a. Variational assimilation

The objective of the variational assimilation is to obtain an optimal solution based on all available observations, background analysis, observation error statistics, background error statistics, and a full-physics forecast model within a certain period of time. The 4DVAR system allows observations to be assimilated in their raw form at any given point in time and space and the model solution is required to fit the observations in a least squares sense. The discrepancy is measured by a cost function J, and the two terms (J^B and J⁰) in the cost function are weighted by the inverse of the background error covariance and the observational error covariance, respectively (Zou et al. 1997):

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where x₀ is the analysis vector consisting of eight model control variables [winds (u, υ), temperature (T), specific humidity (q), pressure perturbation (p′), vertical velocity (w), cloud water (q_c), and rainwater (q_r)] at the model initial time t = t₀. The corresponding forecast background vector is x_b, and 𝗕 the background error covariance matrix. In the present study, 𝗕 is diagonal and derived based on Navon et al. (1992). The weightings are calculated as the inverse square of the maximum differences between two forecasts that are 6 h apart. To assimilate the observed data into the 4DVAR system, J⁰ and the corresponding forward observation operator should be developed by users in advance. The observation operator H transforms the model variables x_r to the observed variables y_r at time t = t_r. Here, the superscript “T” stands for transpose; and n is the number of observed time periods. In this study, an error covariance matrix 𝗢 is defined as a diagonal matrix with invariant elements, which is the inverse of the measurement error variances, under the assumption that measurements at different sites are uncorrelated. Also, we assume a perfect model framework in (2.1).

In the present OSSE study, to understand the impact of GPS water vapor information on short-term prediction of a squall line, we assimilate water vapor measurements—GPS precipitable water or slant wet delay—without any other conventional observations. Therefore, J⁰ will be J^pw or J^swd in this paper.

b. 4DVAR assimilation of slant wet delay

1) GPS slant wet delay retrieval

Microwave radio signals transmitted from GPS satellites to a ground-based receiver are refracted and delayed by the atmospheric gases as they pass through the atmosphere. The delay due to the ionosphere is caused by the charged field of free electrons and can be corrected to millimeter accuracy with the use of a linear combination of measurements at the two GPS frequencies (L1 and L2). In the troposphere, signal delays in transmissions are caused by a variable index of refraction along the ray path. The delay due to the neutral troposphere consists of two components: “hydrostatic delay” and “wet delay.” Wet delay is caused by tropospheric water vapor along the ray path, and hydrostatic delay by all other atmospheric constituents (Rocken et al. 1995). To a good approximation, refractivity in the troposphere is related to atmospheric pressure [P (hPa)], temperature [T (K)], and water vapor pressure [e (hPa)] at microwave wavelengths as in (2.2) (Boudouris 1963):

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where k₁ and k₂ are experimentally determined constants and n is the refractive index. Even though hydrostatic refractivity (N_hydrostatic), the first term on the right-hand side of (2.2), contains water vapor effects through pressure, hydrostatic delay is generally called dry delay to differentiate itself from the wet delay (N_wet), which includes the effects of water vapor only. This expression is considered accurate to about 0.5% under normal atmospheric conditions (Resch 1984).

To simulate slant wet delay along each ray path, we integrate wet refractivity [as defined in (2.2)] from a ground-based receiver to a satellite in view as in(2.3):

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where m is satellite number, i receiver number, and ds has unit of length (m) along the GPS ray path. Slant wet delay can be linearly related to slant water vapor (SWV) using a conversion parameter Π.

(2.4)

where Π = f(T_m) and T_m is a weighted mean temperature of the atmosphere (Davis et al. 1985). Bevis et al. (1994) showed that T_m is strongly related to the surface temperature (T_m = 70.2 + 0.72 T_s) and the conversion parameter Π is approximately 0.15, which implies that 1 mm of integrated water vapor corresponds to a delay of about 6.5 mm. However, the actual value of Π can vary by as much as 20% due to T_m. To avoid errors caused by the assumption regarding T_m, we directly assimilate slant wet delay (instead of slant water vapor) in the present study.

2) Ray integration

In this study, we assimilate SWD data with ray paths above 5° elevation angle only. Based on the radar slant range–height diagram (Fig. 3.2 in Rinehart 1989), the bending effect is negligible for rays with an elevation angle greater than 5°. We evaluated the accuracy of straight-line approximation in the estimate of SWD as compared with a ray-tracing method, and found the approximation to be accurate above 5° elevation angles with less than 1% rms error (Ha et al. 2002). Therefore, we assume that the ray path is a straight line in the ray integration in the 4DVAR system. Zou et al. (1999) used a ray-tracing method, which controls the direction vector of the ray path by calculating the three-dimensional refractivity gradient at each step of the ray integration. The ray-tracing technique has the advantage of accounting for the bending effect of each ray path. But a major shortcoming of this method is the high computational cost, which is about 20 times larger than that of the straight ray-path integration in the present case. In the ray-tracing method, the time-consuming process of ray-path estimation must be included in the minimization procedure, while it only needs to be done once outside the minimization procedure in the straight ray-path integration, for a given elevation and azimuth angle. Thus, the latter is very efficient and very attractive for the 4DVAR system.

A schematic of ray integration is shown in Fig. 1. Wet refractivity (N_wet) is first computed from all gridded pressure, temperature, and specific humidity fields at every half-sigma level and is then horizontally interpolated to the point at which the ray path intercepts each level by a 16-point biparabolic interpolation (Guo and Chen 1994). In order to increase computational accuracy, each sigma level is subdivided into two levels and wet refractivity at the intermediate level is estimated by log-linear vertical interpolation. To calculate slant wet delay along the ray path, wet refractivity is integrated from the ground receiver site [O(i₁, j₁, k₁)] to the model top [E(i_n, j_n, k_n)]. Once we find the start point O and the end point E in the three-dimensional model grid domain, we can integrate the ray path:

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where the subscript m is the vertical-level index, z the reference height (m), and dis the horizontal grid distance of the forecast model (m). Equation (2.5) and the second term (N_wet) in (2.2) correspond to the observation operator H for SWD in (2.1). For simplicity, we ignore the horizontal variation of the reference height field and use the one-point values above the receiver site (i₁, j₁). In the present OSSE study, the hypothetical observation network is located over the southern Great Plains in the United States, which has a relatively flat topography except near the western lateral boundaries (Fig. 2b). Therefore, terrain height at each observation site in the assimilation is fairly close to the “true” topography, which is obtained from the 3-km nature run (which will be explained in section 3). Differences in terrain heights between the 27-km assimilation model and the simulated observation are less than 100 m at all sites, and the effect of terrain differences on slant wet delay is less than 1% on average in this case. The errors in the slant wet delay observation operator introduced by the approximation of a straight ray path and the use of smooth model terrain height are neglected in the present OSSE study, and thus not included in the error covariance matrix in the slant wet delay assimilation.

Slant wet delay observations are first simulated from the nature run through the method described above as instantaneous values valid at every 5 min. To more realistically simulate GPS measurements, we generate random errors that are of uniform distribution with a random number generator and then add these errors to the “perfect” slant wet delay observations. The magnitude of random errors is similar to that of the real measurement errors as stated in the next section.

3) Slant wet delay measurement errors

In the actual observations, the measurement error of slant wet delay is estimated by comparing GPS slant wet delay measurements with those from radiometers pointing to the same direction (at the same azimuth and elevation angles). The slant wet delay in the raw GPS signal is obtained after accurately modeling and accounting for the clock errors, geometric delays, hydrostatic delay, and ionospheric delay. All slant path signals are then projected onto zenith and averaged over a certain period of time under the assumption that the atmosphere is azimuthally homogeneous (Niell 1996); this is called the zenith wet delay. By validating the ZWDs against other independent measurements (e.g., from a radiometer), we can estimate the observation error of ZWD. The error of ZWD is then projected back to the elevation angle of an individual ray path (assuming an isotropic atmosphere around the receiver site). This is called an isotropic error, which is dominated by the systematic error between a ground-based GPS receiver and a water vapor radiometer. Braun et al. (2001) estimated the nonisotropic errors of GPS slant water vapor measurements from the NOAA/FSL network and showed the statistics for bins of 5° elevation angles (see Table 1 in their paper). When we assume that the isotropic and the nonisotropic error are not correlated with each other, total slant wet delay measurement error (SWD_err) can be expressed as in (2.6).

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In this OSSE study, we assume that the “simulated” slant wet delay has errors with magnitudes similar to those of the total real measurement errors (shown by the solid line in Fig. 8b). The magnitude of the nonisotropic error is approximately half of the isotropic component at 10° elevation angle in this study.

c. 4DVAR assimilation of precipitable water

In the previous studies (Kuo et al. 1993, 1996; Guo et al. 2000), precipitable water was calculated using (2.7) under the assumption of hydrostatic balance.

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where q(K) is the analysis of specific humidity at the Kth layer, Δσ(K) is the thickness of the Kth layer, and KX is the total number of layers. Here, p* = p₀ − p_s, where p_s is the surface pressure, p_t is the pressure at the top of the model (100 hPa), and g is the acceleration due to gravity.

In the actual GPS sensing, however, precipitable water is derived from zenith wet delay:

(2.8)

The value Π is the same as that in (2.4). The zenith wet delay is an estimate, commonly based on the principles of least square estimation, which utilizes the zenith-mapped slant wet delays from a specified time window and sampling interval. In the 4DVAR system, the assimilation of ZWD was implemented under the assumption that an average of all slant wet delays would accurately represent this estimation. The average is represented through the following equation:

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where nr is the total number of satellites in view at a given time at each station, and nt is the observation frequency. As we assume that slant wet delay measurements are available every 5 min and precipitable water is estimated at 30-min intervals, nt is 6 in this study.

For convenience, we refer to PW in (2.7) as PW_q, and PW in (2.8) as PW_swd. If the water vapor is homogeneously distributed and the rays scan the atmosphere uniformly in space during the observation period, the error due to the different definition of precipitable water can be neglected to a good approximation. However, due to the inhomogeneity in sampling the data from GPS receivers and the temporal and spatial variation of the atmospheric water vapor, the PW_swd can be quite different from PW_q in (2.7), especially when a squall line is approaching. An estimate of the rms differences between PW_q and PW_swd is 2.9 mm based on the results of the nature run, which is about twice as large as the GPS PW measurement error of 1.5 mm. Therefore, such error cannot be ignored and one should not treat PW_swd as PW_q. Thus, we simulate precipitable water following the same algorithm (2.8) and (2.9) as that used in the real GPS measurement. [In turn, (2.8) and (2.9) correspond to the observation operator for PW in (2.1).] We first calculate slant wet delay along each ray path, scale it to the zenith, and then average all the zenith-mapped slant wet delays for a 30-min time period to calculate PW_swd.

a. Experiment design

We first perform the “nature” run with a 3-km grid mesh and 50 vertical layers using the MM5 (Grell et al. 1994). This run is initialized at 1200 UTC 29 October, and is integrated for 18 h. The computational domain (D3 in Fig. 2) contains an array of 403 × 388 grid points. In this run, the Reisner-1 mixed-phase microphysics and the National Centers for Environmental Prediction (NCEP) Medium-Range Forecast (MRF) model planetary boundary layer (PBL) scheme (Hong and Pan 1996) are employed, but no cumulus parameterization is used. The forecast results from the nature run are used to simulate slant wet delay measurements from a hypothetical network of GPS receivers. This is done by integrating wet refractivity along each ray path under the assumption that the ray path is a straight line from a fixed station to each GPS satellite as explained in section 2b. For the satellite positions, the actual orbit data are used. The hypothetical network consists of a total of 64 GPS receiver sites regularly distributed over the Oklahoma–Kansas region with 90-km spacing where the squall line is initiated (Fig. 2b).

The basic set of our OSSE design includes Perfect_IC run, NO4DVAR run, 4DVAR_SWD run, and 4DVAR_PW run. All 4DVAR experiments are performed at 27-km horizontal resolution (a 45 × 43 grid mesh) using the MM5 4DVAR system (Zou et al. 1997). All forecast runs are performed at 9-km horizontal resolution (133 × 127 grid points) with 23 layers in the vertical. Since the vertical resolution is considered to be important in the slant wet delay calculation, we use the same vertical resolution of 23 layers in 4DVAR experiments as in all forecast runs, while we reduce the horizontal resolution from 3 km (in the nature run) to 9 km (in the forecast run) and 27 km (in the 4DVAR model) due to the high computation cost. In our 4DVAR runs, the CPU cost of the 9-km model was about 20 times larger than the 27-km model for 40 iterations. The 23 vertical layers are defined at σ = 0.05, 0.15, 0.25, 0.35, 0.425, 0.475, 0.525, 0.565, 0.595, 0.625, 0.655, 0.685, 0.71, 0.73, 0.75, 0.775, 0.805, 0.835, 0.865, 0.895, 0.925, 0.955, 0.985. In the 27-km MM5 4DVAR system, Grell cumulus parameterization, Bulk PBL, and an explicit moisture scheme are used, while the 9-km forward runs use Kain–Fritsch cumulus scheme, the Reisner-1 mixed-phase microphysics scheme, and the Blackadar PBL scheme.

In order to provide an upper-bound benchmark in terms of forecast accuracy for the data assimilation experiments, we conduct experiment 1—“Perfect_IC” run. The initial condition for this experiment is obtained by taking the model fields at every third grid point of the nature run at 2100 UTC 29 October. This experiment represents a case in which the best possible initial condition is available for a forecast model at 9-km resolution. The deviation of this run from the nature run is primarily caused by model errors (difference between 3- and 9-km grid, and their associated physical parameterization schemes).

Experiment 2 is a “no data assimilation (NO4DVAR)” run. This experiment represents a lower-bound benchmark, reflecting the accuracy of a forecast model initialized by a typical operational analysis. In order to provide an initial condition for this experiment, we first conduct a 27-km MM5 experiment using the NCEP global analysis at 1200 UTC 29 October as the initial condition. The 9-h forecast from this 27-km run is then linearly interpolated to the 9-km grid and used as the initial condition for experiment 2.

Experiment 3, is the “slant wet delay assimilation (4DVAR_SWD)” experiment. Through the 4DVAR assimilation of slant wet delay over a 3-h time window from 2100 to 0000 UTC 30 October, the optimal initial condition is obtained at 2100 UTC. Slant wet delay measurements are assimilated at 10-min time intervals. The initial condition of experiment 2 at 2100 UTC 29 October is used as the gridded analysis data for the background term of the cost function in the 4DVAR system.

Experiment 4, is the “precipitable water assimilation (4DVAR_PW)” experiment. To investigate the relative impact of SWD to PW_swd, we perform the 4DVAR experiment of PW_swd [as defined in (2.8)]. The observation time interval of SWD measurements for PW_swd estimates is 5 min. For the averaging over a 30-min time period, a temporal projection operator is needed in the cost function for PW_swd.

All forecast runs are conducted at the 9-km horizontal resolution from the four different sets of initial condition to investigate the impact of slant wet delay on short-term precipitation forecast. By using different physics in the 9-km forward run from the ones in the 3-km nature run, we partially remove the identical-twin problem in this study. Figure 3 and Table 1 provide an illustration and a summary of the experiment design, respectively.

b. Case description

A squall line was initiated over the Kansas–Oklahoma panhandle on 29 October 1999 when a surface cold front moved into a convectively unstable environment. The development of mesoscale convective systems and squall lines in this region is often attributed to the special geography and topography of the region: the Rockies to the west, the high Mexican plateau to the southwest, and the Gulf of Mexico to the southeast. The structures and evolutions of the squall lines over the south-central United States have been studied by many authors (Carlson et al. 1980, 1983; Hobbs et al. 1990; Locatelli et al. 1995, 1998; Martin et al. 1995; Martin 1998).

With the advance of the surface cold front, the line convection formed at 2135 UTC 29 October 1999 over northwestern Oklahoma and developed into a prefrontal squall line producing a maximum 24-hr rainfall amount of 89.8 mm around Lamont, Oklahoma. Figure 4 shows National Weather Service radar summaries over the south-central United States during the squall-line passage. In the 3-km nature run, however, the simulated precipitation (which is modeled by explicit moisture scheme) did not occur until 0000 UTC 30 October. Under the OSSE framework, the Perfect_IC run represents the best possible performance of a 9-km model and the goal of the 4DVAR assimilation is to obtain a solution as close as possible to the Perfect_IC. Therefore, we choose to verify the results of our 4DVAR experiments against the Perfect_IC run. It is important to recognize that Perfect_IC run is by no means “perfect.” The Perfect_IC run, when verified against the nature run (not shown) or the observed radar echoes (Fig. 4), contains considerable forecast errors in terms of the onset time and the location of the convective rainband. This is unavoidable, as the treatment of convection in a 9-km model through cumulus parameterization is quite different from that of a 3-km model with explicit microphysics. The difference between the 9-km Perfect_IC run and the nature run provides a realistic estimate of errors associated with a typical mesoscale forecasting model in its handling of convection.

Here, we describe the synoptic pattern before the formation of the squall line as simulated by the Perfect_IC Run (Fig. 5). At 2100 UTC 29 October 1999, with the approach of an upper-level trough at 300 hPa, a strong upper-level jet is coming off the Rockies and the warm and dry air is advected to central Kansas–northwestern Oklahoma (Fig. 5a). At the surface, a trough is located over the northeastern Kansas–northwestern Texas region, where there is a confluence of the cool, moist air from the Gulf of Mexico and the cold, dry air from Colorado (Fig. 5d). At this time, the cool, moist air is being advected from the south along a strong low-level jet (LLJ) with a maximum wind speed of 28.1 m s⁻¹ at 850 hPa. A large shaded area in which relative humidity is greater than 80% is located in the downstream region of the surface cold front (Figs. 5c and 5d). At 700 hPa, a warm and dry tongue along with the southwesterlies from the high Mexican plateau extends to northern Kansas and is superimposed over the low-level moist air to form a convectively unstable region (Fig. 5b). The surface cold front moves southeastward with a speed of 15.5 km h⁻¹ and is gradually enhanced. The surface cold front acts as a barrier to the low-level moist air flowing toward the front and triggers convection.

After a few rain cells first appear in the vicinity of the Kansas–Oklahoma border at 0000 UTC 30 October (not shown), the convection is quickly organized into a squall line ahead of the surface front at 0100 UTC and continues to intensify while moving southeastward with time (Fig. 6). The maximum hourly rainfall at 0600 UTC 30 October (9-h forecast) is 14.3 mm h⁻¹.

As the main objective of this study is to assess the impact of the slant wet delay measurements from a network of ground-based GPS receivers on the forecast of the squall line, the assimilation time window is set from 2100 UTC 29 to 0000 UTC 30 October, a 3-h time period before the squall line is formed.

a. An optimal solution from the 4DVAR assimilation

Figure 7 shows the cost function and the norm of the gradient during the minimization process in the 4DVAR assimilation of slant wet delay (4DVAR_SWD). Both the cost function and the gradient norm decrease very quickly and the minimization process stops at 57 iterations. Since the value of the cost function is reduced by almost 98% and the norm of gradient by about 99% at iteration 40, we take the optimal initial condition at iteration 40 and perform the forward run with it.

To assess the effectiveness of 4DVAR experiments, differences between modeled and observed slant wet delays during the 3-h assimilation window are plotted before and after the minimization process (Fig. 8). Before the minimization process in 4DVAR_SWD, slant wet delays retrieved from the model first guess are much smaller than the “observed” values (simulated from the nature run), which implies that the model initial condition is much drier than the “true” atmosphere. Large differences (ΔSWD) at the low elevation angle (especially around 10°) are due to the long ray path and do not mean that relative errors of GPS slant wet delay depend on the elevation angle of each ray path. After 40 iterations in the minimization process, rms error is reduced by as much as 90% and the large negative bias of −22.57 mm is almost diminished. It is also notable that the errors are greatly reduced for low elevation angles, which implies the accuracy of capturing the low-level moisture content is greatly improved. These results confirm that the model state in the 4DVAR experiment provides a best fit to the observations through the minimization process.

Even though the 4DVAR system can obtain an optimal solution mathematically, it does not guarantee that the solution is physically meaningful. To examine how the 4DVAR assimilation of slant wet delay changes the atmospheric state in the minimization process, we examine the changes of model control variables with iterations. Figure 9 shows the water vapor mixing ratio at the lowest sigma level (σ = 0.985) at iteration 0, 5, 20, 40. With increased iterations, the low-level moisture is increased near the border of Kansas–Oklahoma where the squall line forms in a few hours. Comparing the moisture distribution before minimization (Fig. 9a) and after minimization (40 iterations) (Fig. 9d), we note that the contour of 11 g kg⁻¹ is expanded to southeastern Kansas. As a result, specific humidity in this area is increased by 1.0 g kg⁻¹. The corresponding analyses for the temperature and horizontal wind fields at the same lowest sigma level are shown in Fig. 10. Note that slant wet delay is a function of three-dimensional pressure, temperature, and water vapor. Through the assimilation of SWD, surface temperature is increased by more than 1°C over the squall line area, and the pressure perturbation field is reduced by as much as 0.1 hPa (thick dashed lines in Fig. 10d). Also, it is interesting to note that though SWD data does not contain information on the wind fields, the horizontal wind fields are enhanced by up to 3.5 m s⁻¹ in the surrounding area due to dynamic adjustment. Brooks et al. (1993) showed that the structure of a simulated convective storm changed significantly when the low-level temperature was altered by 1°C. In this case, the changes in temperature and moisture amount near the surface, caused by SWD assimilation, contributed to the on-time initiation of a squall line by enhancing the cross-frontal gradient of the temperature field and the frontal circulation and inducing low-level moisture convergence ahead of the front (which will be shown in Fig. 13a).

Even though slant wet delay is an integrated value along each ray path, it is mostly controlled by the low-level moisture. Therefore, we expect the impact of SWD assimilation to reach the entire boundary layer rather than just being confined to the lowest sigma level. Sensitivity of moist convection to boundary layer thermodynamic parameters has been examined in a few previous studies (Lee et al. 1991; Crook 1996). In particular, Crook (1996) found that convective initiation is most sensitive to surface temperature and moisture perturbations. Moreover, through high-resolution, nonhydrostatic numerical model simulations, he showed that variations in boundary layer temperature and moisture that are within typical observational uncertainties (1°C and 1 g kg⁻¹, respectively) could make the difference between no convection and an intense convective storm. The twice-daily upper-air soundings and the surface measurements are not adequate to provide an accurate initial condition for high-resolution mesoscale models. The assimilation of GPS SWD measurements can potentially address this need. To examine if the assimilation of slant wet delay can improve the atmospheric structure in the boundary layer, we show, in Fig. 11, the differences in water vapor mixing ratio, temperature, and three-dimensional circulation vectors before and after minimization in the vertical cross section along the line AA′ in Fig. 9d. Figure 11 indicates that the assimilation of slant wet delay moistens the entire atmosphere, especially the boundary layer, by as much as 1.5 g kg⁻¹ where the squall line will be initiated. The moistened region (the shaded area) is also overlapped by the region of positive temperature changes (solid line). As a result, the cross-frontal temperature and moisture gradients associated with the differential development of the boundary layer aid to strengthen the surface cold front and to initiate the convection at the correct location. The strong rising motion immediately behind the moistened boundary layer, associated with enhanced frontal circulations, induces strong low-level moisture convergence ahead of the surface cold front and triggers the release of the convective instability. As will be shown later, these changes in the initial state significantly improve the model's skill in forecasting the onset time and the location of the squall line.

b. Forecasting the evolution of the squall line