Region-Specific and Weather-Dependent Characteristics of the Relation between GNSS-Weighted Mean Temperature and Surface Temperature over China

Abstract

Weighted mean temperature of the atmosphere, $T_m$, is a key parameter for retrieving the precipitable water vapor from Global Navigation Satellite System observations. It is commonly estimated by a linear model that relates to surface temperature $T_s$. However, the linear relationship between $T_m$ and $T_s$ is associated with geographic regions and affected by the weather. To better estimate the $T_m$ over China, we analyzed the region-specific and weather-dependent characteristics of this linear relationship using 860,054 radiosonde profiles from 88 Chinese stations between 2005 and 2018. The slope coefficients of site-specific linear models are 0.35~0.95, which generally reduce from northeast to southwest. Over southwest China, the slope coefficient changes drastically, while over the northwest, it shows little variation. We developed a $T_s\sim T_m$ linear model using the data from rainless days as well as a model using the data from rainy days for each station. At half the stations, mostly located in west and north China, the differences between the rainy-day and rainless-day $T_m$ models are significant and larger than 0.5% (1%) in mean (maximal) relative bias. The regression precisions of the rainy-day models are higher than that of the rainless-day models averagely by 28% for the stations. Radiosonde data satisfying $T_m-T_s>10\mathrm{K}$ and $T_s-T_m>30\mathrm{K}$ most deviate from linear regression models. Results suggest that the former situation is related to low surface temperature (<270 K), as well as striking temperature and humidity inversions below 800 hPa, while the latter situation is related to high surface temperature (>280 K) and a distinct humidity inversion above 600 hPa.

1. Introduction

Retrieval of precipitable water vapor (PWV) from Global Navigation Satellite System (GNSS) observations is a major work of ground-based GNSS meteorology [1,2,3,4]. This technique has all-weather and all-time capabilities, and can provide water vapor products with high spatiotemporal resolution as the continuous GNSS observing network grows and the data processing strategy improves [5]. The GNSS-derived PWV are consistent with the radiosonde and ERA5 data [6,7]. Nowadays, the use of ground-based GNSS, effectively complementing the conventional water vapor observations, plays an important role in weather and climate studies, such as water vapor variation [8,9], satellite product validation [10,11], deep convection and rainstorm observation [12,13,14], and monsoon and atmospheric river monitoring [15,16]. In GNSS water vapor retrieving, a crucial step is to convert zenith wet delay (ZWD) into the PWV [17]. The conversion factor between ZWD and PWV is a function of the weighted mean temperature of the atmosphere (hereinafter referred to as $T_m$), which is a quantity defined by Davis et al. [18].

It is worthwhile to obtain the $T_m$ as accurate as possible since the uncertainty of the $T_m$ is the dominant error source affecting the conversion between ZWD and PWV [19]. Since the vertical profiles of temperature and humidity usually cannot be accessed at the GNSS stations, the calculation of the $T_m$ from the integral equation by Davis et al. [18] is not always feasible in practical applications. To overcome this, Bevis et al. [1] took advantage of the significant linear correlation between $T_m$ and surface temperature (referred to as $T_s$) to develop a linear model ($T_m=0.72T_s+70.2$) from the radiosonde profiles observed over U.S. continent. This linear model has been used widely due to its easy implementation. Although the Bevis model is in essence a regional model, it was used as a global model [20,21,22] or as the reference model to validate other global or regional models [23,24,25]. Nevertheless, Ross and Rosenfeld [26] showed that the linear relationship between $T_m$ and $T_s$ varied with geographic location, indicating that a single linear model cannot assure high accuracy of $T_m$ calculation for the whole globe. Wang et al. [27] evaluated the Bevis model using the global ERA-40 data and found that the mean bias of $T_m$ generated from this model has a range of ±10 K (relative bias of ±3.5%). To obtain accurate $T_m$, geodesists and meteorologists developed regional models (with form similar to the Bevis model or a little more complicated) for their concerned regions, e.g., Netherlands [28], Indian [24,29], Brazil [30], Australia [31], European region [8,32], West Africa [33], Greenland [34], Taiwan [35], and Hong Kong [36,37]. Additionally, there are some empirical models that only require the inputs of location and time to estimate the $T_m$ values. The representatives of these models are the GPT model series, GPT2w [38] and GPT3 [39], and the GTm model series, GTm-I [40], GTm-II [21], and GTm-III [41]. These $T_s$-independent empirical models are very useful especially for those GNSS stations without surface temperature observations. However, when the surface temperatures are available, $T_s$-dependent models are better options because that they generally yield more accurate $T_m$ values than the $T_s$-independent models [23,32,42,43].

During the past two decades, the number of GNSS stations grew rapidly in China. The Meteorological Observation Center, China Meteorological Administration collects the data of more than 1000 GNSS stations covering the whole mainland China, and calculates the PWV values on a daily basis serving for weather analysis, forecasting, and scientific studies [44]. These stations are mostly from China Meteorological Administration GNSS Network (CMAGN) and Crustal Movement Observation Network of China (CMONOC), and equipped with meteorological sensors that record the surface pressure and temperature for GNSS PWV retrieving. Since the surface temperature is available, the simple linear $T_s\sim T_m$ models can be used to calculate $T_m$. However, China is a geographically large country with a variety of climate regions. Previous studies [45,46,47,48] showed the coefficients of regional $T_s\sim T_m$ linear models for different areas in China are more or less different. This motivates us to comprehensively study and obtain the accurate linear relationships between $T_m$ and $T_s$ over China with the purpose of providing the basis for high accuracy $T_m$ estimation in the region.

In this study, using 14-year data from 88 Chinese radiosonde stations, we developed a unified $T_s\sim T_m$ linear model for the whole China as well as site-specific linear models for individual stations, and calculated the representativeness errors of the unified model relative to the site-specific models. Then, we analyzed the variation of the linear relation between $T_m$ and $T_s$ with geographic locations and different weather occurrences. We also investigated the regression precision of the generated $T_s\sim T_m$ linear models and the weather conditions related to the data that most deviate from the regressions.

2. Methods and Data

2.1. Role of $T_m$ in GNSS Water Vapor Retrieving

Before the determination of $T_s\sim T_m$ linear relations with radiosonde data, we briefly review the role of $T_m$ in the retrieval of water vapor from GNSS observations. In GNSS high-precision positioning, especially when the precise point positioning (PPP) is applied, the zenith tropospheric delay (ZTD) of L-band signals over a station is estimated simultaneously with the coordinate components. The ZTD includes two parts: zenith hydrostatic delay (ZHD) and ZWD. With a known surface pressure, the ZHD can be estimated with an accuracy of millimeter level. Subtracting the ZHD from the ZTD remains the ZWD. The PWV is obtained from ZWD as [19]

$$\mathrm{PWV}=\Pi \cdot \left(\mathrm{ZTD}-\mathrm{ZHD}\right)=\Pi \cdot \mathrm{ZWD}$$

(1)

where $\Pi$ is a dimensionless mapping factor. Its expression is

$$\Pi =\frac{{10}^6}{\rho R_v\left(\frac{k_3}{T_m}+k_2^{\prime }\right)}$$

(2)

where $\rho$ is the density of liquid water, $R_v$ is the specific gas constant for water vapor, and $k_2^{\prime }$ and $k_3$ are constants that have been evaluated by the actual measurement of refractivity index of the atmosphere [19,49]. Since $\rho$, $R_v$, $k_2^{\prime }$, and $k_3$ are all known constants, $T_m$ is the only quantity to be determined for the conversion of ZWD into PWV.

2.2. Determination of $T_s\sim T_m$ Linear Models

The $T_s\sim T_m$ linear model is expressed as

$$T_m=a{T}_s+b$$

(3)

where all temperatures are in kelvins. $a$ (slope) and $b$ (intercept) are regression coefficients. To determine $a$ and $b$ in regression analysis, a number of $T_s\sim T_m$ data pairs are required to be known. Due to global distribution of sites and long-term data accumulation, radiosonde is a good data source to derive $T_s$ and $T_m$ for establishing $T_s\sim T_m$ linear models over land.

Surface temperatures are obtained from the first level of radiosonde profiles, and weighted mean temperatures are calculated from water vapor pressures and temperatures at all levels. The definition of $T_m$ is [18]

$$T_m=\frac{{\displaystyle {\int }_{H_S}^{H_T}\frac{P_w}{T}dH}}{{\displaystyle {\int }_{H_S}^{H_T}\frac{P_w}{T^2}dH}}$$

(4)

where $P_w$ is the partial pressure of water vapor, $T$ is the temperature of the atmosphere, $H_T$ is the height of the top of the atmosphere, and $H_S$ is the starting altitude. For GNSS meteorology, $H_S$ is the height of GNSS antenna. When calculating $T_m$ from radiosonde profiles, the discrete form of Equation (4) is used as

$$T_m=\frac{{\displaystyle \sum _{i=1}^N\frac{P_{wi}}{T_i}\Delta H_i}}{{\displaystyle \sum _{i=1}^N\frac{P_{wi}}{T_i^2}\Delta H_i}}$$

(5)

where $N$ is the total number of layers of the atmosphere with one layer defined as the atmosphere between two consecutive levels of a radiosonde profile, $T_i$ is the average temperature in the $i$ layer, $\Delta H_i$ is the thickness of the $i$ layer, and $P_{wi}$ is the average water vapor pressure in the $i$ layer. Generally, the radiosonde profiles do not directly provide partial pressure of water vapor $P_w$, and instead provide total atmospheric pressure $P$ and mixing ratio of water vapor $m_x$, from which the $P_w$ can be derived by [50].

$$P_w=\frac{m_x}{m_x+622}P$$

(6)

where $m_x$ is in g/kg.

2.3. Radiosonde Data

The radiosonde data of 88 stations selected from 2005 to 2018 (14 years) are used to analyze the $T_s\sim T_m$ relations over China. Figure 1 shows the geographic distribution of the radiosonde stations and the topography. The station information is shown in

Table A1. Radiosonde station information and site-specific $T_s\sim T_m$ linear models. Columns 1 (C1)–4 (C4) present the information of the 88 radiosonde stations. Column 1 shows the series number of the stations. Column 2 shows the station ID and station name. Column 3 shows the latitude and longitude of the stations. Column 4 shows the altitude of the stations. The site-specific $T_s\sim T_m$ linear models of each station include a weather-independent model (generated from all radiosonde data), a rainless-day model (generated from the data of rainless days), and a rainy-day model (generated from the data of rainy days). Columns 5–10 present the linear regression coefficients of these models.

Radiosonde Station				Weather- Indenpedent Model		Rainless-Day Model		Rainy-Day Model
NO.	ID/STN	Lat/Lon	H (m)	a	b	a	b	a	b
C1	C2	C3	C4	C5	C6	C7	C8	C9	C10
1	45004 Kings Park	22.31 114.16	66	0.57	118.16	0.62	102.43	0.49	139.34
2	50527 Hailar	49.21 119.75	611	0.72	69.55	0.71	72.53	0.76	56.88
3	50557 Nenjiang	49.16 125.23	243	0.77	55.79	0.75	59.88	0.81	42.53
4	50774 Yichun	47.71 128.9	232	0.83	38.75	0.81	44.72	0.88	25.43
5	50953 Harbin	45.75 126.76	143	0.85	33.30	0.85	34.35	0.87	27.88
6	51076 Altay	47.73 88.08	737	0.65	90.53	0.63	96.10	0.70	75.13
7	51431 Yining	43.95 81.33	664	0.65	89.68	0.61	102.63	0.74	63.36
8	51463 Urumqi	43.78 87.62	919	0.60	103.13	0.57	111.42	0.64	90.21
9	51644 Kuqa	41.71 82.95	1100	0.62	97.46	0.61	100.22	0.70	74.88
10	51709 Kashi	39.46 75.98	1291	0.60	102.14	0.58	109.11	0.69	77.31
11	51777 Ruoqiang	39.03 88.16	889	0.58	109.52	0.57	111.79	0.67	84.11
12	51828 Hotan	37.13 79.93	1375	0.61	97.69	0.60	101.21	0.69	75.81
13	51839 Minfeng	37.06 82.71	1409	0.58	108.82	0.56	112.92	0.69	77.39
14	52203 Hami	42.81 93.51	739	0.62	98.06	0.61	99.81	0.68	79.26
15	52267 Ejin Qi	41.95 101.06	941	0.64	92.16	0.63	92.55	0.65	88.61
16	52323 Maz. Shan	41.80 97.03	1770	0.64	89.84	0.63	93.16	0.72	68.73
17	52418 Dunhuang	40.15 94.68	1140	0.59	105.47	0.58	108.71	0.71	69.68
18	52533 Jiuquan	39.76 98.48	1478	0.62	96.06	0.60	101.51	0.72	67.19
19	52681 Minqin	38.63 103.08	1367	0.65	89.38	0.63	93.73	0.74	61.29
20	52818 Golmud	36.41 94.90	2809	0.60	98.88	0.58	104.50	0.69	74.83
21	52836 Dulan	36.30 98.10	3192	0.75	57.22	0.73	63.54	0.80	46.86
22	52866 Xining	36.71 101.75	2296	0.66	86.40	0.62	96.60	0.78	53.66
23	52983 Yu Zhong	35.87 104.15	1875	0.67	84.57	0.64	92.16	0.77	55.08
24	53068 Erenhot	43.65 112.00	966	0.68	78.05	0.67	81.56	0.75	60.08
25	53463 Hohhot	40.81 111.68	1065	0.77	54.02	0.76	57.16	0.82	38.47
26	53513 Linhe	40.76 107.40	1041	0.76	59.71	0.76	59.63	0.81	45.27
27	53614 Yinchuan	38.48 106.21	1112	0.74	63.52	0.74	65.35	0.79	49.17
28	53772 Taiyuan	37.78 112.55	779	0.77	54.03	0.77	55.92	0.82	41.52
29	53845 Yan An	36.60 109.50	959	0.72	68.70	0.71	74.04	0.81	45.08
30	53915 Pingliang	35.55 106.66	1348	0.77	56.72	0.75	61.35	0.83	38.31
31	54102 Xilin Hot	43.95 116.06	991	0.74	64.15	0.71	70.49	0.80	45.91
32	54135 Tongliao	43.60 122.26	180	0.88	24.00	0.88	23.01	0.88	22.81
33	54161 Changchun	43.90 125.21	238	0.87	27.66	0.87	27.91	0.88	24.66
34	54218 Chifeng	42.26 118.96	572	0.85	32.86	0.84	34.16	0.86	28.66
35	54292 Yanji	42.88 129.46	178	0.92	13.44	0.93	12.03	0.93	11.70
36	54342 Shenyang	41.76 123.43	43	0.82	41.59	0.82	41.50	0.85	31.26
37	54374 Linjiang	41.71 126.91	333	0.82	41.93	0.82	44.82	0.87	27.26
38	54511 Beijing	39.93 116.28	55	0.87	25.21	0.87	24.89	0.86	28.63
39	54662 Dalian	38.90 121.63	97	0.96	2.57	0.97	-0.20	0.93	10.08
40	54727 Zhangqiu	36.70 117.55	123	0.83	37.24	0.84	36.95	0.82	40.26
41	54857 Qingdao	36.06 120.33	77	0.96	3.35	0.98	-3.63	0.91	17.59
42	55299 Nagqu	31.48 92.06	4508	0.67	77.83	0.55	109.67	0.75	58.35
43	55591 Lhasa	29.66 91.13	3650	0.63	93.57	0.60	101.16	0.67	82.90
44	56029 Yushu	33.01 97.01	3682	0.73	64.51	0.66	81.31	0.76	56.47
45	56080 Hezuo	35.00 102.90	2910	0.74	63.76	0.69	76.10	0.82	42.92
46	56137 Qamdo	31.15 97.16	3307	0.70	74.10	0.65	87.75	0.71	70.39
47	56146 Garze	31.61 100.00	522	0.72	70.94	0.66	86.30	0.76	58.98
48	56187 Wenjiang	30.70 103.83	541	0.71	72.29	0.69	80.93	0.79	48.56
49	56571 Xichang	27.90 102.26	1599	0.58	110.66	0.55	121.73	0.70	75.35
50	56691 Weining	26.86 104.28	2236	0.62	102.15	0.60	107.06	0.60	105.10
51	56739 Tengchong	25.11 98.48	1649	0.52	130.75	0.53	127.02	0.61	102.11
52	56778 Kunming	25.01 102.68	1892	0.45	148.44	0.41	161.20	0.62	99.25
53	56964 Simao	22.76 100.98	1303	0.35	181.68	0.35	182.07	0.54	124.71
54	56985 Mengzi	23.38 103.38	1302	0.49	138.31	0.49	141.13	0.58	113.11
55	57083 Zhengzhou	34.71 113.65	111	0.81	46.17	0.81	45.67	0.81	44.03
56	57127 Hanzhong	33.06 107.03	509	0.78	54.11	0.75	61.72	0.86	27.99
57	57131 Jinghe	34.43 108.97	411	0.75	60.92	0.74	64.27	0.83	39.08
58	57178 Nanyang	33.03 112.58	131	0.80	48.24	0.80	48.23	0.82	42.57
59	57447 Enshi	30.28 109.46	458	0.77	56.77	0.74	66.97	0.81	45.32
60	57461 Yichang	30.70 111.30	134	0.80	48.09	0.81	47.18	0.80	48.65
61	57494 Wuhan	30.61 114.13	23	0.75	64.14	0.75	63.45	0.74	66.97
62	57516 Chongqing	29.51 106.48	260	0.81	47.57	0.77	58.23	0.83	38.87
63	57679 Changsha	28.20 113.08	46	0.70	78.60	0.71	76.15	0.67	85.54
64	57749 Huaihua	27.56 110.00	261	0.68	83.41	0.68	83.76	0.67	86.91
65	57816 Guiyang	26.48 106.65	1222	0.62	101.20	0.64	94.89	0.59	108.71
66	57957 Guilin	25.33 110.30	166	0.63	99.24	0.67	87.73	0.59	110.69
67	57972 Chenzhou	25.80 113.03	185	0.62	101.09	0.64	96.77	0.60	107.27
68	57993 Ganzhou	25.85 114.95	125	0.63	98.02	0.64	96.53	0.62	102.24
69	58027 Xuzhou	34.28 117.15	42	0.84	37.63	0.85	35.50	0.83	38.18
70	58150 Sheyang	33.76 120.25	7	0.86	32.34	0.87	27.96	0.86	29.31
71	58203 Fuyang	32.86 115.73	33	0.84	37.86	0.85	35.13	0.81	44.24
72	58238 Nanjing	32.00 118.80	7	0.81	44.31	0.82	42.58	0.81	46.42
73	58362 Shanghai	31.40 121.46	4	0.82	42.85	0.82	41.29	0.81	43.50
74	58424 Anqing	30.53 117.05	20	0.79	51.17	0.81	45.75	0.76	61.78
75	58457 Hangzhou	30.23 120.16	43	0.79	50.87	0.80	49.28	0.78	53.00
76	58606 Nanchang	28.60 115.91	50	0.74	67.77	0.76	61.88	0.70	78.62
77	58633 Qu Xian	28.96 118.86	71	0.73	70.12	0.72	73.11	0.74	67.51
78	58665 Hongjia	28.61 121.41	2	0.78	54.31	0.80	50.01	0.77	58.05
79	58725 Shaowu	27.33 117.46	219	0.69	81.32	0.69	82.75	0.71	76.99
80	58847 Fuzhou	26.08 119.28	85	0.73	69.83	0.75	64.35	0.70	78.76
81	59134 Xiamen	24.48 118.08	139	0.68	84.50	0.73	71.76	0.60	108.31
82	59211 Baise	23.90 106.60	175	0.64	95.58	0.64	97.28	0.65	93.27
83	59265 Wuzhou	23.48 111.30	120	0.58	114.12	0.62	104.13	0.54	125.68
84	59280 Qing Yuan	23.66 113.05	19	0.59	111.25	0.64	97.52	0.54	126.00
85	59316 Shantou	23.35 116.66	3	0.63	100.81	0.67	87.97	0.56	120.00
86	59431 Nanning	22.63 108.21	126	0.54	125.08	0.58	114.61	0.50	138.79
87	59758 Haikou	20.03 110.35	24	0.56	121.84	0.61	108.11	0.50	137.63
88	59981 Xisha Dao	16.83 112.33	5	0.47	149.53	0.41	167.98	0.47	147.96

of Appendix A. For these stations, most radiosonde observations are taken at 00:00 UTC and 12:00 UTC daily. All radiosonde data were downloaded from the upper air sounding archive of University of Wyoming (http://weather.uwyo.edu/upperair/sounding.html, accessed on 1 May 2022 ). At most stations, we used the data from 2005 to 2018 for the analyses, while at station ZHANGQIU (ID: 54727), WENJIANG (ID: 56187), and JINGHE (ID: 57131), we only used the data from 2014 to 2018 because the site information is incomplete in data files from 2005 to 2013 for the three stations.

Before data processing, we carried out the quality checking for each radiosonde profile. An accepted radiosonde profile is required to contain the data of the first level, which is the level at the altitude of that station. The pressure of the top level is required to be no more than 300 hPa. In addition, the profile must contain the standard pressure levels and the total number of levels should be no less than 5.

For most radiosonde data, $T_m$ is less than $T_s$, but the difference ($T_s-T_m$) is normally no more than 30 K. In order to detect the profiles with gross error, we manually checked both the profiles with $T_s-T_m>30\mathrm{K}$ and $T_m-T_s>10\mathrm{K}$. Among 724 checked radiosonde profiles, 39 of them were found to contain obvious record errors or unreasonable temperature gradients, and they were then removed from the later data processing.

3. Unified Model

The Bevis model $T_m=0.72T_s+70.2$ was developed using 8712 radiosonde profiles at 13 U.S. stations between 1989 and 1991, and the root mean square (RMS) deviation from the regression is 4.74 K [1]. In this study, we used much more data, 860,054 profiles at 88 stations from 2005 to 2018, to establish a $T_s\sim T_m$ linear model over China (Figure 2), hereinafter referred to as the unified model. The unified model is $T_m=0.79T_s+50.76$, with an RMS about the regression of 4.14 K (14% smaller than the RMS for the Bevis model).

Though using a single $T_s\sim T_m$ linear model to estimate $T_m$ for a large region is convenient in practice, it may introduce large representativeness errors. Ross and Rosenfeld [26] pointed out that, over the U.S., the slope of the Bevis model (0.72) is not that representative of the slopes of site-specific models. Like the U.S., China also has a vast territory, and thus the simply unified model for the whole China may cause significant $T_m$ estimation errors as well. We developed a $T_s\sim T_m$ linear model for each single station and compared the unified model with the 88 site-specific models. The slope $a$ and intercept $b$ of each site-specific model are shown in columns 5 and 6 of Table A1 (Appendix A), respectively. Figure 3 illustrates that some site-specific models show good consistency with the unified model, while others show clear deviation from the unified model. In some situations, the difference between $T_m$ from the unified model and that from some site-specific model reaches >10 K, which is equivalent to 3% $T_m$ relative errors.

To quantify the differences between the unified model and each site-specific model, we calculated the $T_m$ biases ($bias=\left|T_m\_Unified-T_m\_Site\right|$) between them. Figure 4 shows the $T_m$ mean bias, maximal bias, mean relative bias, and maximal relative bias of the unified model relative to each site-specific model. The mean bias is larger than 2 K at 45 stations (51% of all the stations) (Figure 4a), and the maximal bias is larger than 4 K at 46 stations (52%) (Figure 4b). In terms of relative bias, at some stations, the mean relative bias is close to or above 1.5% (Figure 4c) and the maximal relative bias is over 3% (Figure 4d). At more than half the stations, the mean relative bias is larger than 0.5% or the maximal relative bias is larger than 1%. All these results suggest that the unified model is not a good proxy for more than half the 88 site-specific models if the 2 K mean bias, 4 K maximal bias, 0.5% mean relative bias, or 1% maximal relative bias is chosen as the threshold.

4. Region-Specific Characteristics of $T_s\sim T_m$ Linear Relations

The distinct deviation of the unified model from part of the site-specific models lies in the diversity of the $T_s\sim T_m$ linear relations at different areas. We plotted the contours of the slopes (coefficient $a$) of site-specific $T_s\sim T_m$ linear models (Figure 5a), which shows the slope generally increasing from low to middle latitudes. Figure 5a shows that the range of slopes is from 0.35 to 0.96 over China, with large slopes (>0.9) occurring at Bohai bay and small slopes (<0.5) at the south of Yunnan (YN) province (refer to Figure 5b for the area names and their locations). In general, the slope decreases from the northeast to the southwest. In the area of Xinjiang (XJ) and the west of Neimenggu (NM), Gansu (GS), Qinghai (QH), and Xizang (XZ), the slope changes gently. Thus, applying one linear model to the whole area does not give rise to significant representativeness errors. Another similar area where the slope also shows little variation includes Shaanxi (SA), the west of Shanxi (SX), and the north of Sichuan (SC). In contrast, the slope changes drastically over the area of Yunnan (YN) and the border between Sichuan (SC) and Guizhou (GZ), which indicates that using a single $T_s\sim T_m$ linear model for this area will inevitably result in large representativeness errors.

The regression precision of site-specific $T_s\sim T_m$ linear model is related to site location. To investigate the relation between regression precisions of the models and the geographic location, we calculated the RMS deviation from the regression of site-specific model for each station. Figure 6 shows that the stations (TENGCHONG and KINGS PARK) with small RMS (<2 K) are located at low latitudes, while those (DUNHUANG and EJIN QI) with large RMS (>5 K) are located at relative high latitudes. Overall, the RMS tends to increase with latitude.

To better understand the variation of the regression precision with latitude, the distributions of radiosonde $T_s\sim T_m$ data points from two stations with low latitude (TENGCHONG and KINGS PARK) and two stations with relative high latitude (DUNHUANG and EJIN QI) are compared in Figure 7 (refer to Figure 6 for the locations of the four stations). In both the $T_s$-axis and $T_m$-axis dimensions, the data points from high-latitude stations, EJIN QI (Figure 7c) and DUNHUANG (Figure 7d), are much more dispersedly distributed than those from low-latitude stations, KINGS PARK (Figure 7a) and TENGCHONG (Figure 7b). The ranges of $T_s$ at station KINGS PARK and TENGCHONG are much smaller than those at EJIN QI and DUNHUANG. This is because the surface temperatures are relative high at low latitudes all year round, and thus they concentrate on a smaller range. For a certain $T_s$, the variations of $T_m$ at stations with low latitudes are also smaller. A closer inspection of the radiosonde profiles suggests that, in general, the vertical distributions of water vapor and temperature are more uniform for low-latitude stations than for high-latitude stations (not shown), which accounts for the smaller $T_m$ variation at low latitudes.

5. Weather-Dependent Characteristics of $T_s\sim T_m$ Linear Relations

Under different kinds of weather, the vertical distributions of temperature and water vapor can vary considerably. As a result, the $T_s\sim T_m$ relation is likely to change accordingly. For each station, we used the radiosonde data from different weather separately to generate weather-dependent $T_s\sim T_m$ linear models. We then compared the weather-dependent $T_s\sim T_m$ linear models as well as their regression precision. In some $T_s\sim T_m$ plots, such as Figure 2 and Figure 7, some data points are far away from the regression line. We investigated two sets of these points and analyzed their related weather conditions.

5.1. Weather-Dependent $T_s\sim T_m$ Linear Models

For simplification, we only consider two kinds of weather: rain and no rain. All the days involved in the experiment are classified as either a rainy day or a rainless day. If there is no rain for a whole day, that day is defined as a rainless day, and if not, the day is defined as a rainy day. We used the daily precipitation data acquired from the data center of China Meteorology Administration (http://data.cma.cn/, accessed on 16 May 2022) to classify the days. For each station, we generated a $T_s\sim T_m$ linear model using the radiosonde data from rainless days and a model using the data from rainy days. The former model is hereinafter referred to as “rainless-day model” and the latter is referred to as “rainy-day model”. Table A1 of Appendix A shows the slope $a$ and intercept $b$ of the rainless-day models (columns 7 and 8) and rainy-day models (columns 9 and 10) for all the 88 stations.

At some stations, such as station NAGQU, XICHANG, KUNMING, and SIMAO shown in Figure 8, the rainy-day model clearly deviates from the rainless-day model. The difference between the $T_m$ from rainy-day model and that from rainless-day model can be larger than 5 K. Thus, in order to get higher accuracy $T_m$ values for these stations, it is better to use the rainy-day model for rainy days and the rainless-day model for rainless days rather than using a weather-independent model for all days. Meanwhile, at some other stations, such as station HARBIN, BEIJING, WUHAN, and SHANGHAI shown in Figure 9, the rainy-day model shows good consistency with the rainless-day model. Hence, for these stations, it is not necessary to use weather-dependent models for rainy days and rainless days separately.

We compared the rainy-day model with the rainless-day model for each station. At half the stations, the difference between the two models is significant: the mean relative bias of $T_m$ between the two models is larger than 0.5%, or the maximal relative bias is larger than 1%. Figure 10 shows that the stations with significant difference between the rainy-day model and rainless-day model (red dots in Figure 10) distribute over a larger area than the stations with insignificant model difference (blue dots in Figure 10). In general, the stations with significant model difference are at higher altitudes (refer to Figure 1 for the topography), and they are mostly located in the west and north of China. This suggests that, in general, using weather-dependent models for rainy and rainless days separately will effectively improve the $T_m$ accuracy in the west and north of China, but not in the eastern part.

5.2. Comparison of Regression Precision of Weather-Dependent Models

Both Figure 8 and Figure 9 show that the distributions of data points from rainy days (blue dots) are more concentrated than those from rainless days (golden dots), indicating that the temperature and water vapor vertical profiles are more uniform on rainy days than on rainless days. We calculated the root mean squares deviation from the regressions of the weather-dependent models. Figure 11 shows that the RMS for rainy-day model is less than that for rainless-day model at all the stations, and the reduction rate of the rainy-day model RMS relative to the rainless-day model RMS is from 4% to 49% (average: 28%). This result suggests that the rainy-day model yielded $T_m$ values for rainy days are expected to have better accuracy than the rainless-day model yielded ones for rainless days.

While using the weather-dependent models for rainy days and rainless days separately can improve the accuracy of $T_m$ estimates, the benefits for rainy days and rainless days are unequal. Since the number of radiosonde profiles from rainless days is more than that from rainy days for the stations, the weather-independent model of a station is closer to the rainless-day model than to the rainy-day model. If the weather-independent model is used, its representativeness error for the rainy-day model is larger than that for the rainless-day model. Hence, replacing the weather-independent model with the weather-dependent models results in more improvement in the accuracy of $T_m$ estimates for rainy days than for rainless days.

5.3. Weather Conditions Related to Some Specific Data Points

In most cases, as shown in Figure 2, $T_m$ is less than $T_s$, and $T_s$ minus $T_m$ is generally no more than 30 K. For the cases of $T_m$ larger than $T_s$, $T_m$ minus $T_s$ is normally no more than 10 K. However, in some situations, extremes of $T_m-T_s>10\mathrm{K}$ and $T_s-T_m>30\mathrm{K}$ happen. Figure 12 shows the data points from the radiosonde profiles corresponding to the extremes. These data points are the ones that deviate most from the unified $T_s\sim T_m$ linear model, and they are responsible for the large RMS of the linear regression. To better understand the cause of the extremes, the weather conditions related to these specific data points are investigated.

Figure 12 shows that, in the dataset of $T_m-T_s>10\mathrm{K}$ and $T_s-T_m>30\mathrm{K}$, the data from rainy days are much less than that from rainless days: the former only accounts for 4%. Such a small proportion of data points from rainy days suggests that the situations of $T_m-T_s>10\mathrm{K}$ and $T_s-T_m>30\mathrm{K}$, which reduce the precision of the $T_s\sim T_m$ linear regression, seldom occur on rainy days or mostly happen on rainless days. This result partly explains why the rainy-day model has better regression precision than the rainless-day model.

The statistics suggests that 98% of the radiosonde observations with $T_m-T_s>10\mathrm{K}$ were in winter, and 92% of them were observed at 00:00 UTC, while for the situation of $T_s-T_m>30\mathrm{K}$, 97% of the observations were in summer and spring, and 98% of them were observed at 12:00 UTC. For stations in China, the surface temperature is normally low at 00:00 UTC (8:00 Beijing time) in winter, while at 12:00 UTC (20:00 Beijing time) in summer and spring, the surface temperature is relatively high. This indicates that the situation of $T_m-T_s>10\mathrm{K}$ occurs on the condition of low surface temperature, while the situation of $T_s-T_m>30\mathrm{K}$ happens under the circumstance of high surface temperature. This conclusion is confirmed by Figure 12, which shows the data points with $T_m-T_s>10\mathrm{K}$ are all distributed in $T_s<270\mathrm{K}$, and those with $T_s-T_m>30\mathrm{K}$ are all in the range of $T_s>280\mathrm{K}$.

To further investigate the weather backgrounds related to the extremes of $T_m-T_s>10\mathrm{K}$ and $T_s-T_m>30\mathrm{K}$, we checked the temperature and water vapor mixing ratio profiles for the data points corresponding to these extremes. All data with $T_m-T_s>10\mathrm{K}$ have a similar vertical distribution of temperature and also a similar vertical distribution of the mixing ratio. Figure 13 shows the representative temperature and mixing ratio profiles for the situation of $T_m-T_s>10\mathrm{K}$. This situation is related to a temperature inversion and a humidity inversion. The temperature (humidity) inversion is a weather phenomenon that the temperature (humidity) increases with the height [52]. In these profiles, the temperature inversion occurs at a similar height as the humidity inversion, and both of them are striking and are below 800 hPa.

While for the data with $T_s-T_m>30\mathrm{K}$, all temperature (mixing ratio) profiles show a similar vertical distribution, but the vertical distribution is clearly different from that of the data with $T_m-T_s>10\mathrm{K}$. Figure 14 shows the representative temperature and mixing ratio profiles for the situation of $T_s-T_m>30\mathrm{K}$. The profiles show that there is a distinct humidity inversion in each mixing ratio profile, but no obvious temperature inversion occurs at any observation height. For the data with $T_s-T_m>30\mathrm{K}$, the height of humidity inversion is generally above 600 hPa, which is much higher than the height of humidity inversion for the data with $T_m-T_s>10\mathrm{K}$.

6. Conclusions

Analysis of 14 years of radiosonde profiles at 88 stations demonstrates that the $T_s\sim T_m$ linear relation is region specific over China. Using a single $T_s\sim T_m$ linear model to represent all the 88 site-specific models for estimating $T_m$ over the area of the whole of China brings diverse errors of up to 10 K (about 3% relative error), indicating that the region-specific characteristics of the $T_s\sim T_m$ linear relations must be considered in order to accurately estimate $T_m$. The geographical distribution of the slopes (coefficient $a$ of $T_m=a{T}_s+b$) of the site-specific models reflects the variation of the $T_s\sim T_m$ linear relations over different areas. From Bohai bay to the south of Yunnan province, the slope reduces from 0.96 to 0.35. Over Yunnan and the border between Sichuan and Guizhou, the slope changes drastically, while over some other areas, such as Xinjiang and part of Qinghai, the slopes change slowly. This information provides the basis for determining which area can simply use a single $T_s\sim T_m$ linear model to estimate the $T_m$ without bringing in significant representativeness errors and which areas have to use multiple models for highly accurate $T_m$ estimation. The characteristic of slope increasing with latitude can also be found from the study of Ross and Rosenfeld [26] and Yao et al. [53]. However, their work was on a global scale and did not provide detailed analysis specifically for China as this study does. The regression precision of a site-specific $T_s\sim T_m$ linear model is also related to the geographical location. In general, the RMS about the regression of the site-specific model increases with station latitude, and this conclusion is similar to the result of Raju et al. [29] for the Indian subcontinent.

Investigation of the $T_s\sim T_m$ linear models generated from the radiosonde data observed on rainy days (rainy-day model) and on rainless days (rainless-day model) demonstrates that the $T_s\sim T_m$ linear relation is weather dependent. At half the stations, the difference between the $T_m$ estimates from the rainy-day model and those from the rainless-day model is significant (mean relative bias larger than 0.5% or maximal relative bias larger than 1%). These stations are mostly located in the west and north of China. Thus, over these regions, both the region-specific and weather-dependent characteristics of the $T_s\sim T_m$ linear relations should be taken into account for ensuring the accuracy of the $T_m$ estimates. For each station, the regression of rainy-day model has a higher precision than that of the rainless-day model. On average, the RMS of the regression of rainy-day model is 28% smaller than that of the regression of rainless-day model, suggesting that the $T_m$ of rainy days estimated by the rainy-day model is expected to be more accurate than that of rainless days estimated by the rainless-day model.

Another contribution of this study is the exploration of the weather backgrounds for radiosonde data satisfying $T_m-T_s>10\mathrm{K}$ and $T_s-T_m>30\mathrm{K}$ that most deviate from the regression of the $T_s\sim T_m$ linear model. Radiosonde data satisfying $T_m-T_s>10\mathrm{K}$ are related to the weather of low surface temperature (<270 K) and both striking temperature and humidity inversions occurring below 800 hPa, while radiosonde data satisfying $T_s-T_m>30\mathrm{K}$ are related to the weather of high surface temperature (>280 K) and a striking humidity inversion normally occurring above 600 hPa without temperature inversion at any observation height.

Since the calculation of $T_m$ is based on the vertical distribution of temperature and water vapor pressure, it is not difficult to understand that local weather and climate determine the value of $T_m$, the relationship between $T_m$ and $T_s$, and the regression precision of $T_s\sim T_m$ linear models. On the other hand, many evidences provided in this study suggest that some $T_m$-related information reflects the local weather and climate, such as the situation of $T_m-T_s>10\mathrm{K}$ reflecting the (temperature and humidity) inversions and larger RMS of the regression of $T_s\sim T_m$ linear model indicating the poor uniformity of atmospheric profiles. Thus, it is promising to use the $T_m$ as an indicator for weather and climate studies, which deserves further investigation.