forecast experiments were carried out:

  • (i)Experiment o1c (summer)/o1C7 (winter), assimilation based on specific humidity as the assimilation moisture control variable and with one iteration only in the 4D-Var outer loop minimization;
  • (ii)Experiment o22 (summer)/o2C7 (winter), assimilation based on specific humidity as the assimilation control variable and with two iterations in the 4D-Var outer loop minimization; and
  • (iii)Experiment n22 (summer)/n2C7 (winter), assimilation modified to have pseudo-relative humidity as the assimilation control variable and with two iterations in the 4D-Var outer loop minimization.

These three configurations give the possibility to evaluate the effects of a second outer loop in the old formulation and the effects of the new formulation, respectively. The summer experiments all used background-error statistics derived for the summer season and the winter experiments used corresponding winter statistics.

The horizontal resolution of the assimilation increments was set to 60 km in the first iteration of the 4D-Var outer loop minimization and to 40 km in the second iteration. The real observation assimilation experiments were started with an ECMWF 6 h forecast as the background model state for the first assimilation cycles, 0600 UTC on 1 June 2005 and 0600 UTC on 1 January 2007, respectively.

4.4. Observation verification of forecasts

In order to evaluate the relative quality of the analyses and the forecasts from the different experiments, we verified them against SYNOP and radiosonde observations in the list established by the European Working Group on Limited-Area Models (EWGLAM).

Figure 12 shows the time-averaged bias and RMSE verification scores for surface pressure over the European area. For this parameter the impact of the pseudo-relative humidity control variable was essentially neutral, while a small positive impact was found in the old formulation when using two iterations in the outer loop minimization compared with one iteration only. Independent experiments (not shown) have indicated that the positive impact of using two iterations in the outer loop minimization mainly originates from the relinearization applied for the tangent-linear model, for the background-error constraint as well as for the observation operators in the second outer loop iteration. The same analysis and forecast verification scores for surface pressure, but averaged over a Scandinavian area (not shown), indicated that the impact of the second outer loop minimization was a little stronger in the Scandinavian area in the centre of the model integration domain than over the total European area.

Figure 12.

Verification of surface pressure forecasts (RMS errors and biases) against SYNOP stations using the EWGLAM list of 299 stations for the three experiments o1c, o22 and n22 over a European area for June 2005.

Verification figures for 12 h accumulated precipitation forecasts by the Kuipers skill score (KSS) are shown in Figure 13 for the June 2005 period. The KSS is a combination of the hit rate and the false alarm skill scores and it is shown here for the precipitation classes of 0.1, 0.3, 1.0, 3.0, 10.0 and 30.0 mm (12 h)−1. The KSS values of 1.0 indicate a perfect forecast, while values of 0.0 indicate no skill. Figure 13 shows that the impacts of two outer loop iterations as well as the pseudo-relative humidity on precipitation forecasts are essentially neutral.

Figure 13.

Verification of 12 h accumulated precipitation forecasts by the Kuipers skill score against SYNOP stations using the EWGLAM list of 292 stations for the three experiments o1c, o22 and n22 over a European area for June 2005.

Verification scores for vertical profiles of temperature and relative humidity, as verified against radiosonde observations, are presented as averages over +12 h, +24 h, +36 h and +48 h forecasts in Figure 14. The impact of using two iterations in the outer loop minimization is slightly positive, while the impact of using pseudo-relative humidity as the assimilation control variable is close to neutral.

Figure 14.

Verification of (a) temperature and (b) relative humidity forecasts against radiosonde stations using the EWGLAM list of 48 stations for the three experiments o1c, o22 and n22 for June 2005.

The average forecast verification scores for the winter period are shown for surface pressure over the European domain in Figure 15, for precipitation in Figure 16, and for vertical temperature profiles in Figure 17. All scores reveal a similar positive impact when using two iterations in the 4D-Var outer loop minimization compared with the use of a single iteration. The impact of the new moisture control compared with specific humidity as the assimilation control variable was more or less neutral. The impact on precipitation forecasts during January 2007 was marginally positive for two iterations in the outer loop minimization. The impact of the pseudo-relative humidity is again essentially neutral on the whole, although a positive impact is observed for the 10 mm class.

Figure 15.

Verification of surface pressure forecasts against SYNOP stations using the EWGLAM list of 290 stations for the three experiments o1C7, o2C7 and n2C7 for January 2007.

Figure 16.

Verification of 12 h accumulated precipitation forecasts by the Kuipers skill score against SYNOP stations using the EWGLAM list of 271 stations for the three experiments o1C7, o2C7 and n2C7 over a European area for January 2007.

Figure 17.

Verification of (a) temperature and (b) relative humidity forecasts against radiosonde stations using the EWGLAM list of 52 stations for the three experiments o1C7, o2C7 and n2C7 for January 2007.

4.5. Effects of outer loop minimization iterations

In the discussions above we have expressed some concern for the potential negative effects of the small background-error standard deviations near saturation and near zero humidity of the background model state, in particular for the first outer loop minimization iteration when the assimilation increment is assumed zero in the nonlinear background-error transform. In order to examine this problem, three different assimilation experiments were carried out for 0000 UTC on 2 August 2010, utilizing identical analysis background model states. The three experiments were:

  • (i)The old moisture variable, q, with two outer loop iterations;
  • (ii)The new moisture control variable, RH, with two outer loop iterations; and
  • (iii)The new moisture variable, RH, with one outer loop iteration.

From these experiments we examined the statistical distributions of relative humidity assimilation increments for the background relative humidity when close to saturation (above 95%). One example of such a distribution of assimilation increments is presented in Figure 18. From the results in the figure we may notice:

  • The frequency of large positive-valued assimilation increments is reduced with the new moisture control variable;

  • With only one outer loop iteration and the new moisture variable, the frequency of large negative-valued assimilation increments is also reduced, as one could expect from the smaller background-error standard deviations applied during the first outer loop iteration;

  • With two outer loop iterations and the new moisture variable, the frequencies of large negative-valued assimilation increments are closer to the level of the old moisture control variable.

In summary, these experiments confirm that, with the new moisture control variable, the assimilation manages to reduce the frequencies of large positive-valued assimilation increments near saturation (potential supersaturations) but the experiments also confirm our concern with the potential negative effects of small background-error standard deviations near saturation.

Figure 18.

Frequency distribution of relative humidity assimilation increments for the relative humidity background state close to saturation (above 95%), averaged over all model levels between 1000 and 300 hPa within a European domain and for three different single-case (0000 UTC on 2 August 2010) experiments: (i) old moisture variable, q, with two outer loop minimization iterations, (ii) new moisture variable, RH, with two outer loop minimization iterations and (iii) new moisture variable, RH, with one outer loop minimization iteration.

5. Conclusions

We have presented a reformulation of the humidity part of the HIRLAM variational data assimilation, with the purpose of representing some space and time dependencies of moisture variances. A modified pseudo-relative humidity variable has been adapted to the statistical balance background-error constraint, including the associated statistical moisture balance formulation. The transforms applied in the pre-conditioning for the reformulated background-error constraint are nonlinear in terms of the assimilation control variable, thus necessitating the application of relinearizations within the framework of the outer loop minimization. It has been noted that specified standard deviations are substantially reduced for nearly dry and saturated states. It would be desirable to ascertain the degree of realism of this change with further investigations.

The background-error statistics for the new moisture control variable exhibit a clear dependency on season, reflecting the relative importance of different physical processes involved in the moisture–temperature balances during different seasons. Moreover, the seasonal dependency of standard deviations indicates that the formulation and/or its calibration may need to be generalized for an operational implementation, since there are factors other than local background values of temperature and humidity that influence the outcome.

In simulated observation experiments, we illustrated the impact of the new moisture formulation on the assimilation increments, which significantly differed in shape, location and amplitude from the old formulation. These differences could be linked to the background values of humidity and temperature corresponding to the specified flow-dependency. In extended data assimilation and forecast experiments, the impact of the new moisture assimilation control variable was investigated in a 4D-Var framework using real observations. Corresponding forecast verification scores indicated that the impact was essentially neutral. A clearer positive impact was demonstrated for the application of several iterations in the outer loop 4D-Var minimization of the old formulation.

The implications with respect to control of negative and supersaturating humidities have only partially been explored in this study. For example, there is at present no control of supersaturation within the assimilation with specific humidity as the moisture assimilation control variable, and this could result in problems with the vertical distribution of moisture increments when observed quantities represent vertically integrated water vapour, for example ground-based GPS zenith delay measurements. With the new moisture control variable, there is an implicit control of supersaturation as the background-error standard deviations decrease towards background saturations, and this is likely to help the vertical distribution of moisture assimilation increments. On the other hand, it has also been recognized that the substantial reduction of standard deviations for nearly dry and saturated states may be sub-optimal. Therefore it would be interesting to compare this implicit approach with an explicit control of supersaturation, at the end of each inner loop for instance, which can be implemented when using specific humidity as a control variable.

There are several issues which could be investigated in the future. In particular, while the pseudo-relative humidity formulation looks more sophisticated than the formulation based on homogeneous and static variances for q, its neutral impact may require further studies to be understood. For instance, the impacts of the parametrization of the variance as a function of the background state on the one hand, and of the symmetrization transform on the other hand, could be evaluated separately. Moreover, it would be interesting to compare the variance parametrization as a function of background relative humidity and temperature with variance estimates either based on Rabier et al. (1998) or derived from a real-time ensemble. The nonlinearity implied by the symmetrization transform also deserves further studies in terms of advantages but also potential difficulties. These are related to choices in the variance specification (with implications for quality control and minimization results) and to the way the nonlinear analysis may or may not converge to an optimal state. Also remaining to be studied is the extent to which it is more important to define a control variable supposed to be more Gaussian than to specify realistic variances and correlations.

Acknowledgements

We thank the anonymous reviewer B for a very thorough and constructive review that led to significant improvements of the manuscript. S. Thorsteinsson would like to acknowledge the strong support from SMHI in the work reported here.

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