MARCH 2011 C H E N E T A L . 853

Observational Estimation of FORMOSAT-3/COSMIC GPS Radio Occultation Data

SHU-YA CHEN AND CHING-YUANG HUANG Department of Atmospheric , National Central University, Jhongli, Taiwan

YING-HWA KUO University Corporation for Atmospheric Research, and National Center for Atmospheric Research, Boulder, Colorado

SERGEY SOKOLOVSKIY University Corporation for Atmospheric Research, Boulder, Colorado

(Manuscript received 26 October 2009, in final form 20 August 2010)

ABSTRACT

The Global Positioning System (GPS) radio occultation (RO) technique is becoming a robust global ob- serving system. GPS RO refractivity is typically modeled at the ray perigee point by a ‘‘local refractivity operator’’ in a data assimilation system. Such modeling does not take into account the horizontal gradients that affect the GPS RO refractivity. A new observable (linear excess phase), defined as an integral of the refractivity along some fixed ray path within the model domain, has been developed in earlier studies to account for the effect of horizontal gradients. In this study, the error of both observables (refractivity and linear excess phase) are estimated using the GPS RO data from the Formosa Satellite 3–Constellation Observing System for Meteorology, Ionosphere and Climate (FORMOSAT-3/COSMIC) mission. The National Meteorological Center (NMC) method, which is based on lagged forecast differences, is applied for evaluation of the model forecast that are used for estimation of the GPS RO observational errors. Also used are Weather Research and Forecasting (WRF) model forecasts in the East Asia region at 45-km resolution for one winter month (mid- January to mid-February) and one summer month (mid-August to mid-September) in 2007. Fractional standard deviations of the observational errors of refractivity and linear excess phase both show an approximately linear decrease with height in the troposphere and a slight increase above the tropopause; their maximum magnitude is about 2.2% (2.5%) for refractivity and 1.1% (1.3%) for linear excess phase in the lowest 2 km for the winter (summer) month. An increase of both fractional observational errors near the surface in the summer month is attributed mainly to a larger amount of water vapor. The results indicate that the fractional of refractivity is about twice as large as that of linear excess phase, re- gardless of season. The observational errors of both linear excess phase and refractivity are much less latitude dependent for summer than for winter. This difference is attributed to larger latitudinal variations of the specific humidity in winter.

1. Introduction and The´paut 2006; Cucurull et al. 2007; Anthes et al. 2008; Cucurull and Derber 2008; Healy 2008). Several The Global Positioning System (GPS) radio occulta- observables, retrieved from GPS RO , tion (RO) technique has emerged as a robust global which range from raw excess phases to retrieved mois- observing system that provides valuable data to support ture and/or temperature profiles, can be used in data operational numerical weather prediction (e.g., Healy analysis and assimilation (Kuo et al. 2000). Such an observable as the bending angle, defined as the angle between the incoming and outgoing directions of a GPS- Corresponding author address: Ching-Yuang Huang, Dept. of Atmospheric Sciences, National Central University, 300 Jhongda transmitted electromagnetic ray (Kursinski et al. 2000), Rd., Jhongli City, Taoyuan County 32001, Taiwan. can be retrieved under the assumption of spherical sym- E-mail: [email protected] metry of refractivity. The refractivity can be retrieved

DOI: 10.1175/2010MWR3260.1

Ó 2011 American Meteorological Society 854 MONTHLY WEATHER REVIEW VOLUME 139 from the bending angle (using Abel inversion) and as- In this study we estimate observational errors of refrac- signed to the occultation (ray tangent) point. In the tivity and linear excess phase based on FORMOSAT-3/ past, many studies on the assimilation of Abel-retrieved COSMIC data over East Asia and the western Pacific refractivity and bending angle data demonstrated pos- for one winter and one summer month (referring to the itive impacts on regional as well as global weather pre- Northern Hemisphere). We investigate latitudinal de- diction (Kuo et al. 1998; Zou et al. 1999, 2000, 2004; Liu pendence of both refractivity and linear excess phase and Zou 2003; Huang et al. 2005; Healy et al. 2005; observational errors by stratifying the into Healy and The´paut 2006; Cucurull et al. 2006, 2007; different latitudinal bins. A brief introduction of the Healy 2008; Poli et al. 2009; Chen et al. 2009; Huang (including the observational error, local et al. 2010). and nonlocal operators, and the WRF model) and the In most of the regional data assimilation studies, a lo- design is given in section 2. The estimated cal refractivity operator is used to represent observational errors are discussed in section 3. Section 4 the GPS RO Abel-retrieved refractivity as local refrac- concludes this study. tivity. This introduces certain representativeness errors because the Abel-retrieved refractivity derived from 2. Methodologies and experiment design GPS RO observation is influenced by the atmosphere along the ray path. In a grossly simplified approximation, a. Apparent errors, observational errors, the Abel-retrieved refractivity is similar to a density- and forecast errors weighted average over the horizontal path of about Observational errors include errors and 300 km, centered at the ray perigee point. There can be representativeness errors (Daley 1991; Kuo et al. 2004; significant inhomogeneity in the horizontal along the Sokolovskiy et al. 2005a). While the observational errors 300-km averaging path. To reduce this representative- can be estimated theoretically, this is a very difficult task ness error, Sokolovskiy et al. (2005a) suggested a new and commonly they are estimated by comparing to other observable, a linear excess phase, which is defined as observations with uncorrelated error characteristics. In the integrated amount of refractivity along a fixed (e.g., this study we use the model forecast. Then the straight line) ray path, to account for the nonlocal nature 2 of the apparent error sa is related to the of the of the Abel-retrieved refractivity. This follows from the 2 2 observational and forecast errors so and sf as follows: smaller apparent errors normalized by the standard de- 2 2 2 viations of the weather-induced variations of the excess sa 5 so 1 s f . (1) phase than those of the refractivity, as was shown by Sokolovskiy et al. (2005b). Because the nonlocal obser- Note that the apparent errors are the total errors, de- vation operator can achieve better accuracy at reasonable fined as the differences between the model forecast and computational expenses, it has been implemented in the observation. This is equivalent to observation minus data assimilation system of the Weather Research and background (O 2 B), or the innovation, as defined in Forecasting (WRF; e.g., Liu et al. 2008; Chen et al. 2009) standard data assimilation literature (e.g., Daley 1991). and National Centers for Environmental Prediction Given the apparent errors, observational errors can be (NCEP) Gridpoint Statistical Interpolation (e.g., Ma determined once forecast errors have been calculated. et al. 2009). Forecast errors can be estimated using the National The Formosa Satellite 3–Constellation Observ- Meteorological Center (NMC) method (Parrish and Derber ing System for Meteorology, Ionosphere and Climate 1992) based on lagged forecast differences, or by calculating (FORMOSAT-3/COSMIC) mission is now providing the correlation between innovations with uncorrelated substantially more RO soundings (around 2000 sound- observational errors (Rutherford 1972; Hollingsworth and ings per day) with better penetration and smaller mea- Lo¨nnberg 1986). The Hollingsworth and Lo¨nnberg (H-L) surement errors in the lower troposphere (with the method needs complementary observations (e.g., radio- use of the open-loop tracking technique) than other sondes), whereas the NMC method does not. However, relevant missions such as Challenging Minisatellite the NMC method tends to underestimate model forecast Payload (CHAMP), which uses a phase-locked loop errors because of the partial cancellation of the repre- tracking technique [see Anthes et al. (2008)]. To ap- sentativeness errors. In this study, the NMC method was propriately assimilate the FORMOSAT-3/COSMIC used since over the ocean, where error estimates are im- GPS RO data, using either the local refractivity op- portant, there is an insufficient amount of radiosonde data erator or the nonlocal linear excess phase operator, to apply the H-L method. The differences between the one needs the observational error statistics based on forecasts at two different times, 24 h and 12 h, for the these two operators. WRF model are used in the NMC method in this study. MARCH 2011 C H E N E T A L . 855

FIG. 1. The geographical distributions of 2999 GPS RO soundings from COSMIC within the WRF model domain in the winter month (15 Jan–15 Feb 2007). The model grid indices are also shown at the left and bottom sides of the figure.

The observational error covariance matrix is used the error covariance matrix is a diagonal matrix. This in the minimization of the cost function. To reduce the diagonal error covariance matrix is constructed from computational cost, in the WRF three-dimensional vari- the fractional standard deviations (i.e., the standard ational data assimilation (3DVAR) it is assumed that deviations normalized by the observables). In this the observational errors are only autocorrelated and thus study, the observational errors of the two observables

TABLE 1. The numbers of the soundings for linear excess phase (EPH) and refractivity (REF) for both the winter and summer months in different latitudinal zones, and the differences (Diff) in the sets of data that do not pass the criterion (less than 5 standard deviations of the apparent errors).

Before passing the criterion After passing the criterion Winter Summer Winter Summer EPH 5 REF EPH 5 REF EPH Diff REF Diff EPH Diff REF Diff ,58N 324 331 318 26 314 210 331 0 331 0 58–158N 335 324 325 210 315 220 324 0 324 0 158–258N 475 488 470 25 460 215 484 24 485 23 258–308N 211 250 209 22 205 26 248 22 246 24 308–358N 255 246 255 0 254 21 246 0 244 22 358–408N 278 284 275 23 275 23 283 21 280 24 408–458N 413 351 413 0 413 0 350 21 350 21 .458N 708 571 707 21 708 0 571 0 570 21 Total 2999 2845 2972 227 2944 255 2837 28 2830 215 856 MONTHLY WEATHER REVIEW VOLUME 139

FIG. 2. Fractional differences between the observations (Sobs or Nobs) and model forecasts (Sfst or Nfst) at each model level for (a),(c) linear excess phase and (b),(d) refractivity in the (a),(b) winter and (c),(d) summer month. Angle brackets (hi) indicate the mean of the observations at each model mean height. The data before and after the criterion check are shown by the black lines and red lines, respectively.

(refractivity and linear excess phase) are represented in from the model variables and for the line integration of terms of fractional standard deviations for better visu- model refractivity (i.e., linear excess phase), respectively. alization of their vertical variations. c. Observation operators b. Prediction model and preprocessor The retrieved or derived observables are the Abel- The WRF model (version 2.1.2) was used in this study. retrieved atmospheric refractivity and the linear excess A description of the model can be found at the online (see phase (the integral of refractivity along a ray path). At- http://wrf-model.org/index.php) as well as in Skamarock mospheric refractivity N is related to the model pre- et al. (2005). The model is compressible and non- dictive variables (i.e., pressure, temperature, and water hydrostatic, and features multiple dynamical cores with vapor pressure; Bevis et al. 1994). In earlier studies, GPS high-order numerics to improve numerical accuracy. A RO refractivity was often assimilated by assuming that preprocessor is used to map the model variables (i.e., it was representative of a local value at the ray perigee pressure, temperature, and water vapor) from WRF point. Such modeling of GPS RO refractivity, though forecast grid fields to the time and location of the ob- computationally efficient, does not take into consider- servations. The WRF forecasts at 6-h intervals are lin- ation the effects of horizontal gradients and is associated early interpolated into the measurement time of RO with certain representativeness errors. soundings. Then the local and nonlocal observation op- To account for the effects of horizontal gradients, in the erators are used for the calculation of model refractivity nonlocal excess phase operator (Sokolovskiy et al. 2005a), MARCH 2011 C H E N E T A L . 857

FIG. 3. Fractional standard deviations of the apparent error (thick solid lines), forecast error (dotted lines), and observational error (thin solid lines) for (a),(c) linear excess phase and (b),(d) refractivity in the (a),(b) winter and (c),(d) summer months. The counts of GPS RO soundings are presented by circles. the refractivity is integrated along a fixed ray path, which model refractivity, thus accounting for the effects of approximately models the true ray path but does not horizontal gradients, and for the observational counter- depend on refractivity. We note that thus defined linear part, one-dimensional (1D) Abel-retrieved refractivity excess phase used in this study is different from the (where the effects of horizontal gradients are contained commonly used excess phase of GPS RO measurements, in convolved form). Thus, Eq. (2) allows an arbitrary which is the phase between transmitter and receiver in factor, which is to 1. Integration of the refractivity excess to the phase in a vacuum. A geometric illustration along the ray path is stopped once the ray has reached the of the linear excess phase operator is given in Fig. 1 of model boundaries [e.g., upper, lateral, or lower (terrain)]. Sokolovskiy et al. (2005b). In the simplest case (which d. Data processing results in reasonably good accuracy), this fixed ray path is a straight-line tangent to the real ray path at Application for the nonlocal operator is briefly de- the perigee point. The linear excess phase, which is scribed as follows. First, we obtained retrieved profiles, treated as a new observable linear on refractivity, is the atmPrf files, with the highest level of near 60 km defined as from the COSMIC Data Analysis and Archival Center ð (CDAAC; http://www.cosmic.ucar.edu), which include B the longitude, latitude, and height of a tangent point, S 5 Ndl, (2) A the azimuth of the ray, and the refractivity at tangent point. Descriptions of data processing at CDAAC can where dl is the differential ray path, and A and B are be found in Kuo et al. (2004) and Ho et al. (2009). Then, the incoming and outgoing points of the straight line we define the mean model vertical grid by averaging within the model domain, respectively. The linear ex- heights of all model grid points for each vertical level. cess phase S is calculated in the same way for the 3D When interpolating observations to this mean model 858 MONTHLY WEATHER REVIEW VOLUME 139 vertical grid, the observational latitude, longitude, azi- muth, and logarithm of refractivity are vertically aver- aged between the midpoints of the contiguous mean model layers above and below the mean vertical level. For calculation of the linear excess phase, both the ob- servation and the model refractivities are integrated along a ray path according to the observed azimuthal direction with a step size of 5 km, starting from the tan- gent point in both directions and ending at the model boundaries (lateral, upper, or lower). For the integration, both the model and the observational refractivities are log-linearly interpolated between the mean model levels. Although calculation of the linear excess phase for the observational refractivity, which is treated as spherically symmetric, can be reduced to a 1D integral, it is per- formed in exactly the same way as for the 3D model refractivity in order to eliminate the error arising from different discrete representations. The nonlocal opera- tor in WRF 3DVAR may also assimilate refractivity simply by deactivating the ray’s integration (Chen et al. 2009). The horizontal smear of tangent point trajecto- ries has been taken into account both in local and non- local operators; additionally, the nonlocal operator uses the azimuth of the ray at tangent point. e. Model settings and calculation steps In this study, we use the WRF forecast model in a single domain (151 3 151 grids) at horizontal resolution of 45 km, as shown in Fig. 1, covering latitudes from about 108Sto508N and longitudes from 808 to 1608E, with a model top of 50 hPa (31 model vertical levels) with varied vertical resolution. The horizontal resolu- tion of 45 km in this study should be appropriate for FIG. 4. Fractional standard deviations of the observation errors the error estimation according to Chen et al. (2006). of linear excess phase (solid lines) and refractivity (dashed lines) They computed the innovation using the fifth-generation to the south (thin lines) and to the north (thick lines) of 308N in the Pennsylvania State University–National Center for At- (a) winter and (b) summer months. mospheric Research (NCAR) Mesoscale Model (MM5) with different horizontal resolutions and found that the (Fig. 1 shows the distribution of the winter soundings) representativeness error from 30-km resolution is com- available from the FORMOSAT-3/COSMIC mission. patible with that of 10-km resolution. In this study, the The RO data counts decrease near the surface because model physical parameterizations include the cumulus of different penetration of retrieved profiles. Only about parameterization of the Kain–Fritsch scheme (Kain 2004), 500 (800) RO soundings penetrate close to surface for the cloud microphysics of the Purdue–Lin scheme (Chen the summer (winter) month. Table 1 provides a list of the and Sun 2002), and the planetary boundary layer pa- numbers of soundings within different latitudinal bins rameterization of the Yonsei University (YSU) scheme in the model domain. There are several hundred RO (Hong et al. 2006) (for details of the schemes, please soundings in each bin that passed CDAAC quality con- refer to http://wrf-model.org/index.php). trol, sufficient for reliable statistical estimation of the The 12- and 24-h forecasts are taken from the WRF observational errors. Data quality control has been con- model predictions for two periods, each of one month, ducted routinely at CDAAC (see Kuo et al. 2004). In one (15 January–15 February 2007) during the winter and addition, the observations (refractivity and linear excess the other (15 August–15 September 2007) during the phase) are discarded in this study if they deviate from the summer. During the winter (summer) month, there are model forecasts by more than five standard deviations 2999 (2845) RO soundings within the model domain of the apparent errors (we use the same criteria as in MARCH 2011 C H E N E T A L . 859

FIG. 5. Fractional standard deviations of the observational errors stratified in different latitudinal bins (color lines) for (a),(c) linear excess phase and (b),(d) refractivity for the (a),(b) winter and (c),(d) summer months.

WRF 3DVAR), as an additional quality check to re- d Step 3: The observational errors of both refractivity move (Table 1). and linear excess phase are computed using Eq. (1) at The steps for calculation of the observational errors each vertical level for the two months. Although, in are outlined as follows: general, the NMC method tends to underestimate the forecast errors, there are still a limited number of cases d Step 1: The NCEP Aviation (AVN) analyses were when the forecast errors are close to or exceed the used as the initial condition for the WRF model. The apparent errors. When this happens, the observational model was initialized four times (0000, 0600, 1200, and error variance calculated from Eq. (1) is close to zero 1800 UTC) each day during the two months, and each or negative, thus having no physical meaning. These model forecast was integrated for 24 h. Thus, for each cases were considered outliers and removed from the month, there are a total of 125 forecasts, including dataset in this study. 0000 UTC on the last day. d Step 2: The forecast errors are evaluated using the NMC method, which computes the differences be- tween two consecutive forecasts (24- and 12-h fore- 3. Observational error estimation casts) projected to the observation space. The daily a. The fractional errors 12- and 24-h forecasts for the summer and winter months are interpolated to the time and location of The fractional apparent errors before (black curves) RO measurements. The modeled refractivity and lin- and after (red curves) passing the abovementioned cri- ear excess phase are then calculated using the obser- teria for both winter and summer months are shown in vation operators. Thus, the standard deviations of the Fig. 2. We truncate error profiles at ;18 km because forecast errors are derived. close to the model top (50 hPa), the linear excess phase 860 MONTHLY WEATHER REVIEW VOLUME 139

FIG. 6. As in Fig. 5, but for 12-h forecast errors. basically represents the local refractivity (because of the observables (Sokolovskiy et al. 2005b). Calculation of the short ray path). The largest errors (outliers) occur mainly linear excess phase from refractivity (which is reduced in the lower troposphere, which may not necessarily in- to the Abel transform in spherically symmetric case) dicate the largest fractional observational errors since not only reduces the magnitude of the variations and forecast errors may also be large. Although the num- the errors but also increases their vertical correlations. bers of outliers are rather limited after the quality control This follows from the frequency response of the Abel applied by CDAAC, we still find the additional criteria transform (Lohmann 2005). However, in this study we useful. do not consider the vertical error correlations. Figure 3 The fractional apparent errors of refractivity (linear shows the fractional apparent, forecast, and observa- excess phase) after applying the additional criteria, in tional errors (i.e., fractional standard deviations) of linear general, show approximately linear decay with height excess phase and refractivity for both months. The ra- in the troposphere, and their maximum magnitude in tio of the fractional errors of refractivity and linear the lowest several kilometers is about 12% (7%) for excess phase in the lower troposphere, 2:1, appears to both the winter and summer months. Linear excess phase hold true for the apparent errors as well as for the errors are smaller than refractivity errors, especially in forecast errors, and thus also for the observational er- the lower troposphere, by about a factor of 2. This re- rors. Fractional observational errors have minima at duction is to be expected because the linear excess phase approximately 10 km and maxima in the lower tropo- is related to integration of refractivity, and by itself does sphere, 2.2% (2.5%) for refractivity and 1.1% (1.3%) not prove reduction of the representativeness errors re- forlinearexcessphaseinthewinter(summer)month. lated to horizontal gradients. The latter follows from re- Slightly larger errors in the summer are likely related duction of the apparent errors normalized by the standard to a larger amount of water vapor in the lower tropo- deviations of the weather-related variations of the sphere. MARCH 2011 C H E N E T A L . 861

FIG. 7. The monthly zonally averaged specific humidity of 12-h WRF forecasts in different latitudinal bins (color lines) in the (a) winter and (b) summer months. (c),(d) As in (a),(b), respectively, but for the of the forecast errors (24-h forecast minus 12-h forecast). b. Latitudinal dependence fractional observational errors of refractivity for the win- ter month have minima at 8 km (12 km) and maxima of Latitudinal variations of the observational errors of 1.1% (3.0%) near the surface for higher (lower) latitudes, refractivity have been investigated by Kuo et al. (2004) in good agreement with the results of Fig. 13 in Kuo et al. using CHAMP data from December 2001. To investi- (2004). gate latitudinal dependency of the observational errors For further investigation, we stratify the RO obser- of both refractivity and linear excess phase, we stratified vations in different latitudinal bins of 58–108 (with suf- the RO observations by latitude. Figure 4 shows the ficient amount of data in each bin) and show the results fractional observational errors of refractivity and linear in Fig. 5. An additional piece of contained excess phase for latitudes southward and northward of in Fig. 5 compared to Fig. 4 is that the maximal fractional 308N for both months. In the winter month, the frac- observational errors in the lower troposphere for the tional observational errors in the lower troposphere are winter month, reaching maxima of about 3.7% (1.7%) for about 3 times larger in lower latitudes than those in refractivity (linear excess phase), are not at the equator, higher latitudes, with maxima of about 3.0% (1.5%) for but at the 158–258 latitude band. This largest fractional refractivity (linear excess phase). In the summer month, observational error is most likely related to larger errors when the intertropical convergence zone shifts to the of the GPS RO (due to superrefraction and insufficient north and covers most of the model domain, the lat- tracking depth) (Sokolovskiy 2003) and also may be re- itudinal dependency of errors is not pronounced. For all lated to underresolving of the sharp atmospheric bound- months and latitudes the fractional apparent errors of ary layer (ABL) in the subtropical region by the model the linear excess phase in the lower troposphere are (resulting in underestimation of the model errors). For about a factor of 2 smaller than those of the refractiv- both observational errors near the surface, the ratio of ity [in agreement with Sokolovskiy et al. (2005b)]. The the largest to the smallest magnitudes in the considered 862 MONTHLY WEATHER REVIEW VOLUME 139

FIG. 8. The estimated (dashed lines) and fitted (solid lines) fractional standard deviations of the observational errors stratified in different latitudinal bins for (a) linear excess phase and (b) refractivity in the winter month. (c),(d) As in (a),(b), but for the estimated (dots) and fitted (red lines) observational errors in the summer month. latitudinal band is about 5:1 for the winter month but humidity of 12-h WRF forecasts in the winter month, reduces to about 2:1 for the summer month. The vari- as shown in Fig. 7a. Weaker latitudinal dependence of ability of the errors is related to the variability of the the forecast errors in summer is intimately related to real atmosphere and needs to be taken into account in the smaller latitudinal variations of the corresponding assimilation. monthly averaged specific humidity (Fig. 7b). A com- To provide a better understanding of the latitudinal parison of Fig. 6 and Figs. 7c,d shows that the latitudinal structure of errors, the forecast errors, stratified in the dependences of the forecast errors are very similar to same way as the observational errors (Fig. 5), are shown the forecast differences in specific humidity. This sug- in Fig. 6. For the winter month, fractional forecast errors gests that the latitudinal dependency of the observational for both the refractivity and linear excess phase exhibit errors is in fact strongly affected by the atmospheric significant latitudinal variations, which in general are moisture. The latitudinal and vertical variations of the consistent with variations of the fractional observational apparent (total) errors are more similar to the variations errors, except that the largest fractional forecast errors of the observational errors in both summer and winter are found in the band of 58–158 rather than in 158–258 (figures not shown). This is because the forecast errors, on as for the fractional observational errors (Figs. 5a,b). average, are smaller than the observational errors. However, such similarity between the fractional fore- The vertical profiles of the observational errors esti- cast and observational error variations is not as obvious mated in this study contain small-scale structures, es- in the summer month. The latitudinal dependence also pecially in the lower troposphere. We have attempted can be found in the monthly, zonally averaged specific to fit the vertical profiles of observational errors with M ARCH 01CHENETAL. L A T E N E H C 2011

TABLE 2. Fractional standard deviations (%) of the linear excess phase S and refractivity N errors in the winter and summer months after polynomial fitting for different latitudinal bins and heights.

S (Winter) N (Winter) S (Summer) N (Summer) H (km) ,58N58–158N158–258N258–308N308–358N358–408N408–458N .458N ,58N58–158N158–258N258–308N308–358N358–408N408–458N .458N108S–508N108S–508N 0.5 1.055 1.554 1.612 1.215 0.486 0.445 0.428 0.338 2.122 3.771 3.078 2.564 1.265 1.140 0.898 0.727 1.227 2.742 1.0 1.061 1.474 1.568 1.088 0.608 0.520 0.465 0.347 1.863 2.407 2.759 2.067 1.386 1.157 0.937 0.750 1.202 2.242 1.5 1.118 1.469 1.624 1.063 0.696 0.507 0.444 0.364 1.905 2.466 3.167 2.139 1.515 1.128 0.939 0.787 1.135 2.157 2.0 1.132 1.411 1.591 1.034 0.730 0.474 0.413 0.380 1.978 2.786 3.466 2.253 1.606 1.052 0.898 0.807 1.058 2.173 2.5 1.097 1.288 1.449 0.977 0.716 0.446 0.387 0.389 1.998 2.933 3.439 2.243 1.633 0.953 0.833 0.802 0.982 2.160 3.0 1.033 1.136 1.247 0.904 0.671 0.427 0.367 0.386 1.964 2.837 3.146 2.111 1.588 0.855 0.764 0.776 0.912 2.087 3.5 0.964 0.988 1.038 0.828 0.612 0.413 0.351 0.373 1.898 2.576 2.724 1.909 1.480 0.773 0.705 0.734 0.848 1.967 4.0 0.903 0.868 0.860 0.759 0.550 0.401 0.336 0.351 1.822 2.254 2.296 1.691 1.328 0.710 0.660 0.681 0.789 1.822 4.5 0.853 0.779 0.725 0.697 0.494 0.388 0.321 0.326 1.745 1.952 1.932 1.489 1.155 0.663 0.626 0.624 0.733 1.673 5.0 0.810 0.715 0.631 0.640 0.446 0.374 0.305 0.303 1.664 1.707 1.652 1.313 0.982 0.624 0.596 0.565 0.680 1.531 5.5 0.765 0.663 0.563 0.581 0.405 0.362 0.292 0.285 1.574 1.520 1.443 1.161 0.828 0.587 0.565 0.508 0.629 1.401 6.0 0.713 0.613 0.508 0.518 0.370 0.353 0.281 0.275 1.467 1.374 1.274 1.023 0.703 0.549 0.529 0.458 0.580 1.278 6.5 0.651 0.557 0.455 0.452 0.339 0.347 0.276 0.274 1.339 1.244 1.120 0.893 0.612 0.509 0.490 0.419 0.532 1.161 7.0 0.581 0.494 0.399 0.388 0.314 0.346 0.275 0.281 1.193 1.110 0.964 0.773 0.554 0.474 0.452 0.394 0.487 1.045 7.5 0.508 0.428 0.343 0.333 0.294 0.348 0.278 0.294 1.037 0.965 0.806 0.668 0.524 0.448 0.422 0.385 0.444 0.931 8.0 0.440 0.368 0.295 0.296 0.283 0.353 0.284 0.311 0.883 0.811 0.659 0.588 0.515 0.437 0.407 0.393 0.403 0.819 8.5 0.384 0.321 0.260 0.280 0.281 0.359 0.293 0.330 0.742 0.662 0.537 0.541 0.520 0.447 0.412 0.417 0.366 0.715 9.0 0.343 0.291 0.244 0.289 0.291 0.365 0.304 0.347 0.622 0.531 0.454 0.531 0.534 0.477 0.440 0.452 0.333 0.621 9.5 0.319 0.281 0.248 0.318 0.313 0.372 0.317 0.361 0.528 0.431 0.418 0.553 0.553 0.523 0.487 0.493 0.304 0.542 10.0 0.309 0.287 0.267 0.360 0.347 0.379 0.332 0.372 0.460 0.367 0.420 0.596 0.576 0.578 0.546 0.533 0.280 0.481 10.5 0.307 0.301 0.291 0.404 0.390 0.387 0.350 0.380 0.413 0.336 0.447 0.643 0.602 0.633 0.607 0.565 0.262 0.435 11.0 0.308 0.315 0.311 0.441 0.436 0.399 0.370 0.385 0.379 0.330 0.475 0.676 0.631 0.678 0.661 0.583 0.250 0.404 11.5 0.307 0.323 0.319 0.462 0.480 0.413 0.392 0.388 0.354 0.335 0.485 0.682 0.662 0.705 0.697 0.585 0.244 0.383 12.0 0.303 0.321 0.310 0.464 0.515 0.429 0.413 0.391 0.334 0.341 0.466 0.656 0.693 0.710 0.709 0.570 0.243 0.368 12.5 0.298 0.313 0.290 0.451 0.537 0.445 0.431 0.396 0.320 0.343 0.421 0.606 0.718 0.695 0.696 0.543 0.245 0.357 13.0 0.300 0.306 0.270 0.433 0.541 0.458 0.442 0.402 0.321 0.345 0.370 0.553 0.730 0.668 0.664 0.511 0.251 0.350 13.5 0.317 0.312 0.267 0.423 0.528 0.465 0.443 0.413 0.343 0.359 0.342 0.523 0.722 0.637 0.620 0.483 0.258 0.349 14.0 0.356 0.340 0.292 0.434 0.501 0.463 0.435 0.430 0.394 0.396 0.367 0.537 0.691 0.612 0.577 0.467 0.267 0.358 863 864 MONTHLY WEATHER REVIEW VOLUME 139 several fitting curves from a polynomial of tenth order of specific humidity fields, the latitudinal dependence of for each of the latitudinal zones, as shown in Fig. 8. The the observational error is found to be influenced by the fitting curves can represent the gross characteristics of variations of the atmospheric moisture. Such latitudinal the observational errors with respect to latitudes and dependence of the observational errors in winter and seasons. As a result of stronger latitudinal dependence summer must be taken into account in RO data assim- in the winter month, eight fitting curves are derived for ilation. The errors for other seasons may be approxi- different latitudinal bins (Figs. 8a,b). Because the obser- mately evaluated by interpolation between winter and vational errors for different bins do not vary consider- summer. ably in the summer month, we can obtain two well-fitting This study uses the regional WRF model to estimate curves for refractivity and linear excess phase as shown observational errors, and the results are most useful for in Figs. 8c and 8d, respectively. For convenience, a numerical weather prediction applications over this re- lookup table (Table 2) is provided for the fractional gion at comparable resolution. Application of these re- standard deviations of the observational errors below sults for other regions of the world (which may have the height of 14 km for the winter and summer months. different moisture structure and variability) or for models Above 14 km, the fractional observational errors, as with substantially different horizontal and vertical res- shown in Fig. 5, increase with height for linear excess olutions may not be warranted. Because the estimated phase; this increase is less pronounced for refractivity. observational errors depend on atmospheric structures Therefore, the fractional observational errors of re- and their representation by a model, similar studies shall fractivity at 14 km can be used as a proxy above 14 km be conducted for applications related to other regions and/ for both months. 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