**2. Using SPD to monitor hazardous convection**

GNSS-derived PWV is, so to speak, a representative value of PWV within an inverted coneshaped space above each GNSS antenna. GNSS-derived PWV is inherently unsuitable for monitoring water vapor variation under severe storms where several-kilometer-scale phenomena prevail. Is SPD suitable in this case? In this section, we first divide SPD into three components based on an SPD model in GNSS analysis, and discuss the horizontal scale of each component. Next, we propose new indices that represent finer water vapor distribution than the GNSS PWV. Finally, we introduce a procedure to analyze a several-kilometer-scale PWV distribution around each GNSS antenna using SPD.

Following MacMillan [26], the SPD between a GNSS satellite and a receiver at an elevation angle and a direction angle measured clockwise from north can be written in the following form:

$$\text{SPD}(\theta, \phi) = m\text{(\(\theta\)}[\text{ZTD} + \cot \theta (\mathcal{G}\_{\text{\(\(\theta\)}} \cos \phi + \mathcal{G}\_{\text{\(\(\theta\)}} \sin \phi))] + \varepsilon\_{\ell} \tag{1}$$

where ZTD, *m*(*θ*), *Gn* (*Ge* ) and *ε* are the total delay in the zenith direction, the isotropic mapping function that describes the ratio of SPD to ZTD, the delay gradient parameters in the north (east) directions, and the postfit phase residual, respectively. ZTD is vertically integrated refractivity (N) of the atmosphere in the zenith direction. Refractivity is expressed by temperature (*T*), partial dry air pressure (*Pd* ), and partial water vapor pressure (*Pw*):

$$N = (n - 1)\ 10^{6} = K\_{1}\frac{P\_{d}}{T} + K\_{2}\frac{P\_{w}}{T} + K\_{3}\frac{P\_{w}}{T^{2}}\tag{2}$$

where *n* is refractive index, and *K*<sup>1</sup> , *K*<sup>2</sup> , and *K*<sup>3</sup> are constants that have been determined theoretically or by fitting observed atmospheric data. Several studies have evaluated GPS-derived gradient parameters by comparison with those observed by water vapor radiometers (WVR), and found good agreement (e.g., [27–29]).

The gradient parameters represent the horizontal first-order gradients of water vapor. The postfit residuals are expected to contain information on higher order water vapor inhomogeneity (HI). However, other errors that do not originate from the atmosphere are also included (e.g., antenna phase center variation (PCV), signal scattering, multipath, errors in satellite orbit, and errors in clocks of both satellite and receiver). Therefore, to estimate HI, it is necessary to remove all errors not related to atmospheric inhomogeneity. Shoji et al. [1] performed a procedure to eliminate multi-path and satellite clock error-induced residuals to reconstruct HI components. The correlation coefficient of each component retrieved at a different GNSS station and sorted by distance demonstrated that the horizontal scale of the ZTD can be considered as 644 ± 120 km, the gradient parameter Gn, Ge as 62 ± 23 km, and the HI as 2–3 km. This result suggests that ZTD, G, and HI relate to atmospheric motion of the meso- α, meso- β, and meso- γ scales, respectively (**Figure 1**).

the SPD observations at storm scales. For instance, a GNSS signal with a 30° elevation angle at a GNSS receiver travels from the top of the troposphere to the receiver with a horizontal distance of 17 km. Thus, the SPD data cover only two model grid cells, when an assimilation system with 20-km grid spacing is applied. As a result, it is expected that the assimilation effect of SPD data would be similar to the ZTD assimilation. Therefore, it is important for SPD assimilations

In this chapter, the use of SPD data for the monitoring of hazardous convection is described in Section 2, and the first assimilation of SPD data with a DA system with 2-km grid spacing

GNSS-derived PWV is, so to speak, a representative value of PWV within an inverted coneshaped space above each GNSS antenna. GNSS-derived PWV is inherently unsuitable for monitoring water vapor variation under severe storms where several-kilometer-scale phenomena prevail. Is SPD suitable in this case? In this section, we first divide SPD into three components based on an SPD model in GNSS analysis, and discuss the horizontal scale of each component. Next, we propose new indices that represent finer water vapor distribution than the GNSS PWV. Finally, we introduce a procedure to analyze a several-kilometer-scale

Following MacMillan [26], the SPD between a GNSS satellite and a receiver at an elevation angle and a direction angle measured clockwise from north can be written in the following form:

ping function that describes the ratio of SPD to ZTD, the delay gradient parameters in the north (east) directions, and the postfit phase residual, respectively. ZTD is vertically integrated refractivity (N) of the atmosphere in the zenith direction. Refractivity is expressed by

> *P*\_\_*d <sup>T</sup>* + *K*<sup>2</sup> *P*\_\_\_*w <sup>T</sup>* + *K*<sup>3</sup>

retically or by fitting observed atmospheric data. Several studies have evaluated GPS-derived gradient parameters by comparison with those observed by water vapor radiometers (WVR),

The gradient parameters represent the horizontal first-order gradients of water vapor. The postfit residuals are expected to contain information on higher order water vapor inhomogeneity (HI). However, other errors that do not originate from the atmosphere are also included (e.g., antenna phase center variation (PCV), signal scattering, multipath, errors in satellite

[ZTD + cot*θ*(*Gn* cos*φ* + *Ge* sin*φ*)] + *ε*, (1)

), and partial water vapor pressure (*Pw*):

*P*\_\_\_*w*

are constants that have been determined theo-

*<sup>T</sup>*<sup>2</sup>, (2)

) and *ε* are the total delay in the zenith direction, the isotropic map-

to use assimilation systems with high grid spacings, hopefully less than 5 km.

(storm scale) is given in Section 3. The chapter is summarized in Section 4.

**2. Using SPD to monitor hazardous convection**

PWV distribution around each GNSS antenna using SPD.

SPD(*θ*, φ) = *m*(*θ*)

146 Multifunctional Operation and Application of GPS

(*Ge*

temperature (*T*), partial dry air pressure (*Pd*

where *n* is refractive index, and *K*<sup>1</sup>

and found good agreement (e.g., [27–29]).

*N* = (*n* − 1) 106 = *K*<sup>1</sup>

, *K*<sup>2</sup>

, and *K*<sup>3</sup>

where ZTD, *m*(*θ*), *Gn*

Sato et al. [30] compared zenith-scaled SPD using a mapping function and that retrieved from radiosonde observations. They found that the zenith-scaled SPD, in which the path is closest to a radiosonde path, exhibited better agreement than zenith total delay (ZTD) retrieved by standard GNSS analysis (i.e., a representative value of the inverted-cone-shaped space above the GNSS antenna).

Although it requires some careful effort to retrieve, GNSS SPD possesses information on local-scale atmospheric activity. Shoji [31] proposed procedures for retrieving two indices indicating the degree of inhomogeneity of water vapor using the GNSS SPDs. One index (WVC) describes the spatial concentration of water vapor (Eq. 3), whereas the other (WVI) indicates higher order water vapor inhomogeneity (Eq. 4). The horizontal scales of the two indices are considered to be approximately 60 km and 2–3 km, respectively.

$$\text{WVC} = -\nabla^2 \text{PWV}\_{\text{$$

where <sup>∇</sup> PWVG is the horizontal gradient of PWV estimated from the atmospheric gradient parameter (G). *i* − ¯

$$\begin{array}{ll} \stackrel{\circ}{\text{parameter (G)}} & \stackrel{\circ}{\text{ }} & \stackrel{\circ}{\text{ }} \\\\ \text{WVI = } \sqrt{\frac{1}{n} \sum\_{i=1}^{n} \left( \underline{H}\_{\text{PVV}}^{\prime} - \overline{\underline{H}\_{\text{PVV}}}^{\prime} \right)^{2}} & \end{array} \tag{4}$$

**Figure 1.** Correlation coefficients of (a) ZTD, (b) gradient component, and (c) postfit residuals as a function of distance. Gray-filled circles are from GEONET, and black-filled circles are from the Tsukuba GPS dense net campaign. Each data point is based on 51 days of data from July 14 to September 2, 2001. (Modified from **Figures 9**–**11** of Shoji et al. [1]).

where HIPWV is the inhomogeneity component of SWV normalized as a vertical value using a mapping function. To extract HIPWV from the postfit phase residual (*ε*), it is essential to eliminate the effects of GNSS antenna phase center variation (PCV), multipath effects, and errors in satellite orbits and clocks.

The statistical examination between these indices and precipitation (**Figure 2**) illustrate that the inhomogeneity indices show stronger correlation with rainfall amount than with PWV. It seems that PWV related to precipitation of less than 10 mm h−1, but was not connected to precipitation greater than 10 mm h−1. It is also true for both present and imminent precipitation. Furthermore, the spatiotemporal variations of the indices were also examined on a particular thunderstorm on August 11, 2011. Both the WVC and WVI indices increased ahead of the convective initiation (**Figure 3**). The WVI index is based on the standard deviation of the SPDs measured at a GNSS station, so the directional information for each SPD is neglected.

Shoji et al. [2] proposed a new method for estimating PWV distribution around each groundbased GNSS station on a scale of several kilometers (**Figure 4**).

**Figure 3.** Longitude–time cross sections on August 11, 2011. (a) 1-h accumulated precipitation, (b) PWV deviation from

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**Figure 4.** Schematic illustrating the difference in integrated refractivity (N) at height (Z) between the zenith direction and signal direction. The difference between ZTDSPD and ZTDEST is assumed to be caused by the horizontal gradient of

wet refractivity. Shoji et al. 2014 [2].

1-month average, (c) WVC index, and (d) WVI index. Modified from **Figure 14** of Shoji 2013 [31].

**Figure 2.** Frequency of precipitation in the last hour against PWV, WVC, and WVI index. (a) Frequency of 1-h precipitation (FD1P) against temperature (abscissa) and PWV (ordinate). Precipitation is less than 1 mm (a1), between 1 and 10 mm (a2), and more than 10 mm (a3). (b) FD1P against WVC (abscissa) and WVI indexes (ordinate). Modified from **Figure 10** of Shoji 2013 [31].

where HIPWV is the inhomogeneity component of SWV normalized as a vertical value using a mapping function. To extract HIPWV from the postfit phase residual (*ε*), it is essential to eliminate the effects of GNSS antenna phase center variation (PCV), multipath effects, and errors

The statistical examination between these indices and precipitation (**Figure 2**) illustrate that the inhomogeneity indices show stronger correlation with rainfall amount than with PWV. It seems that PWV related to precipitation of less than 10 mm h−1, but was not connected to precipitation greater than 10 mm h−1. It is also true for both present and imminent precipitation. Furthermore, the spatiotemporal variations of the indices were also examined on a particular thunderstorm on August 11, 2011. Both the WVC and WVI indices increased ahead of the convective initiation (**Figure 3**). The WVI index is based on the standard deviation of the SPDs

measured at a GNSS station, so the directional information for each SPD is neglected.

based GNSS station on a scale of several kilometers (**Figure 4**).

Shoji et al. [2] proposed a new method for estimating PWV distribution around each ground-

**Figure 2.** Frequency of precipitation in the last hour against PWV, WVC, and WVI index. (a) Frequency of 1-h precipitation (FD1P) against temperature (abscissa) and PWV (ordinate). Precipitation is less than 1 mm (a1), between 1 and 10 mm (a2), and more than 10 mm (a3). (b) FD1P against WVC (abscissa) and WVI indexes (ordinate). Modified

in satellite orbits and clocks.

148 Multifunctional Operation and Application of GPS

from **Figure 10** of Shoji 2013 [31].

**Figure 3.** Longitude–time cross sections on August 11, 2011. (a) 1-h accumulated precipitation, (b) PWV deviation from 1-month average, (c) WVC index, and (d) WVI index. Modified from **Figure 14** of Shoji 2013 [31].

**Figure 4.** Schematic illustrating the difference in integrated refractivity (N) at height (Z) between the zenith direction and signal direction. The difference between ZTDSPD and ZTDEST is assumed to be caused by the horizontal gradient of wet refractivity. Shoji et al. 2014 [2].

In this approach, the following three assumptions were set.


$$\mathbf{g}\_{\rm net}(Z) = \mathbf{g}\_{\rm net}(0) \exp\left(-\frac{z}{H}\right),\tag{5}$$

**Figure 5.** Variation in RMS difference as a function of distance from virtual GNSS stations. The gray thick line is PWVEST, the colored thin lines are PWVSPD at various elevation angles, and the red dots are the minimum RMS difference at each elevation angle. The number of samples is different according to satellite elevation, which ranges from 326,836 to

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**Figure 6.** PWV fields around a tornadic supercell. (a) NHM50m. (b) Both PWVEST and PWVSPD. (c) Only PWVEST. Black lines in (a) are the rainfall intensity of 40 mm h−1 reproduced by NHM50m; those in (b) and (c) are the reflectivity intensity of 40dBZ at the 0.5° elevation angle of the MRI C-band radar. Although the timing differs by about 30 minutes, the NHM simulation succeeded in reproducing the movement and development of the parent storm. (Modified from **Figure 7** of Shoji et al. [2]).

1,307,344. (Modified from **Figure 3** of Shoji et al. [3]).

where gwet(z) is the horizontal water-vapor gradient at altitude *Z*, and *H* is the scale height of gwet.

Under these assumptions, the horizontal PWV gradient can be expressed as the following equation.

$$
\nabla \text{PWD} = \Pi \,\nabla \text{ZTD} \approx \Pi \,\nabla \text{ZWD} = \Pi \frac{\text{ZTD}\_{\text{SO}} - \text{ZTD}\_{\text{ESI}}}{H \cot \{\theta\}},\tag{6}
$$

where Π is the proportionality coefficient. The horizontal gradient of PWV is expressed as a function of the ZTD difference, elevation angle, and scale height.

For practical monitoring of severe convection, we must carefully assess the accuracy of PWV distribution using the new method. Shoji et al. [3] quantitatively evaluated the method of Shoji et al. [2] using the simulation results of a high-resolution NWP model performed by Mashiko [32] for a tornadic supercell case, which generated an F3 tornado.

**Figure 5** plots the evaluation results at a particular moment in time (1208 JST, May 6, 2012). The thick gray lines represent the root mean square (RMS) difference of PWVEST against the true PWV field around each virtual GNSS station. PWVEST is the GNSS-derived PWV (not equal to the true PWV at virtual GNSS stations). In the figure, the RMS difference of PWVEST is 0.5 mm at the GNSS site. The RMS increased with distance and the value reached 2.5 mm at a distance of 3 km, when PWVEST was extrapolated as the PWV value near a virtual GNSS station. The thin colored lines in **Figure 5**, which illustrate distances at where the RMS difference was the smallest, show the distance increases as the elevation angle decreases. The RMS was the smallest within 1 km range from a virtual GNSS station for PWVSPD with a 77.6° elevation angle. In case of PWVSPD with a 17.6° elevation angle, RMS was the smallest of 1.5 mm at a 4.5 km distance. The distance and elevation dependency of minimum RMSE are illustrated by the red dots (**Figure 5**). Overall, the conventional procedure causes about RMS of 0.5 mm at the GNSS site, and the error by simple extrapolation increases with distance, reaching 1.5 mm at 1 km. The distance dependency of PWV errors in PWVSPD differs in each elevation angle. From this result, we can estimate PWV with RMSE of less than 2 mm within 6 km from a GNSS station by using SPD with an elevation angle over 15°. Essentially, with PWVSPD, it is able to estimate the PWV distribution around each GNSS station with better than half the RMSE of that obtained by the conventional GNSS PWV retrieval method (PWVEST).

In this approach, the following three assumptions were set.

150 Multifunctional Operation and Application of GPS

<sup>g</sup>*wet*(*Z*) <sup>=</sup> <sup>g</sup>*wet*(0) exp(−\_\_*<sup>z</sup>*

∇*PWV* = Π ∇*ZTD* ≈ Π ∇*ZWD* = Π

conventional GNSS PWV retrieval method (PWVEST).

function of the ZTD difference, elevation angle, and scale height.

Mashiko [32] for a tornadic supercell case, which generated an F3 tornado.

gwet.

equation.

**1.** The horizontal gradient of dry refractivity (Ndry) is small enough to be negligible.

**3.** The horizontal Nwet gradient (gwet) decreases exponentially with height.

**2.** The difference between ZTDEST (retrieved value through standard GNSS analysis) and ZTDSPD (normalized SPD into the zenith direction using a mapping function) is caused by the several-kilometer-scale horizontal gradient of water vapor refractivity (Nwet) alone.

where gwet(z) is the horizontal water-vapor gradient at altitude *Z*, and *H* is the scale height of

Under these assumptions, the horizontal PWV gradient can be expressed as the following

where Π is the proportionality coefficient. The horizontal gradient of PWV is expressed as a

For practical monitoring of severe convection, we must carefully assess the accuracy of PWV distribution using the new method. Shoji et al. [3] quantitatively evaluated the method of Shoji et al. [2] using the simulation results of a high-resolution NWP model performed by

**Figure 5** plots the evaluation results at a particular moment in time (1208 JST, May 6, 2012). The thick gray lines represent the root mean square (RMS) difference of PWVEST against the true PWV field around each virtual GNSS station. PWVEST is the GNSS-derived PWV (not equal to the true PWV at virtual GNSS stations). In the figure, the RMS difference of PWVEST is 0.5 mm at the GNSS site. The RMS increased with distance and the value reached 2.5 mm at a distance of 3 km, when PWVEST was extrapolated as the PWV value near a virtual GNSS station. The thin colored lines in **Figure 5**, which illustrate distances at where the RMS difference was the smallest, show the distance increases as the elevation angle decreases. The RMS was the smallest within 1 km range from a virtual GNSS station for PWVSPD with a 77.6° elevation angle. In case of PWVSPD with a 17.6° elevation angle, RMS was the smallest of 1.5 mm at a 4.5 km distance. The distance and elevation dependency of minimum RMSE are illustrated by the red dots (**Figure 5**). Overall, the conventional procedure causes about RMS of 0.5 mm at the GNSS site, and the error by simple extrapolation increases with distance, reaching 1.5 mm at 1 km. The distance dependency of PWV errors in PWVSPD differs in each elevation angle. From this result, we can estimate PWV with RMSE of less than 2 mm within 6 km from a GNSS station by using SPD with an elevation angle over 15°. Essentially, with PWVSPD, it is able to estimate the PWV distribution around each GNSS station with better than half the RMSE of that obtained by the

*<sup>H</sup>*), (5)

*<sup>H</sup>* cot(*θ*) , (6)

*ZTDSPD* <sup>−</sup> *ZTDEST* \_\_\_\_\_\_\_\_\_\_\_\_

**Figure 5.** Variation in RMS difference as a function of distance from virtual GNSS stations. The gray thick line is PWVEST, the colored thin lines are PWVSPD at various elevation angles, and the red dots are the minimum RMS difference at each elevation angle. The number of samples is different according to satellite elevation, which ranges from 326,836 to 1,307,344. (Modified from **Figure 3** of Shoji et al. [3]).

**Figure 6.** PWV fields around a tornadic supercell. (a) NHM50m. (b) Both PWVEST and PWVSPD. (c) Only PWVEST. Black lines in (a) are the rainfall intensity of 40 mm h−1 reproduced by NHM50m; those in (b) and (c) are the reflectivity intensity of 40dBZ at the 0.5° elevation angle of the MRI C-band radar. Although the timing differs by about 30 minutes, the NHM simulation succeeded in reproducing the movement and development of the parent storm. (Modified from **Figure 7** of Shoji et al. [2]).

The new method demonstrated the capability to capture a strong PWV gradient associated with the parent storm of the F3 tornado that struck Tsukuba City in Ibaraki Prefecture, Japan, on May 6, 2012, with a numerical model simulation at a super-high resolution of 50 m (NHM50m; **Figure 6**). An area of large PWV contrast centered on strong precipitation (**Figure 6a**) implies a strong upward wind in front of and a strong downdraft behind the parent storm. The PWV contrast toward the tornado is also seen in **Figure 6b**, whereas, no such PWV gradient is seen at all in **Figure 6c**.

with each weight set according to the distance. Finally, the slant path delays are calculated by integrating each delay in model grid cells from the receiver to the top boundary of the model.

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The observations over elevation angles of 5° or more were assimilated in their study. The operator works on the World Geodetic System 1984 (WGS84; National Imagery and Mapping

First, three experiments were performed using NHM-4DVAR with a 2-km horizontal grid spacing in which three SPD observations (SO\_SPD), one ZTD observation (SO\_ZTD), or one PWV observation (SO\_PWV) from a single observation site were assimilated. Note that the ZTD and PWV observations were derived from the SPD observations originally. These experiments were conducted to confirm the effects of SPD assimilation on a single analysis and to examine the differences between SPD, ZTD, and PWV assimilations. The assimilation window was set at 10 min, and the observations were assimilated every 5 min (at 0, 5, and 10 min in the assimilation window). The observational data set was chosen by considering the horizontal distributions, elevation angles, and the first guess field from an experimental data set. The propagation paths of GNSS signals from three satellites to a receiver in the model atmosphere in both the horizontal (**Figure 7a**) and the vertical plane (projected from the south; **Figure 7b**) are illustrated. Path I with the smallest elevation angle is also the longest, while path II at a near 90° elevation angle is the shortest. The large amount of delays is illustrated

The distributions of the increments (analysis minus first guess) of PWV in SO\_SPD (**Figure 8a**) are different from that in SO\_ZTD (**Figure 8b**) at the end of the assimilation window. In SO\_ZTD, the increment distribution (**Figure 8b**) is axisymmetric and elliptical mostly (i.e., isotropic). Although 4D-Var captures analysis increments shaped flow-dependent (anisotropic) in general, it is not

**Figure 7.** Propagation paths of GNSS signals from satellites (I, II, and III) to a receiver in the simulated atmosphere, (a) viewed in the horizontal plane and (b) the vertical plane. Colors illustrate the values of delay in each model grid cell (each cell is shown by one pixel). The open circle in (a) displays the GNSS receiver (the observational site). After

Agency 1997) [39].

Kawabata et al. [4].

**3.2. Assimilation experiment**

*3.2.1. Single set of SPD observations at a single site*

mostly in low altitudes area by warm colors.
