a. Hypothetical network design

Our goal is to recover high-resolution (mesobeta) three-dimensional water vapor from a network of ground-based GPS receivers. To resolve this scale, the network must have a spacing comparable to the desired resolution. A grid of stations over the United States (excluding Alaska) at 40-km spacing would include about 5000 installations, while a grid of 60-km spacing would require a little more than 2000 installations. It is clear that there is a trade-off to be made between the number of the stations in the surface network and the cost of the system.

The operational concept includes a typical ground installation with a dual-frequency GPS receiver and an automated surface weather station measuring winds, temperature, and humidity. The number of GPS satellites in view from the surface would average about eight. We assume that with suitable time integration and averaging, there is one observed slant range between each station and each satellite every 5 min. Thus, for each station, there are approximately 100 observations of slant range moisture per hour. The observations are distributed around the hemisphere, with the exception of the northern quadrant, as discussed in Rocken et al. (1997). The geometry of the system allows more observations in the 45° closest to the zenith than in the 45° above the horizon, but there are typically several satellites rising or setting during the 1-h period. We refer to the “splay” of slant ranges as the dataset taken during an observing period, which in our case is taken to be 1 h. In addition to the integrated slant water vapor measurements, there are two other types of observations used in the 3DVAR. The inclusion of an accurate moisture sensor at each station allows the integrals of slant water vapor to start from the surface with a known value, which is a very valuable constraint in the variational analysis. In addition, a low density of water vapor soundings (e.g., from microwave radiometers, Solheim et al. 1998) is needed to constrain the analysis.

An important characteristic of the network would be to resolve moisture in the lowest layers of the atmosphere. It is an observational fact that moisture is generally greatest near the surface, and typically decreases upward with a scale height of about 2500 m. The boundary layer, which is normally 1000–3000-m deep, is very important in weather prediction. For example, showers and thunderstorms are usually driven by moisture-laden plumes originating in the boundary layer (Emanuel et al. 1995). The use of the low angles allows resolution of the moisture to within approximately 250 m above the surface, with resolution of 500 m above. The density of stations and the lowest usable slant angle both determine the minimum altitude that the GPS network would be sampling. At lower slant ranges, say, less than 6°, the measurement of slant range water vapor has some associated difficulties. Most important are blockage due to the terrain horizon, reduced signal strength, and multipath effects in which the signal is reflected from surface objects. Despite these difficulties, there is evidence that usable slant range water vapor can be obtained down to the horizon (Ware et al. 1997). A recent test that measured GPS bending angles from a surface receiver (Sokolovskiy et al. 2001; Lowry et al. 2001) showed that the GPS-measured refractivity agreed well with collocated radiosondes all the way to the surface.

b. Observing system simulation

An OSS begins with the creation of a dataset with the goal of emulating nature. Ideally, if there were an experiment where moisture soundings were taken at 10-km spacing over a large domain (e.g., 1000 or more kilometers on a side), this dataset could be used to represent nature for a simulation. Unfortunately, such a dataset does not exist. Since we do not have such realistic field measurements, a technique must be used to create a nature field that is reasonable. By reasonable, we mean that the characteristics of the field, such as its power spectrum, spatial distribution, including typical gradients, etc., must be like those found in the real atmosphere. We assumed in this experiment that state-of-the-art high-resolution mesoscale models could create water vapor fields that are similar to those found in the real atmosphere.

In the OSS described here, we used two mesoscale models to create nature atmospheres over an approximately square domain with sides of 1460 km centered in the west-central United States. The domain was chosen to include the lower, flatter topography of the Great Plains on the east side and the complex terrain of the Rocky Mountains. This also meant a gradient of moisture, with high specific humidities in the eastern area, and in the west the low humidities characteristic of the mountains. The two models used were the quasi-nonhydrostatic (QNH) model of MacDonald et al. (2000) and the Pennsylvania State University–National Center for Atmospheric Research fifth generation Mesoscale Model (MM5; Dudhia 1993). Both models were run at 10-km resolution for 36 h, with the three-dimensional water vapor at the end of the period taken as the nature dataset. Each model has a full microphysics package, with at least five water forms (vapor, rain, snow, cloud water, and cloud ice) and the associated phase changes. Both models develop complex moisture fields in three dimensions, with high gradients in both the horizontal and vertical directions. As has been seen in aircraft measurements of water vapor fields (e.g., Palmen and Newton 1969), the spatial variance on mesobeta scales is quite large. The reasons for this variability are well known: they include nonlinear scale collapse, and the fact that when high gradients develop in water vapor fields, generally only diffusive processes operate to even them out. This characteristic is in contrast to the dynamic fields in which a high gradient (e.g., of temperature) causes strong adjustment processes that operate to decrease spatial variability. Another process that operates to generate complex moisture fields is the combination of phase change, such as condensation, and gravity. A dry lower atmosphere can be penetrated by a rain shaft, with the resulting evaporation generating strong moisture gradients. In the two cases used for nature runs, all of these processes operated to create complex and realistic-looking three-dimensional moisture fields.

The QNH model was initialized in April 1999, while the MM5 run was from a late January 1999 case. Thus the results are representative of a moist springtime moisture field and a drier, colder winter moisture field. Both models showed the results of small-scale precipitation processes, which had generated small areas of enhanced water vapor, an important test for the hypothetical system. As will be shown in section 5, the technique was able to recover this structure, albeit slightly smoothed.

The next step in the OSS is to use the nature field to generate pseudo-observations of the slant water vapor, the surface moisture fields, and the soundings. These pseudo-observations from the network are used as input to the variational analysis that attempts to approximate the nature field. A number of precautions were taken to make the data realistic. The pseudo-observations of water vapor were randomly perturbed by a 5% error function, which was based on estimates of random error by BRW. For lower angles, below 6°, the error was increased to 7%. Also, those slant paths that penetrated the lateral boundary were perturbed with 7% errors. Note that ground-based GPS stations do not measure integrated water vapor, but rather integrated delay, which includes effects due to atmospheric mass between the receiver and the satellite, and ionospheric delay, as discussed in section 2. The chosen error bounds include errors due to these effects, and thus makes it reasonable to analyze the system in terms of water vapor delays. It was not assumed that the stations could be located precisely on a grid; a busy highway would not be a good place to site a station. The station locations were perturbed in a ring around the nominal station grid points, with an average “location change radius” of 5 km. The surface moisture readings were assumed to have random errors of 5%, and the vertical soundings were given average moisture errors of 8% based on statistical comparison with radiosondes (Güeldner and Spänkuch 2001).

In order to collect enough pseudo-observations, there is an assumption of stationarity for the water vapor field. All observations for an hour are treated in the variational analysis as if they represent a single field. This implies that for the very small scale (e.g., less than 20 km), rapid changes in the moisture field will not be resolved. To make the OSS realistic, we used the actual ephemeris data to locate the satellites every 5 min. The significant motion of the satellites causes each measurement to be made in a different direction, making the measurements somewhat independent. Furthermore, the model moisture data were allowed to change through the hour by use of the actual 5 min data from the model run.

The OSS is properly defined if realistic errors for the observations are included. In particular, if a technique for recovery of slant water vapor is tested against another observing system and the average error of the water vapor is determined, then no other information is necessary for the simulation. Slant water vapor measurements using GPS have been validated by comparison with microwave radiometers to determine the errors used in this simulation (BRW). This does not imply that this approach would be used in an operational system. It is likely that additional information could be gleaned by using total delay in an operational variational analysis, which would provide temperature as well as moisture information. We have isolated the moisture in this simulation to simplify the exposition.

There are two techniques possible for recovery of water vapor. First, the 3DVAR could use total delay, and be formulated to recover both temperature and water vapor. This approach is currently being investigated by the authors. A second technique is to take the mass field as a given, with an assumed error. In the second technique, a future operational system for water vapor would use the latest and best analysis of the mass field available to calculate the dry delay. For example, the Rapid Update Cycle (RUC) model now has hourly analysis (Smith et al. 2001) of the mass field and could be used to isolate the dry delay due to mass from the unknown wet delay. This procedure would work well except in those cases where the delays are strongly correlated and not accommodated in the model analysis. For example, an area of heavy rain showers falling into a dry lower layer can result in cooling and moistening of the layer. Both the cooling and moistening result in an increased signal delay between the satellite and the receiver. Thus, an operational analysis system would be best if it used measurements such as hourly surface pressure changes, and estimates of evaporative cooling to refine the mass field before use in the moisture-based 3DVAR analysis. The separation of dry delay from moist delay constitutes a significant assumption in this analysis and will need to be further tested, both theoretically and experimentally.

With the pseudo-observations from the nature atmosphere, a 3DVAR is used to determine the three-dimensional distribution of water vapor. Here, we use a simple version of 3DVAR in which it is assumed that the errors in the pseudo-observations of slant range water vapor are randomly distributed and uncorrelated with one another. We do not use a model background. In a more realistic future analysis, background errors would also need to be considered. Theoretically, the assumption of random errors is not too bad (Kaplan 1996), and should not invalidate the main conclusions of the study.

Figure 1 is a schematic of the network and grid used for the 3DVAR analysis. The small ovals on the bottom are the symbols for the GPS receivers and associated surface humidity sensors. The regularly spaced dots on the grids are the locations of the control points. At each control point, the 3DVAR has a control variable, the estimated water vapor (specific humidity), that will be iterated through a series of estimates toward the best estimate field. Each GPS receiver has an associated splay of pseudo-observations of integrated slant water vapor, and an estimated slant water vapor from the control grid (e.g., the forward model) is obtained by interpolating and using a Riemann sum. The number of control points in the grid must be exceeded by the number of observations in order for the 3DVAR problem to be well determined. In the cases presented here, a 40-km resolution network and control grid can be well determined if there are approximately 100 observations per station, as discussed in section 3a.

The 3DVAR control grid is formulated with 40-km resolution in the horizontal and 500 m in the vertical. Our first attempt at an OSS 3DVAR used 40-km model data. This had very rapid convergence and excellent correspondence to the nature run. However, we thought that it was unrealistic to use such a low-resolution nature run, because it would not perturb that analysis with smaller scale variability. When we tried to run the 3DVAR analysis using 10-km model data, it was much more difficult to obtain convergence. As discussed in section 4, we needed to use a multigrid technique to get convergence when the high-resolution model data were used. We think that the 16-fold increase in model data points, which enters the problem through the pseudo-observations, realistically accommodates the variability that will be seen in the real atmosphere. It should be noted that although it was more difficult to obtain convergence with the high-resolution data, the improved techniques resulted in excellent convergence, as will be shown in section 6. However, it is possible that mesogamma-scale variability, associated with clouds, may cause difficulties. These could be tested with a higher resolution nature model.

a. GPS line integral

We compute an estimated integrated water vapor using the grid values along each GPS line, c_m,l, which starts from a ground station G^m located at (x^m, y^m) and points to a satellite S^l. Denote this quantity as Q^e(x^m, y^m, l). That is,

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which can be approximated using the values of q_i,j,k, the specific humidity for a grid box. First the integral can be approximated by

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where c_m,l,n is a sequence of equally spaced points on the curve c_m,l and Δs is the length between two adjacent points. Therefore, we have the following approximation:

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where the αs are the bilinear interpolation coefficients and [i₀(n), j₀(n), k₀(n)] are the lowest indices of a grid cell containing the point c_m,l,n.

b. The cost function and its gradient

The cost function is used to represent the difference between the slant water vapor observations and the approximations of slant water vapor calculated from the control grid. The cost function is defined as

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This typical optimization problem can be solved by numerical optimization algorithms. The technique we have chosen requires the gradient of the cost function.

c. Optimization technique

To minimize the cost function, a fairly large-scale minimization problem, the limited memory BFGS (L-BFGS) method (Liu and Nocedal 1989), is used. The BFGS method is a quasi-Newton method which has faster convergence rate than the steepest descent algorithms. Suppose at the nth iteration, we have an approximation Gⁿ of the solution G*, where G is a vector with elements of q_i,j,k, and an approximation of the Hessian of f at Gⁿ, Bⁿ ≃ ∇²f. The BFGS method obtains new approximations by

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where

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To further restrict the solution, we solve this minimization problem with a simple bound, q_{i,j, k} ≥ 0. In the numerical tests, we use the L-BFGSB algorithm developed by Byrd et al. (1995).

The 3DVAR analysis is initialized with a first guess. This was constructed by taking the surface values of the water vapor specific humidity and assuming an exponential drop-off with altitude, with an e-fold radius in the vertical of 2500 m. With an initial guess, the L-BFGSB algorithm starts iterating until a certain termination criterion is met.

d. Multigrid technique

We were able to solve the 3DVAR problem with the above formulation and algorithm. However, to counter a very noisy moisture field, we had to smooth the field in order to see the large-scale features. A very slow reduction of the cost function after the initial 10–20 iterations convinced us that the optimization algorithm spent most of the iterations converging the small-scale features and improved the long waves very little. This suggested that a multigrid technique could be helpful in improving the convergence of the longer waves. Thus, we combined the multigrid technique and the L-BFGSB algorithm in our numerical experiments. The multigrid technique is implemented with two multigrid levels, 40- and 80-km grids, over the same domain. We ran 10 iterations of the L-BFGSB algorithm to reduce the cost function over the 40-km grid and then projected the solution of the 40-km grid to the 80-km grid. On the 80-km grid, we ran 20 iterations of the L-BFGSB algorithm to reduce the new cost function over the 80-km grid and obtained an approximate solution. Then we interpolated the solution to the 40-km grid and ran 10 iterations of L-BFGSB algorithm again. This procedure, one multigrid cycle, can be repeated. We repeated this cycle 10 times. As a result, the solutions showed good convergence in the long waves without any smoothing. The multigrid and the uniqueness of the solution are discussed in MacDonald and Xie (2000).

a. Depiction of the nature and 3DVAR fields

Several depictions of fields were obtained from the 3DVAR analysis for comparison with the nature fields. For the horizontal depictions, we display the specific humidity. In this OSS, we used 16 moisture soundings distributed evenly over the domain, a surface moisture observation from each station in the 40-km network, and the integrated slant water vapor observations.

Figure 2 shows a horizontal plot of specific humidity (g kg⁻¹) at the 750-m above ground level (AGL) for the test area of the western United States. Note that the area goes from central Montana in the north to southern New Mexico in the south, and from central Utah to central Kansas in the east–west direction. Fig. 2a is the nature run, and Fig. 2b is the field recovered from the 3DVAR analysis. Starting with the nature run, a number of features are evident. A strong gradient of water vapor, the “dryline,” is evident from the Texas Panhandle up through central Kansas. An area of higher moisture values, with a maximum of 12.6 g kg⁻¹, is located just to the east of the dryline on the Oklahoma–Kansas border. Comparing the nature run with the 3DVAR, it is seen that the dryline has been recovered in the right place, with a decrease in the horizontal gradient. This is what would be expected from a 40-km grid spacing, where the nature field is created at 10-km resolution. Notice that the 3DVAR analysis recovered a moisture value of 12.9 g kg⁻¹ on the Oklahoma–Kansas border moisture maximum. A large portion of the southwestern part of the domain has rather low values of specific humidity. Compare the minimum of moisture of 0.592 g kg⁻¹ near the Colorado–Wyoming–Nebraska triple point with the 3DVAR minimum of 0.505 g kg⁻¹ slightly to the south. Similarly, compare the value of the maximum in western Colorado of 2.10 g kg⁻¹ in the nature run with the 2.19 g kg⁻¹ located slightly to the north in the 3DVAR analysis.

Across the northern portion of the nature run are a series of small-scale maxima of moisture due to precipitation, which had fallen earlier. For example, note the maximum of 14.3 g kg⁻¹ near the Nebraska–South Dakota border, two maxima on either side of the Wyoming–South Dakota border, and the maxima near the Wyoming–Idaho–Utah triple point. These should be compared with the same maxima in the 3DVAR analysis. Again, it is seen that the gradients are somewhat smoothed and the values are typically within 10%–20% of the nature run. In summary, at the 750-m level, typically in the middle of the boundary layer, the moisture analysis is far superior to that which could be obtained from the current network of radiosondes. Furthermore, this type of analysis would be available hourly, in contrast to the radiosondes, which only go up twice a day.

Figure 3 shows a horizontal plot of specific humidity at 3250 m AGL for the nature run, 3DVAR, and a Barnes (1981) analysis. The Barnes analysis is presented to show the moisture diagnosis available from the current observing system. The black dots mark the location of radiosonde sites that were used in the analysis. A striking difference between the Barnes analysis (Fig. 3c) and the nature and 3DVAR analyses is its smoothness. The Barnes analysis only captures the synoptic-scale moisture variability. This is a reflection of the fact that the current upper-air observing system undersamples the moisture field (Emanuel et al. 1995). Again, similar to the discussion of the 750-m level, it can be seen from Figs. 3a and 3b that the 3DVAR captures the small-scale mesobeta moisture fields with reasonable accuracy.

Figure 4 shows plots of specific humidity at the 3250-m-level from the January MM5 case. The nature run, Fig. 4a, shows higher moisture values along the eastern border and a lobe of moisture from the Texas Panhandle into southern Colorado. These features are in the 3DVAR, Fig. 4b, but are not as well defined. The small features over the mountains have corresponding features in the 3DVAR analysis. In general, the 3DVAR does not do as well for drier parts of the atmosphere because it is minimizing on the rms of the field—an rms of 0.5 g kg⁻¹ is a bigger percentage error when the field value is 1 g kg⁻¹ than it is when the field is 10 g kg⁻¹.

A skew T–logP moisture sounding for the April QNH case is presented in Fig. 5. There are two curves: the solid curve from the nature run and the dashed curve from the 3DVAR. The location of the sounding, picked to be in a moist area, is marked with an asterisk in Fig. 2. The 3DVAR captures the moist layer from the surface to 850 mb and the transition from moist to dry between 850 and 750 mb. The 3DVAR is drier in the layer from 700–400 mb. The lower accuracy seen in the upper troposphere is discussed in the next section. This sounding is an excellent example of the rationale for using the slant water vapor observations. The strong vertical gradient of moisture between 850 and 750 mb is very important, and could never be gleaned from the use of vertically integrated water vapor measurements.

b. Quantitative comparison

The OSS approach allows comparison of different analysis techniques against the “truth,” that is, the nature field. In this study, the 3DVAR analysis derived from the hypothetical 40-km-spaced network of GPS slant water vapor, surface stations, and a few vertical soundings can be compared with the Barnes analysis that would be obtained from the existing radiosonde network. Note that the radiosonde locations are shown in Fig. 3c.

We calculated three quantities: the percent error (pce), the rms error, and the bias. These quantities are defined as

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Table 1 shows the rms error of the two simulations. In both cases, the rms error of the 3DVAR slant water vapor network is smaller by a factor of 2 than the Barnes analysis of the radiosonde network. This substantial reduction should make a difference in model predictions and analysis of fields such as the lifted index. Similarly, the pce, shown in Table 2, indicates that the 3DVAR approach is better by a factor of two in both simulations.

We also studied the rms and bias as a function of height above ground. Figure 6 shows the rms for the April case. It can be seen that the rms of the Barnes analysis of radiosondes starts out quite large, about 2.8 g kg⁻¹ near the surface, and decreases with height up to 8 km. The general decrease is because the lower atmosphere is far more moist, with a typical scale height for moisture of between 2 and 3 km. The 3DVAR has an error of about 0.6 g kg⁻¹ near the surface, with similar values all the way to 8 km. Note that the goal of the 3DVAR algorithm is to equalize rms errors over the whole domain. It is clear from Fig. 6 that it was quite successful in this case, converging to a similar error regardless of altitude. As a result, the 3DVAR slant water vapor network analysis is far superior in the lower levels, but actually inferior at levels above 6.5 km. This suggests that if the moisture of the upper troposphere were more important than that of the lower troposphere, the 3DVAR could be weighted to do a better job at higher altitudes. In fact, the low- and midtropospheric moisture analysis is more important for weather prediction simply because the amount is so much larger in the lower levels.

The 3DVAR slant water vapor bias for the April case is shown in Fig. 7. The bias is near zero (unbiased) up to about 6 km, and negative (a moist bias) above that level. Evidently the 3DVAR is insensitive to the actual very dry air in the upper levels as it attempts to reach a minimum difference over the entire domain.

The estimate of rms errors as a function of height in Fig. 6 allows comparison with error estimates from other sources. The National Centers for Environmental Prediction (NCEP) recently published estimated errors for radiosondes (Zapotocny et al. 2000), and Smith et al. (2001) have estimated moisture errors from the RUC model. The rms errors at two levels, 850 mb and 500 mb, were interpolated from Fig. 6 and compared with the NCEP and RUC errors. The results, shown in Table 3, indicate that the radiosonde is most accurate at both levels, with the simulated GPS network showing a significantly reduced error compared to the RUC first-guess field and the Barnes analysis. Although inferences from a single case should be limited, it is encouraging to see the GPS network-derived errors much closer to the radiosonde at 850 mb than the model guess.

An important distinction must be made between the hypothetical analysis obtained from the radiosonde network versus the GPS slant water vapor network. Now, and in the foreseeable future, the radiosondes are only released twice a day in the United States. The hypothetical slant water vapor network would deliver a three-dimensional analysis every hour. Thus, the comparisons shown in this section would be significantly stronger in favor of the slant water vapor network since a typical analysis would be “off time,” that is, other than the radiosonde time. Continuous hourly moisture analysis would be helpful in severe convective weather prediction and other short-range forecast problems where rapid change is important (Emanuel et al. 1995).