The motivation for using the absolute moist-air entropy in atmospheric science was first described by , and then fully formalized by . The method comprises taking into account the absolute value for dry air and water vapour and to define a moist-air entropy potential temperature variable called θs.

However, the version of θs published in was not really synonymous with the moist-air entropy. This problem has been solved with the version of by imposing the same link with the specific entropy of moist air (s) as in the dry-air formalism of , leading to 1s=cpdln⁡θsT0+sd0(T0,p0), where cpd≈1004.7 JK-1kg-1 is the dry-air specific heat at constant pressure, T0=273.15 K is a standard temperature and sd0(T0,p0)≈6775 JK-1kg-1 is the reference dry-air entropy at T0 and at the standard pressure p0=1000 hPa. Because cpd, T0 and sd0(T0,p0) are constant terms, θs defined by Eq. () is synonymous with, and has the same physical properties as, the moist-air entropy s.

The conservative aspects of this potential temperature θs and its meteorological properties (in e.g. fronts, convection, cyclones) have been studied in , and . The links with the definition of the Brunt–Väisälä frequency and the PV are described in and , while the significance of the absolute entropy to describe the thermodynamics of cyclones is shown in and .

Only the first-order approximation of θs, denoted (θs)1 in , will be considered in the following, written as 2θs≈(θs)1=θexp⁡-Lvapql+LsubqicpdTexp⁡Λrqt, where θ=Tp/p0κ is the dry-air potential temperature, p the pressure, κ≈0.2857 and Lvap(T) and Lsub(T) the latent heat of vaporization and sublimation respectively. The explanation for Λr follows later in the section.

The first term θ on the right-hand side of Eq. () leads to a first conservation law (invariance) during adiabatic compression and expansion, with joint and opposite variations of T and p keeping θ constant. Here lies the motivation for using θ to describe dry-air convective processes, also used in data assimilation systems by and .

The first exponential on the right-hand side of Eq. () explains a second form of conservation law. Indeed, this exponential is constant for reversible and adiabatic phase changes, for which d(cpdT)≈d(Lvapql+Lsubqi) due to the approximate conservation of the moist static energy cpdT-Lvapql-Lsubqi, and therefore has joint variations of the numerator and denominator and a constant fraction into the first exponential. It should be mentioned that the product of θ by this first exponential forms the conservative variable θl, which is presently used together with qt to describe the moist-air turbulence in general circulation models (GCMs) and NWP models.

While the variable θl was established with the assumption of a constant total water content qt in , the second exponential in Eq. () sheds new light on a third and new conservation law, where the entropy of moist air can remain constant despite changes in the total water qt. This occurs in regions where water-vapour turbulence transport takes place, or via the evaporation process over oceans, or at the edges of clouds via entrainment and detrainment processes.

We consider here “open-system” thermodynamic processes, for which the second exponential takes into account the impact on moist-air entropy when the changes in specific content of water vapour are balanced, numerically, by opposite changes of dry air, namely with dqd=-dqt≠0. In this case, as stated in , the changes in moist-air entropy depend on reference values (with subscript “r”) according to d[qd(sd)r+qt(sv)r], and thus with (sd)r and (sv)r being constant and with the relation qd=1-qt, leading to [(sv)r-(sd)r]dqt.

This explains the new term Λr=[(sv)r-(sd)r]/cpd≈5.869±0.003, which depends on the absolute reference entropies for water vapour (sv)r≈12671 JK-1kg-1 and dry air (sd)r≈6777 JK-1kg-1. This also explains that these open-system thermodynamic effects can be taken into account to highlight regimes where the specific moist-air entropy (s), θs and (θs)1 can be constant despite changes in qt, which may decrease or increase on the vertical seefor such examples.

Although it should be possible to use (θs)1 as a control variable for assimilation, it appeared desirable to define an additional approximation of this variable for a more “regular” and more “linear” formulation, insofar as tangent-linear and adjoint versions are needed for the 1D-Var system. Considering the approximation exp⁡(x)≈1+x for the two exponentials in Eq. (), neglecting the second-order terms in x2, also neglecting the variations of Lv(T) with temperature and assuming a no-ice hypothesis (qi=0), the new variable is written as 3(θs)a=θ1+Λrqt-Lvap(T0)qlcpdT,4(θs)a=1cpdp0pκcpd(1+Λrqt)T-Lvap(T0)ql, where Lvap(T0)≈2501 kJkg-1. This formulation corresponds to Sm/cpd, where Sm is the moist static energy defined in Eq. 73 and used in the European Centre for Medium-Range Weather Forecasts (ECMWF) NWP global model by .

The new potential temperature (θs)a remains close to (θs)1 (not shown) and keeps almost the same three conservative properties described for (θs)1. This new conservative variable (θs)a will be used along with the total water content qt=qv+ql in the data assimilation experimental context described in the following sections.

The general framework describing the retrieval of atmospheric profiles from remote-sensing instruments by statistical methods can be found in . In the following we present the main equations of the one-dimensional variational formalism. Additional details are given in , who developed the first 1D-Var inversion applied to satellite radiances using the adjoint technique.

The 1D-Var data assimilation system searches for an optimal state (the analysis) as an approximate solution of the problem minimizing a cost function J defined by 5J(x)=12(x-xb)TBx-1(x-xb)+12[y-H(x)]TR-1[y-H(x)]. The symbol T represents the transpose of a matrix.

The first (background) term measures the distance in model space between a control vector x (in our study, T, qv and ql profiles) and a background vector xb, weighted by the inverse of the background error covariance matrix (Bx) associated with the vector x. The second (observation) term measures the distance in the observation space between the value simulated from the model variables H(x) (in our study, the RTTOV-gb model) and the observation vector y (in our study, a set of microwave TBs from a ground-based radiometer), weighted by the inverse of the observation error covariance matrix (R). The solution is searched iteratively by performing several evaluations of J and its gradient: 6∇xJ(x)=Bx-1(x-xb)-HTR-1[y-H(x)], where H is the Jacobian matrix of the observation operator representing the sensitivity of the observation operator to changes in the control vector x (HT is also called the adjoint of the observation operator).

The 1D-Var assimilation defined previously with the variables x=(T,qv,ql) can be modified to use the conservative variables z=((θs)a,qt). A conversion operator that projects the state vector from one space to the other can be written as x=L(z). In the presence of liquid water ql, an adjustment to saturation is made to separate its contribution to the total water content qt from the water-vapour content qv. This is equivalent to distinguishing the “unsaturated” case from the “saturated” one. Therefore, starting from initial conditions (TI,qI)=(T,qv) and using the conservation of (θs)a given by Eq. (), we look for the variable T∗ such that 7T∗+αqsat(T∗)=TI+αqI, where 8α=Lvap(T0)cpd(1+Λrqt), and qsat(T∗) is the specific humidity at saturation.

For the unsaturated case (qv<qsat(T∗)), we obtain the variables (T,qv,ql) directly from Eq. (): ql=0,qv=qt, and 9T=(θs)app0κ11+Λrqt.

For the saturated case (qv≥qsat(T∗)), we write ql=qt-qsat(T∗), and 10qv=qsat(T∗).

In this situation, it is necessary to implicitly calculate the temperature T∗, given by Eq. (). We numerically compute an approximation of T∗ by using Newton's iterative algorithm.

Taking into account this change of variables, the cost function can be written as 11J(z)=12(z-zb)TBz-1(z-zb)+12[y-H(L(z))]TR-1[y-H(L(z))].

Then, its gradient given by Eq. () becomes 12∇zJ(z)=Bz-1(z-zb)-LTHTR-1[y-H(L(z))], where LT is the adjoint of the conversion operator L.

The second term on the right-hand side of Eqs. () and () indicates that the conversion operator L is needed to compute the TBs from the observation operator H. Indeed RTTOV-gb requires profiles of temperature, specific humidity and liquid water content as input quantities. This space change is required at each step of the minimization process. For the computation of the gradient of the cost function ∇zJ, the linearized version (adjoint) of L is also necessary. In practice, the operator LT provides the gradient of the TBs with respect to the conservative variables, knowing the gradient with respect to the classical variables.

This section presents the experimental context of 9 February 2020 at the Saint-Symphorien site characterized by (i) a radiative fog event observed in the morning and (ii) the development of low-level clouds in the afternoon and evening.

Figure  shows a time series of cloud radar reflectivity profiles (W-band at 95 GHz) measured by the BASTA instrument in the lowest hundred metres (top panel) between 9 February 2020 at 00:00 UTC and 10 February 2020 at 00:00 UTC. The instrument reveals a thickening of the fog between 00:00 UTC and 04:00 UTC (9 February 2020). The fog layer thickness is located between 90 and 250 m. After 04:00 UTC, the fog layer near the ground rises, lifting into a “stratus” type cloud (between 100 and 300 m). After 08:00 UTC, the stratus cloud dissipates. In the bottom panel, BASTA observations up to 12 000 m (≈ 200 hPa) indicate low-level clouds after 14:00 UTC, generally between 1000 m (≈ 900 hPa) and 2000 m (≈ 780 hPa), with a fairly good agreement with AROME short-range (1 h) forecasts (see Fig. f). Optically thin (reflectivity below 0 dBZ) high-altitude ice clouds are also captured by the radar.

Figure  depicts the diurnal cycle evolution in terms of the vertical profiles of (a) absolute temperature T, (b) dry-air potential temperature θ, (c) water-vapour specific content qv, (d) entropy potential temperature (θs)a, (e) cloud liquid water specific content ql and (f) relative humidity (RH), from 1 h AROME forecasts (background) of 9 February 2020 at Saint-Symphorien. At this stage, it is important to note that the AROME model has a 90-level vertical discretization from the surface up to 10 hPa, with high resolution in the planetary boundary layer (PBL) since 20 levels are below 2 km.

Figure e and f, for ql and RH, show two main saturated layers: a fog layer close to the surface between 00:00 and 09:00 UTC with the presence of a thin liquid cloud layer aloft at 850 hPa at 00:00 UTC, and the presence of a stratocumulus cloud between 14:00 UTC and midnight (24:00 UTC) at 850 hPa. During the night, the near-surface layers cool down, with a thermal inversion that sets at around 01:00 UTC and persists until 07:00 UTC. After the transition period between 06:00 and 09:00 UTC, when the dissipation of the fog and stratus takes place, the air warms up and the PBL develops vertically (see the black curves plotted where vertical gradients of θ in Fig. b are large). Towards the end of the day, the thickness of the PBL remains important until 24:00 UTC, probably due to the presence of clouds between 800 and 750 hPa, which reduces the radiative cooling (see Fig. c and f for qv and RH).

Figure d reveals weaker vertical gradients for the (θs)a profiles, notably with contour lines often vertical and less numerous than those of the T, θ and qv profiles in panels (a), (b) and (c), as also shown by more extensive and more numerous vertical arrows in panel (d) than in panel (b). Here we see the impact of the coefficient Λr≈5.869 in Eqs. ()–(), which allows the vertical gradients of θ in Fig. b and qv in Fig. c to often compensate each other in the formula for (θs)a. This is especially true between 980 and 750 hPa in the morning between 04:00 and 10:00 UTC, and also within the dry and moist boundary layers during the day.

Note that the dissipation of the fog is associated with a homogenization of (θs)a in Fig. d from 04:00 to 05:00 UTC in the whole layer above, in the same way as the transition from stratocumulus toward cumulus was associated with a cancellation of the vertical gradient of (θs)1 in Fig. 6 of . This phenomenon cannot be easily deduced from the separate analysis of the gradients of θ and qv in Fig. b and c. Therefore, three air mass changes can be clearly distinguished during the day. The vertical gradients of (θs)a are stronger during cloudy situations, first (i) at night and early morning before 04:00 UTC and just above the fog, then (ii) at the end of the day above the top-cloud level at 800 hPa and (iii) turbulence-related phenomena in between that mix the air mass and (θs)a, up to the cloudy layer tops that evolve between 950 and 800 hPa from 13:00 to 17:00 UTC.

The observations to be assimilated are presented in the following. The HATPRO MicroWave Radiometer (MWR) measures TBs at 14 frequencies between 22.24 and 58 GHz: 7 are located in the water-vapour absorption K-band and 7 are located in the oxygen absorption V-band (see the Table ). For our study, the third channel (at 23.84 GHz) was eliminated because of a receiver failure identified during the campaign. In this preliminary study, we have only considered the zenith observation geometry of the radiometer for the sake of simplicity.

The H RTTOV-gb model needed to simulate the model equivalent of the observations, which, together with the choice of the control vector and the specification of the background and observation error matrices, are presented in the next section.

In 1D-Var systems, the integrated liquid water content, liquid water path (LWP), can be included in the control vector x as initially proposed by and more recently used by . A first experimental set-up has been defined where the minimization is performed with the control vector being (T,qv,LWP). It will be considered as the reference, named REF. The 1D-Var system chosen for the present study is the one developed by the EUMETSAT NWP SAF (Numerical Weather Prediction Satellite Application Facility), where the minimization of the cost function is solved using an iterative procedure proposed by with a Gauss–Newton descent algorithm. During the minimization process, only the amount of integrated liquid water is changed. In this approach, the two “moist” variables qv and LWP are considered to be independent (no cross-covariances for background errors between these variables). The second experimental framework, where the control vector is z=((θs)a,qt), corresponding to the conservative variables, is named EXP. The numerical aspects of the 1D-Var minimization are kept the same as in REF.

Then, a set of reference matrices Bx(T,qv) was estimated every hour using the EDA system of the AROME model on 9 February 2020. These matrices were obtained by computing statistics from a set of 25 members providing 3 h forecasts for a subset of 5000 points randomly selected in the AROME domain to obtain a sufficiently large statistical sample. Then, matrices associated with fog areas, denoted Bx(T,qv)fog, were computed every hour by applying a fog mask (defined by areas where ql is above 10-6 kg kg-1 for the three lowest model levels), in order to select only model grid points for which fog is forecast in the majority of the 25 AROME members. The background error covariance matrices Bz((θs)a,qt) and Bz((θs)a,qt)fog were obtained in a similar way.

The observation errors are those proposed by with values between 1 and 1.7 K for humidity channels (frequencies between 22 and 31 GHz), values between 1 and 3 K for transparent channels affected by larger uncertainties in the modelling of the oxygen absorption band (frequencies between 51 and 54 GHz) and values below 0.5 K for the most opaque channels (frequencies between 55 and 58 GHz).

The RTTOV model is used to calculate TBs in different frequency bands from atmospheric temperature, water vapour and hydrometeor profiles together with surface properties (provided by outputs from the AROME model). This radiative transfer model has been adapted to simulate ground-based microwave radiometer observations (RTTOV-gb) by .

Figure  displays for the selected day at 06:00 UTC the cross-correlations between T and qv (top) and between (θs)a and qt (bottom), with (right) and without (left) a fog mask. For the classical variables the correlations are strongly positive in the saturated boundary layer with the fog mask from levels 75 to 90 (between 1015 and 950 hPa), while with profiles in all-weather conditions the correlations between T and qv are very weak in the lowest layers. On the other hand, the atmospheric layers above the fog layer exhibit negative correlations between temperature and specific humidity along the first diagonal.

When considering conservative variables, the correlations along the diagonal show a consistently positive signal in the troposphere (below level 20 located around 280 hPa). Contrary to the classical variables, which are rather independent in clear-sky atmospheres as previously shown by , the Bz matrix reflects the physical link between the two new variables (shown by Eq. ) as diagnosed from the AROME model. The correlations are positive with and without a fog mask. This result shows that the matrix Bz((θs)a,qt) is less sensitive to fog conditions than the Bx matrix. It could therefore be possible to compute a Bz((θs)a,qt) matrix without any profile selection criteria that would be nevertheless suitable for fog situations, resulting in a more robust estimate. This result is key for 1D-Var retrievals which are commonly used in the community of ground-based remote-sensing instruments to provide databases of vertical profiles for the scientific community. In fact, the accuracy of 1D-Var retrievals is expected to be more robust with less flow-dependent B matrices.

We also note that these background error statistics are less dependent on the diurnal cycle and on the meteorological situation (e.g. in the presence of fog at 06:00 UTC and low clouds at 21:00 UTC), contrary to the Bx(T,qv) matrix, where there is a reduction in the area of positive correlation in the lowest layers between 06:00 and 21:00 UTC (Fig. ).

The 1D-Var results are now assessed in observation space by examining innovations (differences between observed and simulated TBs) from AROME background profiles and residuals. In the following, we have only used background error covariance matrices estimated at 06:00 UTC with a fog mask, for a simplified comparison framework of the two 1D-Var systems.

Figure  presents both (a) innovations and (b, c) residuals obtained with the two 1D-Var systems (Fig. b: REF and Fig. c: EXP) for the 13 channels (1, 2, 4–14) and for each hour of the day. The innovations are generally positive for water-vapour-sensitive channels during the day, and negative for temperature channels, especially in the morning. The differences are mostly between -2.5 and 5 K. For channels 8, 9 and 10, which are sensitive to liquid water content, the innovations can reach higher values exceeding 10 K (in the afternoon) or around -5 K (in the morning).

In terms of residuals, as expected from 1D-Var systems, both experiments significantly reduce the deviations of the observed TB from those calculated using the background profiles, especially for the first eight channels sensitive to water vapour and liquid water. We can note that the residuals are not as reduced for channel 9 (52.28 GHz) compared to other channels. Indeed, channels 8 and 9 (51.26 and 52.28 GHz) suffer from larger calibration uncertainties and larger forward model uncertainties dominated by oxygen line mixing parameters than other temperature-sensitive channels. However, by comparing simulated TB with different absorption models , or through monitoring with simulated TB from clear-sky background profiles , larger biases are generally observed only at 52.28 GHz. Consequently, the higher deviations observed in Fig.  for channel 9 mostly originate from larger modelling and calibration uncertainties, which are taken into account in the assumed instrumental errors (prescribed observation errors of about 3 K for these two channels compared to <1 K for other temperature-sensitive channels) and also possibly from larger instrumental biases.

The temperature channels used in the zenith mode are only slightly modified as the deviations from the background values are much smaller than for the other channels. During the second half of the day, characterized by the presence of clouds around 800 hPa (see Fig. e and f), the residual values are largely reduced in the frequency bands sensitive to liquid water for channels 6, 7 and 8, especially for EXP as shown by the comparison of the pixels in the dashed rectangular boxes in Fig. b and c. Residuals are also slightly reduced for EXP in the morning and during the fog and low temperature period for the first five channels (1, 2, 4–6) between 02:00 and 08:00 UTC.

In order to quantify these results for the 9 February 2020 dataset (all hours and all channels), the bias and root mean square error (RMSE) values are computed for the background and the analyses produced by REF and EXP. The innovations are characterized by a RMSE of 3.20 K and a bias of 1.32 K. Both assimilation experiments reduce these two quantities by modifying model profiles. The RMSEs are 0.71 K for EXP and 0.72 K for REF and the biases are -0.17 K for EXP and -0.11 K for REF. These statistics have also been calculated by restricting the dataset to the two dashed rectangular boxes presented in Fig. b and c. A significant improvement is observed for the most sensitive channels to liquid water in the afternoon with the RMSE decreasing from 4.3 K in the background to 0.57 K in REF and 0.37 K in EXP. For all computed statistics, EXP always provides the best performance in terms of RMSE. Table  summarizes the bias and RMSE values obtained for the different samples.

After examining the fit of the two experiments to the observed TBs, we assess the corrections made in model space. Figure  shows the increments of (a, b) temperature, (c, d) specific humidity and (e, f) liquid water for the two experiments REF (left panels) and EXP (right panels). In addition, the increments of (θs)a are shown in panels (g)–(h).

The temperature increments are mostly located in the lower troposphere (below 650 hPa) with a dominance of negative values of small amplitude (around 0.5 K). This is consistent with the negative innovations observed in the temperature channels, highlighting a warm bias in the background profiles. The areas of maximum cooling take place in cloud layers (inside the thick fog layer below 900 hPa until 09:00 UTC and around 700 hPa after 12:00 UTC). The increments are rather similar between REF and EXP, but the positive increments appear to be larger with EXP (e.g. at 08:00 and 20:00 UTC around 800 hPa).

Concerning the profiles associated with moist variables, the structures show similarities between the two experiments but with differences in intensity. During the night and in the morning, the qv increments near the surface are negative. These negative increments are projected into increments having the same sign as T by the strong positive cross-correlations of the Bfog matrix up to 900 hPa (Fig. ). Thus, the largest negative temperature and specific humidity increments remain confined in the lowest layers.

Liquid water is added in both experiments between 03:00 and 07:00 UTC, close to the surface, where the Jacobians of the most sensitive channels to ql (6 to 8) have significant values in the fog layer present in the background (see Fig. e). After 14:00 UTC, values of qv between 850 and 700 hPa and ql around 800 hPa are enhanced in both cases, with larger increments for the REF case, in particular at 20:00 UTC and around 24:00 UTC (midnight). Most of the liquid water is created in low clouds. Additionally, increments of ql above 600 hPa are larger and more extended vertically and in time in EXP, where condensation occurs over a thicker atmospheric layer between 500 and 300 hPa after 12:00 UTC. In the REF experiment, the creation of liquid water above 500 hPa only reaches values of 0.3 g kg-1 sporadically, for example at 21:00 UTC. In this experimental set-up, condensed water can be created or removed over the whole column by means of the supersaturation diagnosed at each iteration of the minimization process (since RTTOV-gb needs (T,qv,ql) profiles for the TB computation). This is a clear advantage of EXP over REF, which keeps the vertical structure of the ql profile unchanged from the background. In REF, liquid water is only added where it already exists in the background because once the LWP variable is updated, the analysed ql profile is just modified proportionally to the ratio between the LWP of the analysis and of the background, as explained in more details by .

The profiles of increments for (θs)a show structures similar to the increments of qv around 800 hPa and to the increments of T below, where temperature Jacobians are the largest (see Fig. 7 in ). The conversion of T, qv and qt changes obtained with REF into (θs)a increments (Fig. g) highlights the main differences between the two systems. They take place around 800 hPa with larger increments produced by the new 1D-Var particularly between 11:00 and 14:00 UTC.

Some radiosoundings (RSs) have been launched during the SOFOG3D IOPs. As only one RS profile was launched at 05:21 UTC in the case study presented in the article, no statistical evaluation of the profile increments can be carried out. However, we have conducted an evaluation of the analysis increments obtained at 05:00 and 06:00 UTC (the 1D-Var retrievals were performed at a 1 h temporal resolution in line with the operational AROME assimilation cycles) around the RS launch time. As the AROME temperature background profile extracted at 06:00 UTC was found to have a vertical structure closer to the RS launched at 05:21 UTC, Fig.  compare the AROME background profile and 1D-Var analyses performed with the REF and EXP experiments valid at 06:00 UTC against the RS profile.

The temperature increments are a step in the right direction by cooling the AROME background profile in line with the observed RS profile. The two 1D-Var analyses are close to each other, but the EXP analysis produces a temperature profile slightly cooler compared to the REF analysis. In terms of absolute humidity (ρv=pv/(RvT), with pv the partial pressure and Rv the gas constant for water vapour), the background profile already exhibits a similar structure compared to the RS profile. The 1D-Var increments are thus small and close between the two experiments. However, we can note that the EXP profile is slightly moister than the REF profile from the surface up to 3500 m, which leads to a somewhat better agreement with the RS profile below 1500 m. In terms of integrated water vapour (IWV), a significant improvement of the background IWV with respect to the RS IWV is observed with the difference reduced from almost 1 kgm-2 in the background to less than 0.4 kgm-2 in the analyses. These analyses confirm the improvement brought to the model profiles by both the REF and EXP analysis increments, with some enhanced improvement for EXP.