Centre Nationale de Recherches Meteorologique, Groupe de Meteorologie Moyenne Echelle, Modélisation et Initialisation de la Convection en synergie avec l'Analyse des Données Observées, Météo-France, Toulouse, France
Laboratoire de Géophysique Interne et de Tectonophysique, Maison des Géosciences, CNRS, Université Joseph Fourier, Grenoble, France
Centre Nationale de Recherches Meteorologique, Groupe de Meteorologie Moyenne Echelle, Modélisation et Initialisation de la Convection en synergie avec l'Analyse des Données Observées, Météo-France, Toulouse, France
Centre Nationale de Recherches Meteorologique, Groupe de Meteorologie Moyenne Echelle, Modélisation et Initialisation de la Convection en synergie avec l'Analyse des Données Observées, Météo-France, Toulouse, France
 Estimations of zenith total delays (ZTD) were obtained during postprocessing of a high-resolution (2.4 km) nonhydrostatic atmospheric model (Méso-NH). These estimations were used to determine their sensitivity with respect to formulations of atmospheric refractivity, the approximation of zenith hydrostatic delays (ZHD) deduced from ground pressure, and the contributions of hydrometeors. The factor κ for the conversion of zenith wet delay (ZWD) to integrated water vapor (IWV) was examined. Méso-NH is applied here to the extreme flash flood event of 8–9 September 2002 in southeastern France. The use of the hydrostatic formulation (to infer ZHD) leads to an overestimation of up to 18 mm with respect to the vertical integration of refractivity. Delay contributions of hydrometeors simulated by the high-resolution model reached more than 70 mm (≈11 kg/m2 IWV) in the heart of the convective cells in the case of the extreme flood event. The mean variations of IWV due to the use of different conversion factors (κ used to transform ZWD to IWV) are evaluated to be less than 0.3 kg/m2. This is less than the mean underestimation of IWV by 0.6 kg/m2 relative to the GPS-like evaluation of IWV using the hydrostatic formulation and the ground temperature. In this study we also use GPS ZTD observations to validate three different numerical simulations of this extreme flood event. The simulation with the best fit to the GPS observations is also in best agreement with the surface rainfall measurements.
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 This paper contributes to the studies of the “Observatoire Hydrométéorologique Méditerranéen-Cévennes-Vivarais” (OHM-CV) [Delrieu et al., 2005], which aims to understand and improve the forecast of frequent flash flood events over the Cévennes-Vivarais region close to the Mediterranean coast in southeastern France (100 km to the northwest of Marseille, Figure 1). In this framework, the 8–9 September 2002 extreme flash flood event has been simulated with a high-resolution nonhydrostatic model (Méso-NH) [Ducrocq et al., 2004]. This extreme flash flood event took place over southeastern France (Gard region) and was characterized by an extreme precipitating convective event, exhibiting both high hydrometeor contents and nonhydrostatic effects due to convective upward and downward motions. Méso-NH simulations of this event have been postprocessed to quantify of the sensitivity to the formulation of the tropospheric refractivity, from which the GPS ZTD and IWV are derived. We have especially examined the impact of nonhydrostatic effects and of hydrometeors on ZTD estimation.
 The evaluation and the understanding of atmospheric refractivity has been studied by several authors [e.g., Boudouris, 1963; Bean and Dutton, 1966; Owens, 1967; Saastamoinen, 1973a, 1973b, 1973c]. Generally, they describe microwave propagation with three atmospheric refractivity coefficients (N = N(k1, k2, k3)), in an atmospheric medium evolving with a temperature, T, an absolute pressure, P, and a water vapor partial pressure, e. Several sets of (k1, k2, k3) values have been proposed in the past [Smith and Weintraub, 1953; Essen and Froome, 1963; Thayer, 1974; Hasegawa and Stokesberry, 1975; Bevis et al., 1994]. Therefore, as a first step in our study, uncertainties of ZTD estimations referring to different physical expressions of refractivity have been evaluated. An expression for the atmospheric coefficients that depends on pressure and temperature, following Saastamoinen [1973b], has also been tested. Then, as proposed by [Kursinski et al., 1997; Solheim et al., 1999; Hajj et al., 2002], a contribution induced by the hydrometeors has been added in the refractivity formulation and its impact on the ZTD estimation evaluated. In a second step, sensitivity of the relationships between ZTD, ZWD, and IWV have been studied referring to Bevis et al.  and Emardson and Derks . The conversion formulae are dependent on site or region. With the high-resolution nonhydrostatic simulations we can examine the validity of the formulae used to derive IWV from ZTD and their associated assumptions (hydrostatic state, surface temperature dependency). Then, knowing the sensitivity to the refractivity formulations and to the IWV-ZWD-ZTD formula, section 3 illustrates how GPS measurements from a mesoscale network may be helpful in validating high-resolution simulations, with differing initial conditions.
 In the following, we introduce the GPS data and analysis, the flash flood event, the Méso-NH simulations and the assessments of ZTD, ZWD and IWV from model outputs. Then, the results of the sensitivity tests are shown as one outcome of our study. The second outcome is the comparison of simulated ZTD with GPS measurements for the validation of the simulations of 8–9 September 2002 event.
2. Data and Numerical Simulations
2.1. GPS Network and Data Analysis
 Data from 35 GPS stations have been processed (Figure 1): stations AIGL, CHRN, FCLZ, GINA, JOUX, MICH, MTPL, NICE, SJDV and SOPH from the REGAL (Réseau GPS permanent dans les Alpes) network (http://kreiz.unice.fr/regal/), stations AJAC, EGLT, GRAS, MARS and TLSE from the national French permanent network RGP (http://lareg.ensg.ign.fr/RGP/index.html), stations SJDS and VERC from the semipermanent network VENICE in south of France [Masson et al., 2003], stations BELL, CREU, EBRE, LLIV from the regional Spanish network CATNET (http://draco.icc.es/geofons/catnet/en/home.php), and station MAHO in the Baleares maintained by the Royal Observatory of Spain; stations ALAC, BOR1, BRUS, CAGL, GRAZ, MALL, MATE, ONSA, POTS, TORI, VILL, WTZR and ZIMM from the EUREF network (http://www.epncb.oma.be/index.html) are included in the data analysis as fiducial stations for the realization of the reference frame. The GPS data analysis has been performed using the GAMIT software (version 10.07, King and Bock ). The primary analysis provides precise coordinates for the local stations for each 24 h of measurements. In this step, tropospheric parameters have been estimated with a 2 hourly resolution. The repeatabilities of the unconstrained daily GAMIT solution for all baseline components are 1.2 mm, 2.1 mm and 4.7 mm for the north, east and vertical components, respectively. The final positions of the stations in the ITRF2000 reference frame [Altamimi et al., 2002] are obtained in a global solution using the Kalman filter, GLOBK [Herring et al., 1990]. The reference frame is established by constraining the positions of 13 fiducial GPS stations to their ITRF2000 values. In a secondary analysis, zenith delays are calculated every 30 minutes and horizontal tropospheric gradients (NS and EW components) are estimated hourly. Baselines greater than 2000 km have been used in order to decorrelate the tropospheric parameters from vertical position estimations [Tregoning et al., 1998]. Loose constraints have been applied on the station coordinates obtained in the primary analysis (1 m on horizontal and 2 m on vertical). The tropospheric parameters of the ambiguity free solution have been used. ZTD measurements have been produced using a sliding window strategy with sessions of 24 hours of data shifted by 12 hours. Only the middle 12 hours of each session have been retained (for more details, see Champollion et al. ).
2.2. The 8–9 September 2002 Flash Flood Event
 On 8–9 September 2002, a heavy precipitating system affected the Gard region (the region including Nîmes, Anduze, and Orange; see Figure 1): 24 people were killed during this event and the economic damage is estimated at 1.2 billion euros [Huet et al., 2003]. Delrieu et al  proposed a detailed description of the meteorological and hydrological event. This paper describes a brief overview. The meteorological environment was characterized by an upper cold low pressure, centered over Ireland, that extended medicinally to the Iberian Peninsula and generated a southwesterly diffluent flow over southeastern France on 8 September 2002 (Figure 2), that progressively evolved in a southerly flow during the night of 8–9 September. Associated with this upper level flow a surface front undulated over western France. Convection formed well ahead of the surface cold front, in the warm sector, where a moist low-level southeasterly flow prevailed (see the window in Figure 2a for the location of the heavy precipitation). Prior to the development of the convection, the atmosphere was conditionally unstable in this region as shown by the midnight Nîmes radiosounding of the 8 September (CAPE of the most unstable parcel about 850 J/kg). IWV computed from this radiosounding shows that the water vapor content of the atmosphere was already high (33 kg/m2). This value is in the upper ten percent of the distribution of IWV for the Nîmes sounding for September months between 1994 and 2003, the average value being of 22 kg/m2. After the onset of convection, the 1200 UTC sounding is almost saturated with a water precipitation value of 39 kg/m2 reaching the upper five percents of the distribution of IWV for the 1994–2003 Nîmes soundings.
 Triggered over the Mediterranean Sea during the early morning, the convective cells progressed northward to form inland a quasi-stationary mesoscale convective system (MCS) over the Gard region after 0800 UTC on 8 September. The quasi-stationary MCS stood over the same region until the following morning and then evolved eastward with the surface front. High surface rainfall was recorded over the Gard department, with a maximum of recorded daily precipitations that has reached about 700 mm. Figure 3 presents the accumulated rainfall from the Nîmes radar for the three phases of the rainfall event as identified by Delrieu et al : (1) during phase I (prior to 2200 UTC on 8 September, Figure 3a), the precipitation induced by the MCS were mainly over the plain region of the Gard department; (2) then phase II (between 2200 UTC on 8 September and 0400 UTC on 9 September, Figure 3b) was characterized by a shift of the MCS toward the upper regions at the limit of the mountain ridge (near Anduze), where it merged with the surface front that had progressed eastward during the same period; (3) during phase III (after 0400 UTC on 9 September, Figure 3c), the front with the embedded convection moved eastward and again swept over the Gard plain region. Figure 4 shows the temporal evolution of the hourly rainfall for two rain gauges; one (Anduze station) corresponds to the region where the maximum daily surface rainfall was recorded. The second one (Orange station) is situated 60 km eastward and north of the CHRN GPS station (see Figure 1 for locations). The Orange station recorded significant precipitation during phase I, then rainfall weakened when the precipitating system moved westward over the upper regions (phase II), before again showing rainfall peaks corresponding to the front passage (phase III). The temporal evolution of ZTD at CHRN station is well correlated with the precipitation evolution at Orange and also shows the three phases. At Anduze, significant rainfall occurred mainly from the end of phase I to the beginning of phase III. In less than 9 hours, 500 mm were recorded at Anduze. No rainfall was observed after 0900 UTC on the 9 September, the front and associated convection had already evacuated the Anduze region.
2.3. Characteristics of the Méso-NH Simulations
 The 8–9 September event has been simulated with the nonhydrostatic Méso-NH model. A comprehensive description of this model is given by Lafore et al. . The simulations were performed using two nested grids (Figure 1) interacting with each other according to a two way interactive grid-nesting method [Clark and Farley, 1984; Stein et al., 2000]. The horizontal resolution of the two domains are 9.5 and 2.4 km, respectively. In the following, only the delays and IWV for the 2.4-km domain are discussed. The vertical grid is defined by a stretched vertical coordinate [Gal-Chen and Sommerville, 1975], with 40 vertical levels spaced by 75 m in the lowest levels to 900 m at the top of the model which is at about 20 km. The prognostic variables are the three-dimensional wind components, the potential temperature, the mixing ratios of six water variables (vapor, cloud water, rainwater, primary ice, graupel, snow) and the turbulent kinetic energy. A bulk microphysical scheme [Caniaux et al., 1994; Pinty and Jabouille, 1998] governs the equations of the six water species. Convection is explicitly resolved for the inner domain (no convective parameterization scheme).
 In this study, three experiments have been considered. The simulations differ only by their initial conditions [Ducrocq et al., 2002; Chancibault et al., 2006]. The first one (ARP12 experiment) starts from the analysis of the large-scale global ARPEGE system (ARPEGE for Research Project on Small and Large Scales, Météo-France NWP system). For the second one (RAD12 experiment), the mesoscale initialization procedure of Ducrocq et al.  has been applied. It is composed of a mesoscale surface observation analysis and an adjustment of water vapor and hydrometeor contents based on the radar reflectivity and infrared Météosat brightness temperature valid for 8 September at 1200 UTC. The mesonet surface observations, which are on average spaced by about 30 km, are analyzed by an optimal interpolation analysis that has been tuned for the mesoscale [Calas et al., 2000; Ducrocq et al., 2000]. Then, a cloud and precipitation analysis based on the radar and satellite data updates the water vapor by imposing saturation inside cloudy regions, and adds rainwater[snow] below[above] the freezing level according to the reflectivity values. For the AMA12 experiment, the water vapor and hydrometeor adjustment is not applied; its initial state is simply obtained from the mesoscale surface data analysis. The background to the mesoscale initialization procedure is provided by the 1200 UTC ARPEGE analysis, so that the initial conditions of ARP12 and AMA12 differ only in the boundary layer, whereas middle and upper tropospheric moisture and hydrometeors are added inside the observed cloudy and rainy regions in the initial conditions of RAD12 with respect to AMA12. Chancibault et al.  have performed a hydrological validation of these experiments and showed that high-resolution simulations improve the amount of surface rainfall compared to the actual operational models. In addition, the mesoscale initialization procedure improves significantly the location of the MCS during the phase I of the event.
2.4. Assessments of Zenith Delays and Integrated Water Vapor Content From Méso-NH Outputs
 Computation of zenith delays and IWV has been incorporated in the postprocessing of the Méso-NH model. The synthetic delays and IWV have been computed at each column of the 2.4 km domain, providing 2D fields of these parameters. An estimation of the synthetic delays at GPS station locations inside the Méso-NH domain has also been developed. For that purpose, a bilinear interpolation between the four closest grid columns is applied and the differences between the model orography and the true station altitude are also taken into account. For the eighteen GPS stations inside the inner Méso-NH domain (Figure 1b), only VERC is below the model orography of the 2.4-km domain, with a departure of only 25 m. All the other stations are above the model orography, with a maximum difference for AIGL (436 m). An altitude correction has to be considered (for GPS sites in mountainous regions essentially). However, such corrections are not straightforward, especially when extrapolation below the model orography is required. We have applied an altitude correction only when the real station height is above the model orography, by removing the contributions to the vertical integration below the height of the GPS sites. In our simulations, such corrections may induce a delay reduction of up to 13.5 cm for AIGL. No correction is proposed for the stations below the model orography in this study (only VERC is concerned). However, a correction can be considered for future work. Extrapolation methods such as the one proposed by Vedel et al.  that assumes a hydrostatic state of the atmosphere, constant relative humidity and a constant temperature lapse rate could be retained.
2.4.1. Retrieval of Zenith Delay
 The zenith total delay may be expressed as
where Rd = (287.0586 ± 0.0055) J/(kmol K) is the specific molar gas constant for dry air, Rw = (461.525 ± 0,013) J/(kmol K) the specific molar gas constant for water vapor, ρh, ρw are the densities of moist air and water vapor, P, e are the total pressure and the partial pressure of water vapor, Tv is the virtual temperature, k1, k2, k3 and k′2 = k2 − k1 are refractivity coefficients, and Nlw, Nice are respective contributions of liquid and ice water to refractivity. The first term after the second equals sign represents ZHD, the second term represents ZWD, and the third term represents zenith hydrometeors delay (ZHMD). For the Méso-NH model, mixing ratios of cloud and water rain are available for liquid water components, and mixing ratios of pristine ice, snow and graupel for ice water components.
 Zenith hydrostatic delay (ZHD) represents the contribution of the total atmospheric density to ZTD (including water vapor density). Zenith wet delay (ZWD) is the specific additional contribution of atmospheric water vapor to ZTD. The Méso-NH microphysics permits to describe five classes of hydrometeors. The zenith hydrometeors delay (ZHMD) depends on their total density. Appendix A presents the expressions of Nlw and Nice which allow the assessment of the ZHMD contribution to ZTD. Note that the hydrostatic assumption has not been used to establish Zenith hydrostatic delay (ZHD) in equation (1), according to the Méso-NH equation system.
 In the case of the analysis of GPS data, the hydrostatic equilibrium is generally assumed so that ZHD can be evaluated from observations of surface pressure [Saastamoinen, 1972; Davis et al., 1985]:
where k1 can be fixed to a constant value (0.7760 ± 0.0005 K/Pa), PS is the surface pressure and gm is the gravity in the center of the atmospheric column following Saastamoinen  (gm = 9.784 × (1 − 0.0026 cos (2λ) − 0.000279 H) with H the height of the GPS station and λ its latitude). When the model vertical column is in hydrostatic equilibrium, ZHD from equation (1) is reduced to equation (2), except that the gravity component in the center of the column gm is replaced by the gravity used in the Méso-NH model (i.e., g0 = 9.807 ms−2).
 The Méso-NH model provides prognostic variables (pressure, temperature and mixing ratio) at the middle of its vertical layers. The vertical integration to estimate ZHD, ZWD and ZHMD is made up by the accumulation of each layer's contribution from ground surface to the uppermost layer in the model. Beyond the uppermost layer (i.e., Ztop ≈ 20 km), the mixing ratio of water vapor is weak, as are the hydrometeor contents. Therefore neglecting the contribution of ZWD and ZHMD to ZTD above the uppermost layer of the model is legitimate. However, the contribution of ZHD outside the model (ZHDout = ∫Ztop∞(k1) dz) cannot be neglected as it provides significant contributions up to an altitude of approximately 80 km [Vedel et al., 2001]. The hydrostatic equilibrium can be assumed above the top of the model and thus ZHDout is reduced to the hydrostatic formulation (equation (2)) using the pressure and the gravity acceleration at the top of the model.
χ and β can be considered in good approximation as constant values for the given frequencies of the L band (L1, L2 or LC). We suggest χ = 0.7755 K/Pa and β = 1.3 × 10−7 K/Pa. Constant values of k2 = (0.704 ± 0.022) K/Pa and k3 = (373900 ± 1200) K2/Pa have been used [Bevis et al., 1994].
2.4.3. ZTD-ZWD-IWV Relations
 Integrated water vapor retrieval from GPS data (IWVGPS) is commonly done by isolating zenith wet delays (ZWDGPS) from GPS-measured zenith total delays (ZTDGPS). For that, the zenith hydrostatic delay is assessed using surface pressure measurements equation (2)), and then subtracted from ZTDGPS. The inferred ZWD is converted into IWV via a proportionality factor κ [Hogg et al., 1981]:
with ZWD in m, and IWV in kg/m2.
 The proportionality factor κ is a function of the atmospheric temperature profile [Askne and Nordius, 1987] (referred to as κA&N here after):
The Méso-NH model offers the possibility to compute directly the “true” integrated water vapor (IWVMeso-NH) by vertical integration of the water vapor content (ρw) through the model vertical layers:
3. Sensitivity Tests on Zenith Delay Formulation
 In this section, the sensitivity of the zenith delay formulations to various factors (expression of refractivity coefficients, hydrometeor contributions, hydrostatic assumption, conversion of ZWD into IWV) is evaluated, based on the simulations of the 8–9 September 2002 event. To help synthesizing the results, statistical parameters have been computed (mean biases and standard deviations). They concern the calculation of delays and IWV, evaluated at each column within the model and at an hourly step between 1200 UTC, 8 September 2002, and 0600 UTC, 9 September 2002, resulting in a population of 240 × 240 × 18 elements. A reference for the delay formulations is defined (equation (1) with refractivity constants from Bevis et al.  and no hydrometeor contributions) and the statistics aim at documenting the departures from this reference by a given sensitivity test. The statistical parameters are presented for the AMA12 simulation; results for the two other experiments (ARP12 and RAD12) do not differ significantly from AMA12 ones. When analyzing these statistical parameters, one must bear in mind that from a meteorological point of view, 6 millimeters of ZWD correspond approximately to 1 kg/m2 of IWV which is the limit of resolution of standard meteorological water vapor measurements.
3.1. Refractivity Coefficients
 For a complete analysis of the sensitivity of the ZTD evaluation to the formulation of refractivity, different sets of refractivity constants k1, k2, k3 proposed in the literature have been examined (two expressions of Smith and Weintraub , two and three coefficients from Essen and Froome , Thayer , Hasegawa and Stokesberry , and Bevis et al. ). Our tests show that there are no significant differences in zenith delay evaluations between each set of constants (a mean ZTD bias of less than 2 mm and a maximum difference of less than 3.5 mm) except for the expression of ZTD from Smith and Weintraub  with only two refractivity coefficients which is commonly used in GPS applications. The mean bias reaches nearly 12 mm and the maximum bias is 12.6 mm (overestimation). The significant overestimation is due to the approximation which has been made in obtaining one coefficient rather than two to formulate the water vapor contribution to the refractivity. The refractivity constants are given with uncertainties by the different authors. Considering the upper bounds of the uncertainty range leads to average biases of 2 mm for the Bevis et al.  constants. Note that the ZTD estimates with different sets of refractivity constants are all included in the uncertainty range of Bevis et al.  estimations, except for the two constants expression of Smith and Weintraub .
Figure 5 presents the temporal evolutions of ZHD and ZTD at CHRN station for k1(Pd, T) given by equation (3) and the Bevis et al.  set of constants based on the RAD12 experiment. Departures between the use of a constant value of k1 and the expression k1(Pd, T) reach at most 5 mm. The statistical parameters computed on the AMA12 experiment (Table 1, column 2) show that the mean bias is approximately 2 mm, for a maximum departure of 5 mm (6 mm for RAD12 and 5 mm for ARP12).
Table 1. Statistical Results for ZHD From AMA12 Simulationa
The second column shows ZHD from equation (1) with k1(Pd, T) given by equation (3). The third column shows from equation (2), with gm from Saastamoinen . The fourth column shows from equation (2), but with gm replaced by g0. The mean bias (ΔZHDmean), the standard deviation (δZHD), and the minimum and maximum departures and are evaluated using as reference ZHD from equation (1) with the Bevis et al.  constants set. The statistics consider the 18 hours of simulations and the 240 × 240 grid points.
 To sum up, results show a weak sensitivity to the refractivity coefficients, with differences in the domain of uncertainty of the water vapor measurements. In the following, k1(Pd, T), k2 and k3 of Bevis et al.  are used.
3.2. Hydrostatic Formulation Versus Nonhydrostatic Formulation of ZHD
 ZHD is computed by vertical integration through the model grid thermodynamic profiles (equation (1)). It is compared with the commonly used formulation of ZHD deduced from ground pressure data equation (2)). These formulations of ZHD differ in two aspects: the gravity constant and the hydrostatic assumption. On the one hand, assuming an hydrostatic state, ZHD from equation (2) scales by gm/g0 the vertical integrated ZHD value. On the other hand, the nonhydrostatic equation system of Méso-NH allows us to fully study the impact of using or not an assumed hydrostatic equilibrium. Given the same gravity constant, it is therefore possible to isolate this impact of using an assumed hydrostatic equilibrium. For the heavy precipitation event simulated, the atmosphere departs from the hydrostatic equilibrium inside the vigorous deep convective cells as well as, to a lesser extent, within the gravity waves induced by the convection. The difference between the two formulations of ZHD (assuming the same gravity constant) is shown at 1500 UTC, 8 September 2002, on Figure 6a. Figure 6b presents the model reflectivity at the same time; only high values of reflectivity are shown in order to highlight the convective part of the rainfall system. Inside the intense convective cells (i.e., areas with reflectivity above about 40 dbZ), the pressure tends to be weaker than the hydrostatic pressure, resulting in larger delays for the hydrostatic formulation than for the integrated model one. The maximum difference of ZHD reaches 18 mm at 1500 UTC. The pressure is also larger than the hydrostatic pressure in some locations affected by gravity waves, leading to weaker delays for the hydrostatic formulation than for the integrated model one. The absolute values of the departure from the hydrostatic equilibrium are however weaker than inside the convective cells.
Figure 5 displays the temporal evolution of ZHD at the CHRN station (bottom part of the graph) for the hydrostatic formulation including two sources of differences (i.e., gravity constant and hydrostatic assumption) from the model integrated estimation (equation (1)). The ratio between gm and g0 is about 0.997 for the CHRN station, which leads to a mean shift between the curve and the model integrated ZHD one of about 6 mm, except between 1400 and 1700 UTC on 8 September 2002 where the departure is slightly larger. Indeed, between 1400 and 1700 UTC, 8 September 2002, the Méso-NH model simulates convective cells in the region of CHRN, leading to nonhydrostatic vertical profiles at CHRN. Therefore, for that period, the departure between and the model integrated ZHD is increased, reaching 9 mm. The statistical parameters confirm the overestimation by the hydrostatic formulation. Table 1 presents statistical results for both sources. Column 3 evaluates the differences between the hydrostatic formulation with a gravity constant defined by gm [Saastamoinen, 1972] and the integrated formulation of ZHD in Méso-NH. The mean departure is about 6 mm. Column 4 evaluates the differences using g0 for the hydrostatic formulation, thus isolating the impact of the hydrostatic assumption. The departures due to the hydrostatic assumption range from −3.7 mm to 18 mm. This highlights potential errors in the estimation of ZWD and IWV extracted from GPS ZTD measurements inside observed convective systems. Using a constant gravity in ZHD computation allows to be consistent with the NWP model equations, but it induces a bias with respect to the estimation of ZHD using a more realistic variable g like the one proposed by Saastamoinen (i.e., gm). As a first approach, the bias can be corrected by scaling the model ZHD, by even though gm is also based on assumptions derived from climatology which may be not valid for such extreme situations as the one considered here.
3.3. Hydrometeor Contributions to Zenith Delay (ZHMD)
 In this section, the contributions of liquid water and icy hydrometeors to zenith delays (ZHMD) are estimated. Figure 5 (upper plot) shows the ZTD at the CHRN station including the ZHMD. It can be seen that ZHMD can reach more than 20 mm in the afternoon of 8 September 2002. These large contributions are mainly located inside the convective part of the MCS as can be seen on Figure 7 which displays ZHMD at 2100 UTC on 8 September 2002. ZHMD reaches more than 50 mm in the heart of the convective part of the simulated precipitation system which is materialized by the synthetic reflectivities displayed on Figure 7b. During the 18 hours of simulation, the maximum of ZHMD attains 70 mm at 1500 UTC 8 September. The contributions induced by liquid water species are generally 10 times larger than the icy species ones.
Figure 8 shows the temporal evolution of the surface in the model concerned with significant hydrometeor contributions, splitting in four classes (5–10 mm, 10–15 mm, 15–20 mm and >20mm). Clearly the areas concerned by large ZHMD contributions are correlated with the convective activity of the system. Between 1400 and 1800 UTC, 8 September 2002, and after 0300 UTC, 9 September 2002, the surface with ZHMD contributions larger than 20 mm reach more than 1000 km2. A surface of about 5000 km2 is concerned by ZHMD contributions larger than 10 mm during the afternoon of 8 September. Therefore, for such torrential rain event, the hydrometeor contributions are far from being negligible.
3.4. Relationship Between ZWD and IWV
 Several authors have proposed to approximate κ by a linear function of the surface temperature, which is more attractive for GPS analyses as only observations of surface temperature are required instead of vertical temperature profile observations (see section 2.4.3 for the definition of κ and the formulation of κA&N). Bevis et al.  established such a relation depending on surface temperature from a global climatology of radiosoundings (referred to as κBevis), while Emardson and Derks  determined site- and region-dependent relations taking into account more than 120000 radiosoundings from 38 sites in Europe, with among them a specific relation for the Mediterranean region which is our region of interest (referred to as κE&D):
We now examine the differences between, on the one hand, IWV deduced from model ZWD and applying κA&N, κBevis or κE&D as defined by equations (5), (7) and (8), and, on the other hand, from the model value of integrated water vapor (IWVMeso-NH). The three first lines of Table 2 display the mean bias, the standard deviation as well as the maximum and minimum differences between the different IWV estimations with κA&N, κBevis or κE&D and IWVMeso-NH. IWV obtained by the conversion with the κ given by Askne and Nordius  corresponds almost exactly to IWVMeso-NH. Figures 9a and 9b show the differences between IWVMeso-NH and and between IWVMeso-NH and respectively, i.e. the two formulations depending on surface temperature. Figure 9c presents the model IWVMeso-NH. These values are calculated for the AMA12 experiment at 1500 UTC on 8 September 2002. IWVMeso-NH reaches more than 45 kg/m2, implying ZWDMeso-NH values of more than 300 mm in the Gard region. We can see weaker values over the relief, and a moist area over the Mediterranean Sea and its littoral feeding the convective system. Figure 9a considers κ calculated with the expression of Tm (TS) given by Bevis et al.  based on a global climatology. It exhibits a highly variable differential field covering an amplitude of 2.7 kg/m2. The high variability of this field may arise from surface temperature variability induced by orography and land cover. Over the sea, the smoother surface temperature and the zero orography result in a smoother differential field. The mean difference is about 0.08 kg/m2 (Table 2), whereas the maximum positive shift reaches 2 kg/m2 and the maximum negative shift −0.6 kg/m2. Figure 9b considers κ calculated with the expression of Emardson and Derks , a Mediterranean specific climatology. With respect to the conversion with κBevis, this one provides a smoother differential field with lower amplitudes of the variations (an interval of 1.5 kg/m2 is covered). The conversion over the continental area yields IWV values close to IWVMeso-NH. However, over almost all the sea surface, the differences of the IWV evaluations reach more than −0.5 kg/m2. This results in a mean bias of −0.24 kg/m2 (higher than that of Bevis et al.), but the maximum positive shift reaches only 0.45 kg/m2 and the maximum negative shift only −0.86 kg/m2 (lower than that of Bevis et al.). The ZWD to IWV conversion formula of Emardson and Derks  appears more adapted to our study than the one of Bevis et al. . However, the comparison of these two conversions show very weak differences in IWV. Only sparse sites present more than 1 kg/m2 of IWV differences between the Bevis et al.  and the Emardson and Derks  conversion, without correlation with the location of the IWV maximum.
Table 2. Statistical Results of Different Methods of Conversions of ZWD Into IWV or Standard (GPS Stand-Alone) Extractions of IWV From ZWD (via ZHD Estimated From Ground Pressure)a
Type of Conversion
Statistics have been computed for the AMA12 experiment, with the set of refractivity constants of Bevis et al. . Estimation of the mean difference ΔIWVmean and the standard deviations δIWV between deduced IWV and model IWV has been performed. The maximum positive and negative and shifts are also presented. Values are in in kg/m2.
(ZWD + ZHMD) IWV
(ZWD + ZHMD) IWV
ZWDGPS (with ZHMD) IWV
ZWDGPS (with ZHMD) IWV
ZWDGPS (without ZHMD) IWV
ZWDGPS (without ZHMD) IWV
 In rows 4 and 5 of Table 2, the conversion of the sum of ZWD and ZHMD into IWV has been estimated, simulating the fact that in GPS stand-alone IWV measurements ZWD and ZHMD cannot be distinguished. The statistical results show mean biases of less than 0.2 kg/m2, but the extreme values reach more than 10 kg/m2. That means there is a risk of bad conversion of ZWD into IWV with κ inside intense precipitation systems when a distinction between ZWD and ZHMD is not available, as is generally the case for GPS measurements of IWV.
 The different error sources of the standard GPS IWV extraction are now compared in more detail. Model estimated ZTD has been reduced by determined from ground pressure to ZWD, then the extracted ZWD is converted into IWV by κBevis or κE&D. Four more statistical results concerning this standard GPS IWV extraction are presented in Table 2 (rows 6–9). Model ZTD is always the sum of ZWD, ZHMD and ZHD. In rows 6 and 7, IWV extractions from (for GPS stand-alone measurements inseparable) ZWD + ZHMD contributions are estimated (IWV = κ (ZWD + ZHMD)). Extraction with κBevis and κE&D are quasi equivalent: the mean biases are less than 0.6 kg/m2. This degradation with regard to rows 4 and 5 is due to the ZHD determination from PS with the hydrostatic formulation. However, as for rows 4 and 5 of Table 2, including hydrometeor contributions in the IWV conversion induces high extreme values of the differences with respect to model IWV (close to 8 kg/m2). This lower value with respect to the approximately 11 kg/m2 in rows 4 and 5 (where integrated ZHD was used to separate ZWD from ZTD) is due to the overestimation of ZHD and the subsequent underestimation of ZWD by the hydrostatic formulation. In rows 8 and 9, IWV extractions from ZWD without ZHMD contribution (IWV = κ ZWD) are proposed. Information about ZHMD necessary to separate it from GPS deduced ZWD as suggested in this test could be provided by polarimetric radar measurements. The mean biases are similar to the previous values in lines 6 and 7, but the extreme values are limited to a little more than 3 kg/m2.
 The determination of IWV from GPS ZTD is sufficiently precise in average to yield significant observations for the assimilation of IWVGPS in NWP, considering that 1 kg/m2 of IWV is the limit of resolution of standard meteorological water vapor measurements. All mean differences of the different retrieval strategies with respect to model IWV presented in Table 2 are below 0.6 kg/m2. The larger part of these differences in the estimation of IWV from GPS-like strategy is due to the ZHD approximation with the hydrostatic formulation, not to the ZWD to IWV conversion with κ(TS). Special attention has to be paid in strong precipitation areas as large contributions to the delay due to the hydrometeors may be included in the GPS deduced ZWD.
4. Validation of the Méso-NH Simulations With GPS ZTD
 In this section, the zenith delays simulated by the three numerical experiments (ARP12, RAD12 and AMA12) are compared with the observed GPS ZTDs. This will allow us to quantify the impact of the three different initial conditions of the experiments on the delay estimation, and therefore on the value of integrated moisture throughout the troposphere. We will also verify that the simulation with the best fit to the GPS measurements is also the one simulating the best precipitation field. Figure 10 shows the accumulated surface rainfall during phase I of the event from the three numerical experiments, superimposed with the rain gauge data. When comparing to the rain gauge data and to the radar rainfall estimations (Figure 3a), the RAD12 and AMA12 simulations clearly provide a better localization of the heaviest precipitation during phase I of the event than the ARP12 simulation. RAD12 performs slightly better than AMA12 concerning the localization of the heaviest precipitation and the estimation of the maximum amount. An objective validation of these three simulations is given by Chancibault et al. . The observed and simulated mean areal rainfall depths have been compared for nine watersheds of the region. For the phase I, the relative error can reach 400% for the ARP12 simulation, whereas the relative error for RAD12 and AMA12 experiments never exceed a fifth of this value. Even though the differences in terms of relative error between RAD12 and AMA12 are weak, RAD12 performs in most cases better than AMA12, in particular for the Gard watersheds. For the two other phases of the event, the benefit of using a mesoscale data analysis as initial conditions decays; the three simulations have the same drawback which is an underestimation of rainfall over the Gard plain [Chancibault et al., 2006].
Figure 11 shows the ZTD assessments for the three simulations and observations at some of the GPS stations (see Figure 3 for locations). The GPS measurements are plotted from 0600 UTC on 8 September to 1700 UTC on 9 September, whereas the simulated ZTD are plotted from 1200 UTC on 8 September to 0600 UTC on 9 September. Note that GPS observations are missing at the MARS station at the end of the period. The mean differences between observed GPS ZTD and Méso-NH ZTD have been computed for all the GPS stations inside the 2.4 km domain, except for those too close to the borders of the model domain (mean scores, Figure 12). ZTD from simulations are based on equation (1) with Bevis et al.  coefficients k2 and k3 and coefficient k1(Pd, T) from equation (3); the contributions from hydrometeors are included.
 The GPS sites of Chateau-Renard (CHRN) and Vercoiran (VERC) were affected by the convective precipitation during phase I of the event (before 2200 UTC on 8 September 2002; see Figure 3) and later during phase III with the passage of the front and the embedded convection (after 0400 UTC, 9 September 2002). Clearly, the two precipitating periods correspond to the highest observed ZTD values (Figures 11a and 11b). The observed ZTDs increase quickly in the morning of 8 September for the two stations (by 75–80 mm in less than 8 hours). Then, when the MCS moves toward the crests of the Massif Central (phase II of the event), the observed delays decrease before increasing again at the passage of the front with embedded convection. ZTDs computed from the ARP12 experiment show clearly an underestimation for the two stations during phase I of the event, which reduces slightly for the two subsequent phases. Using a mesoscale surface observation analysis as initial conditions (AMA12 experiment) improves the simulation of ZTD for VERC. However, clearly, adding to the surface observation analysis an adjustment of the moisture and hydrometeors based on radar and satellite data (RAD12 experiment) gives the best simulation of ZTD with a significant reduction of the bias for the two stations (Figure 12).
 In the western part of the network, heavy precipitation has been recorded only during the beginning of phase I at the Montpellier station (MTPL), whereas the Aigoual (AIGL) station was affected by heavy precipitation during phase II only (Figure 3). For these two stations, all simulations underestimate the ZTD values (Figure 11c for MTPL), with biases of more than 18 mm (Figure 12). This underestimation of ZTD cannot be explained by an underestimation of ZHD due to a bad forecast of surface pressure: differences between the GPS ZHD and the modelled ones are less than 6 mm. Therefore, the underestimation of ZTD for the three simulations are mainly linked to an underestimation in ZWD. So, the simulations starting from the mesoscale initial conditions do not succeed in improving the integrated water content of the atmosphere for the western stations.
 For the eastern sites, i.e., the stations of Ginasservis (GINA), St. Michel l'Observatoire (MICH), Marseille (MARS), Grasse (GRAS), Nice (NICE) and Sophia Antipolis (SOPH), no precipitation during the first two phases has been recorded (Figure 3). Again, the GPS observations show an increase of the ZTD values during the morning and beginning of the afternoon of 8 September, as it can be seen on the temporal evolution of ZTD for MICH and MARS (Figures 11d and 11e). However, the temporal increase is not as important as inside the heavy rain area, it does not exceed 30–50 mm. For all these sites, ARP12 underestimates the ZTD whereas the RAD12 and AMA12 experiments give systematically the best estimation of ZTD (Figure 12). RAD12 and AMA12 are close to the observations in the area of MARS, GINA and MICH (Biases < 5 mm). RAD12 and AMA12 ZTD values do not differ significantly. This is not astonishing as the adjustment added for RAD12 was introduced only over the observed rainy regions at 1200 UTC.
 The southern station, i.e., Cap de Creus (CREU) in Spain, was outside the region of the flood event; nevertheless, some convective cells have also passed over Cap de Creus during the studied period as for example at 0400 UTC on 9 September as evidenced by a peak in the observed ZTD time series (Figure 11f). As for the eastern stations, RAD12 and AMA12 significantly reduce the biases. ARP12 largely underestimate the observations with a bias of more than 35 mm.
 For the three experiments the use of k1(Pd, T) reduces the ZTD biases (except for MICH in the AMA12 and RAD12 experiments). This reduction is weak for high-altitude stations (AIGL and GRAS), but it reaches 5 mm for CHRN and MTPL. Hydrometeor contributions decrease the biases between observed and simulated ZTD (up to 3 mm of diminution for VERC), except for CHRN in the RAD12 experiment. The bias is 1.5 mm with k1(Pd, T) and becomes −3 mm with the additional ZHMD contribution. This highlights the fact that the contribution of hydrometeors is occasional in time and space, and that no time average has been taken into account in the model ZTD assessments. For this reason, we have simulated ZTD every 15 minutes for the RAD12 experiment, and calculated an average ZTD over one hour. The comparison of the averaged ZTDs to GPS observations presents a mean bias of −1 mm. When the contribution of ZHMD is very important a time average can be introduced in ZTD assessments to obtain more realistic simulations.
 For all the stations, whether or not taking into account the hydrometeor contributions or the k1(Pd, T) does not alter the superiority of RAD12 over the two other experiments. Clearly, RAD12 is the simulation that best fits the observed ZTD and consequently the integrated water vapor content of the troposphere inside the region covered by the convective system during phase I. This is also the simulation that best fits the rainfall observations. The ARP12 simulation, which gives the worst precipitation forecast, is the simulation that underestimates ZTD most (Figure 12). For phase II, when the system moves northwestward, the precipitation area does not extend southward enough in the simulations, even for the RAD12 experiment. Results at the GPS sites in this region clearly show an underestimation of ZTD for all the simulations during the afternoon and the night of 8–9 September.
 We used a high-resolution (2.4 km) nonhydrostatic atmospheric model (Méso-NH) to simulate GPS tropospheric observables during an extreme flash flood event that occurred the 8–9 September 2002 in southeastern France.
 Integrated evaluations of ZTD performed for the first time in such a high-resolution nonhydrostatic model permit us to quantify contributions of hydrometeors to zenith delays (up to 70 mm), and the extension of overestimations made by the hydrostatic formulation in ZHD evaluations inside strong convective cells (up to 18 mm). Several atmospheric refractivity coefficient sets proposed by the literature have been tested. Results show a weak sensitivity to the set chosen, except the one with only two coefficients from Smith and Weintraub . The use of a more precise pressure- and temperature-dependent expression for the refractivity coefficient k1 = k1(Pd, T) yields ZTD differences from 1 to 6 mm.
 The comparison of the hydrostatic formulation of ZHD (its evaluation based on ground pressure measurements with the Saastamoinen  formula) and an integrated reference evaluation shows differences related to two aspects, namely, (1) the use of two different terms for the gravity (gm in the Saastamoinen formula, g0 in the Méso-NH integration) which leads to an approximately 6 mm mean overestimation of ZHD by the hydrostatic evaluation from ground pressure and (2) the departure from hydrostatic equilibrium which induces an overestimation of up to 18 mm of ZHD by the evaluation based on ground pressure. These large differences are located inside the strong convective cells, where pressure departs from the hydrostatic equilibrium.
 The errors associated with ZWD conversions into IWV have been evaluated for ZWD inferred from ZTD using integrated ZHD and ZHD from ground pressure, and with or without separation of hydrometeor delay from ZWD. When using a conversion factor κ dependent on surface temperature, the κ given by Emardson and Derks  shows the best performance in our study case. IWV from model-integrated ZWD converted by this κE&D has a low mean bias of 0.2 kg/m2 with respect to model IWV and maximum differences of 1 kg/m2. The mean bias between IWV from ZWD inferred using ZHD from ground pressure and model IWV yields 0.6 kg/m2 with maximum differences of 8 kg/m2. These large differences are locally confined and due to the ZHMD contribution contained in the value of ZWD. If ZHMD could be provided in an operational way inside the heavy rainfall areas, the maximum differences could be decreased to less than a quarter of the previous value. The increase of the mean IWV bias when using ZWD inferred by a GPS-like strategy using surface pressure and temperature is mainly due to the ZHD overestimation with the hydrostatic formulation, and less to the impact of ZHMD included in the ZWD. However, the unmodeled ZHMD contribution is responsible for the localized extreme differences inside the heavy rainfall event with respect to the model IWV field. Results on the impact of the hydrostatic assumption and of the hydrometeor contributions lead us to recommend caution when using IWV inferred from GPS ZTD inside vigorous convective and precipitating cells. Results show that the induced IWV error can reach 15% in the convective cells.
 The comparison of the three different model simulations with GPS observations of ZTD shows that the differences in simulated and observed ZTD are essentially due to underestimations of the wet delay and therefore to the water vapor content by the simulations. The simulation integrating the most precise information about the distribution of water vapor and water in the atmosphere obtains the best score (RAD12). Taking into account the contribution of ZHMD (more than 20% of the ZWD estimation in extreme cases) has a positive impact on the shape of the ZTD time series (up to 18% of bias diminution). The expression k1(Pd, T) present also a significant positif impact (up to 60% of bias diminution). The mesoscale network of GPS stations used for this study has allowed a detailed validation of the simulations. In particular, comparison with GPS ZTD for the western sites shows that all simulations suffer from an underestimation of the delays, as well as for the CHRN and VERC stations during phases II and III of the event. The assimilation of these masoned GPS data is envisaged in the near future with the hope of improving the simulation of precipitation during phases II and III of the event. Our study favors assimilation of GPS ZTD measurements, rather than of GPS-inferred IWV, to avoid the significant errors due to hydrometeors and the ZHD overestimation made by the hydrostatic formulation in extreme weather situations.
Appendix A:: Liquid Water and Ice Refractivity
 The propagation in the neutral atmosphere can be considered independent of the signal frequency (approximation with constant values for atmospheric refractivity coefficients). However, a contribution to neutral atmospheric refractivity can arise from some polar atmospheric gases [Owens, 1967] and hydrometeors [Solheim et al., 1999]. In general, the total refractivity is expressed as N(f) = N0 + N′(f) + iN″(f), where N0 and N′(f) are the nondispersive and dispersive parts of refractivity related to the real part of the permittivity (phase of signal), and N″(f) is dispersive attenuation, related to the imaginary part of the permittivity (amplitude of signal). However, in this study attenuation of the signal has not been considered (N″(f) = 0). Therefore only the delay induces by the neutral part of the atmosphere is considered. Influence of atmospheric carbon dioxide on refractivity has been studied by Edlén [1953, 1966], Owens , and Thayer . In our work, we do not consider the influence of CO2 on GPS signal propagation, because its atmospheric refractivity coefficient is badly constrained and the Méso-NH model does not provide any CO2 information.
 However, dispersive propagation in the neutral part of atmosphere can be, among others, caused by particles formed by the condensation of water vapor (such as rain, hail, pristine ice, snow and graupel) [Solheim et al., 1999; Hajj et al., 2002]. Phase delays induced by these suspended forms of water can be approximated using calculations based on permittivity. A thin strip approximation for non gaseous and nonscattering particles and the Clausius-Mossoti equation for refractivity [Debye, 1929] can be applied in obtaining equation (A1) and equation (A4).
Liebe et al.  proposed, for a signal frequency less than 100 GHz, an approximate formulation (single Debye model) for the liquid water additional contribution to the real part of the refractivity (Nlw):
where ρlw is the density of liquid water (≈1 g/cm3), Mlw is the mass content of the liquid water particles per unit of air volume, and lw is the permittivity of liquid water (function of T and f) defined as follows.
where ɛs = (77.66 − 103.3 θ) is the static dielectric coefficient, with θ = (1 − ) and T the temperature expressed in K (T ∈ [250–330]); f is the microwave frequency in Hz and fD = (20.27 + 146.5 θ + 314 θ2)109 is the relaxation frequency in Hz. The specific refractivity contribution of liquid water to delay can be approximated by
Hufford  proposed, for 1 MHz ≤ f ≤ 1 THz, an approximate formulation for the ice additional contribution to the real part of the refractivity (Nice):
where ρice is the density of ice (≈0.916 g/cm3 [Huining et al., 1999]), Mice is the mass content of the solid water particles per unit of air volume, and ɛice = 3.185 [Mätzler, 1996] the permittivity of ice. Specific refractivity contribution of ice to delay can be approximated by
These additional refractivity expressions for hydrometeors are an approximation to the mix formulae of Garnett . This formulation of delays caused by hydrometeors is valid for any medium considering inclusion of disjunct spherical particles (case of Méso-NH), with sizes of particles clearly smaller than the wavelength of the signal (case of hydrometeors size versus GPS wavelengths).
where χ = and β = , for temperature T0, partial pressure of dry air compressibility factor of dry air and refractive index of dry air n0 given in standard conditions, and compressibility factor of dry air Zd at Pd and T [see Owens, 1967; Birch and Downs, 1993]. The ratio can be approximated by unity considering perfect gas. For the refractive index of air n0, the correction to the updated Edlén equation is used [Edlén, 1966; Birch and Downs, 1994]:
where σ is wave number in μm−1. χ and β can be considered in a first good approximation like constant values. In fact, considering wavelengths of GPS (L band), in this expression of the refractive index of air n0, we have actualized the expression of k1(Pd, T) derived by Saastamoinen [1973b] to GPS frequencies. These values are constant, independent of the GPS signal frequency (L1, L2 or LC), but these values are also constant because the proportion of dry air in the atmosphere is quasi-constant. For a wavelength of 0.574 μm [Jordan et al., 1970], it gives the value of χ = 0.788828 K/Pa and β = 1.315 × 10−7 K/Pa as suggested by Saastamoinen [1973b]. We suggest χ = 0.7754713 K/Pa and β = 1.2925 × 10−7 K/Pa considering the L band (GPS frequency domain).
 The current study was performed under the PATOM and PNRH research programs of the CNRS-INSU (the National French Institute of Universe Sciences). This collaboration between the LGIT and the CNRM results from an ACI Ph.D. grant in the frame of the “Observatoire Hydrométéorologique Méditerranéen Cévennes-Vivarais” (OHM-CV). We thank Laurent Labatut for his informative management as well as Paddy O'Brien and Anne Holford for improving the English of the manuscript. We thank Paul Tregoning and two anonymous reviewers for their advice and their helpful contributions for improving this work.