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In the atmosphere, hydrometeors and water vapor are intimately related. Hydrometeors refer to any water or ice particles that form in the atmosphere as a result of condensation or desublimation of water vapor. Some wellknown hydrometeors are clouds (cloud liquid water and cloud ice), fog, rain, snow, hail, graupel, dew, rime, glaze, blowing snow, and blowing spray. In this paper, hydrometeors mainly refer to the cloud water (liquid and solid); other types of hydrometeors are not considered. Water vapor turns into hydrometeors when its phase is changed from gaseous to liquid or solid, during which the mass of water is conserved. Based on the atmospheric water substance budget and water mass conservation (Trenberth and Guilemot, 1995), we hereby propose the concepts, definitions, and calculation formulas for the CWR and its components/contributors as well as associated characteristic variables.
For any region over any period, the budget equation of atmospheric hydrometeors is expressed as:
$$ \hspace{30pt} {M_{\rm{h2}}}  {M_{\rm{h1}}} = {Q_{\rm{hi}}}  {Q_{\rm{ho}}} + {C_{\rm{vh}}}  {C_{\rm{hv}}}  {P_{\rm s}}.$$ (1) Here, M_{h1} and M_{h2 }are mass of atmospheric hydrometeors at time moments t_{1} and t_{2}. They are instantaneous/state quantities that can be observed at a specific moment, with the same physical meaning as CWP (comparable after unit being transformed into column water depth). The advection items, Q_{hi} and Q_{ho}, are influx and outflux of atmospheric hydrometeors along the boundaries of the study area. For a specific region and period, advection along an individual boundary at various time moments and vertical levels can be positive or negative, representing influx and outflux of hydrometeors, respectively. The sum of the positive and negative advection is the accumulative Q_{hi} and Q_{ho} during the study period. C_{vh }is the mass of hydrometeors converted from water vapor through condensation or desublimation, while C_{hv }is the mass content of water vapor converted from hydrometeors through evaporation or sublimation. P_{s }is the surface precipitation. C_{vh}, C_{hv}, and P_{s} are the source/sink terms; they accumulate with extension of the study time and expansion of the study area.
Sorting the positive and negative terms in Eq. (1) separately, the equation can be rewritten as:
$${M_{\rm{h1}}} + {Q_{\rm{hi}}} + {C_{\rm{vh}}} = {M_{\rm{h2}}} + {Q_{\rm{ho}}} + {C_{\rm{hv}}} + {P_{\rm s}} = {\rm G{M_h}}.$$ (2) Here, we define GM_{h} as the gross mass of atmospheric hydrometeors, representing the total amount of hydrometeors over the time period t_{2} − t_{1 }within the study region. From Eq. (2), it is inferred that GM_{h} also equals half of the sum of the 7 terms on the left and righthand sides of the equation.
As for water vapor, a similar equation is derived as follows:
$${M_{\rm{v1}}} + {Q_{\rm{vi}}} + {C_{\rm{hv}}} + {E_{\rm s}} = {M_{\rm{v2}}} + {Q_{\rm{vo}}} + {C_{\rm {vh}}} = {\rm G{M_v}},$$ (3) where E_{s}, the source item, is the surface evaporation, and GM_{v }is the gross mass of atmospheric water vapor.
The surface precipitation P_{s} is absent in Eq. (3), suggesting that water vapor is not directly associated with precipitation. The terms C_{vh} and C_{hv} in Eq. (3) represent the phase changes between water vapor and hydrometeors. It is the hydrometeors produced by water vapor through condensation that are directly linked to precipitation, and the relationship is quantified in Eq. (1).
By adding Eqs. (2) and (3), the terms of mass conversion between water vapor and hydrometeors offset each other, and the balance equation for the atmospheric water substance, which is the summation of water vapor and hydrometeors, is obtained as follows:
$${M_{\rm{w1}}} + {Q_{\rm{wi}}} + {E_{\rm s}} = {M_{\rm{w2}}} + {Q_{\rm{wo}}} + {P_{\rm s}} = {\rm G{M_w}},$$ (4) where GM_{w} is the gross mass of water substance in the atmosphere.
Equations (2)–(4) are collectively called the atmospheric water balance equations, in which the values of the 12 independent items are positive with a unified unit of kg. Their values are determined by the size of the study area and by the length of the study period. These items/terms can also be converted into column water per unit area (kg m^{–2}, equivalent to mm of water depth) for comparative analysis.

For a specific region during a certain period of time, among all the atmospheric hydrometeors, those that have participated in the water cycle, yet have not formed precipitation, and remain in the air, are targeted for investigation in this paper, with the mass concentration of these hydrometeors defined as the quantified CWR. According to Eq. (2), the CWR is written as:
$$\begin{split} {\rm CWR} =& \,\,\,{\rm G{M_h}}  {P_{\rm s}} \\ =& \,\,\,{M_{\rm {h1}}} + {Q_{\rm {hi}}} + {C_{\rm {vh}}}  {P_{\rm s}} \\ =& \,\,\,{{M_\rm {h2}}} + {Q_{\rm {ho}}} + {C_{\rm {hv}}}. \\ \end{split} $$ (5) In Eq. (5), M_{h1} and M_{h2 }are important instantaneous/state terms for cloud water resources. For other nonrenewable resources on the earth, such as minerals, these terms become the only source of origin, meaning that the more used, the less remains; in this case, the change of the state terms (M_{h2 } − M_{h1}) is always negative (M_{h2 }<M_{h1}). However, the atmospheric hydrometeors are constantly replenished and renewed, so the changes of the state terms can also be positive (M_{h2 }> M_{h1}).
Different from other contributors of CWR, the state terms do not accumulate over time; it becomes less important for longterm (e.g., monthly, yearly, etc.) CWR estimation. Previous studies largely focused on the state quantity/term of cloud water (Chen et al., 2005; Li X. Y. et al., 2008; Peng et al., 2010). Although termed as CWR, it is actually only a small part of the CWR, because the two important terms—influx (Q_{hi}) and condensation (C_{vh}), are ignored. In fact, the influx of hydrometeors and local condensation of cloud water contribute significantly to the CWR.

Precipitation efficiency (PE) is the ratio of surface precipitation to all available atmospheric water substance. It is a metric for measuring how much atmospheric water substance can be converted into real precipitation received at the ground surface. PE has significant implications for precipitation analysis and weather modification. Hobbs et al. (1980a; hereafter Hb80) defined PE as the ratio of total ground precipitation rate to total condensation rate in a rainbelt. Similar to Hb80, Sui et al. (2005) defined the ratio of the ground precipitation rate to the sum of water vapor condensation rate and water vapor deposition/desublimation rate as cloud microphysical precipitation efficiency (CMPE). Sui et al. (2007; hereafter Su07) further improved the definition of CMPE by treating the denominator as the sum of water vapor condensation rate, water vapor desublimation rate, and local hydrometeors loss and convergence. To ensure that CMPE is within a reasonable range of 0%–100%, the hydrometeors loss and convergence terms are considered only when they are positive, otherwise they are treated as zero.
In the present study, PE_{h} is defined as the ratio of total ground precipitation to the gross mass of hydrometeors, expressed by:
$${\rm{P}}{{\rm{E}}_{\rm{h}}} = {P_{\rm{s}}}/{\rm{G}}{{\rm{M}}_{\rm{h}}}.$$ (6) Here, PE_{h} reflects the efficiency of cloud microphysical processes in transforming small cloud droplets into precipitable particles. It can be applied over any time period and region, and the value is between 0 and 1. Compared to the definition of Hb80, PE_{h} calculated by Eq. (6) is more reasonable since it considers the initial state quantity/term and influx of hydrometeors, in addition to condensation and desublimation, in the denominator. Compared to Su07, additional consideration of the initial state term and the influx of hydrometeors is physically more rational than only considering the positive convergence and local change of hydrometeors. For a longterm and largescale precipitation process, the initial value and the influx of hydrometeors become smaller than the cloud condensation term, thus the results from the three definitions [Hb80, Su07, and Eq. (6) of this study] are relatively close to each other.
Similarly, precipitation efficiency of water vapor (PE_{v}) and that of atmospheric water substance (PE_{w}) are calculated by:
$$\hspace{62pt} {\rm{P}}{{\rm{E}}_{\rm{v}}} = {P_{\rm{s}}}/{\rm{G}}{{\rm{M}}_{\rm{v}}},$$ (7) $$\hspace{62pt} {\rm{P}}{{\rm{E}}_{\rm{w}}} = {P_{\rm{s}}}/{\rm{G}}{{\rm{M}}_{\rm{w}}}.$$ (8) Furthermore, when water vapor in the atmosphere undergoes phase transformation to form liquid or solid water particles through condensation or desublimation, the ratio of the condensed water particles to the total water vapor is termed as condensation efficiency of water vapor (CE_{v}), which is calculated by:
$${\rm{C}}{{\rm{E}}_{\rm{v}}} = {C_{{\rm{vh}}}}/{\rm{G}}{{\rm{M}}_{\rm{v}}}.$$ (9) 
The mean mass of atmospheric hydrometeors (MM_{h}) is the average mass content of atmospheric hydrometeors over the study period within the study region, expressed by:
$${\rm M{M_h}} = \frac{1}{N}\sum\limits_{i = 1}^{N} {{M_{\rm h}}_i} ,$$ (10) where M_{h}_{i} is the mass of atmospheric hydrometeors at time moments t_{i} and N is the number of observed M_{h} at a specific station or grid within the study period.
In the atmosphere, clouds scatter discontinuously and are constantly changing. Thus, the mass concentration of hydrometeors (M_{h}) may vary greatly with time and space. For longterm CWR investigations, MM_{h} becomes a more suitable physical variable.
The atmospheric water substance (i.e., water vapor and hydrometeors) experiences phase changes and spatiotemporal variations constantly. The time needed for the phase changes is an important metric that reflects combined effects of the environmental dynamics and physical mechanisms. We hereby define it as hydrometeor renewal time (RT_{h}) to depict the cycle and transition features of hydrometeors over the study time and in the study area, which is calculated by:
$${\rm{R}}{{\rm{T}}_{\rm{h}}} = {\rm{M}}{{\rm{M}}_{\rm{h}}}/({P_{\rm{s}}}/T),$$ (11) where P_{s}/T is the mean precipitation intensity/rate. The unit of RT_{h} is the same as the unit of the study period T. That is, if the unit of T is day or h or second, the unit of RT_{h} varies correspondingly.
Similarly, the mean mass of water vapor (MM_{v}) and renewal time of water vapor (RT_{v}) are calculated by
$$ \hspace{62pt} {\rm M{M_v}} = \frac{1}{N}\sum\limits_{i = 1}^{N} {{M_{\rm v}}_i},$$ (12) $$ \hspace{62pt} {\rm{R}}{{\rm{T}}_{\rm{v}}} = {\rm{M}}{{\rm{M}}_{\rm{v}}}/({P_{\rm{s}}}/T).$$ (13) Together with GM_{h}, GM_{v}, and GM_{w}, the variables defined in this section represent the characteristics of the CWR in a specific region over a certain time period.

The CWR cannot be observed directly, but can be quantified through numerical simulations and observationbased diagnostic methods. The cycle and change of the atmospheric water substance is an overall balanced process, in which both atmospheric hydrometeors and water vapor participate. To quantify the CWR in a certain region and time period, it is necessary to consider the three balance equations [Eqs. (2)–(4)] of hydrometeors, water vapor, and atmospheric water substance mentioned above as a whole. On this basis, the state quantities/terms, the advection terms, the sources/sink terms, as well as GM_{h}, GM_{v}, GM_{w} can be derived, and the CWR and its related characteristic variables [Eqs. (5)–(13)] can be quantitatively calculated according to their definitions and algorithms. The input (known) variables to these definitions and algorithms include wind field, water vapor content, cloud water content, cloud condensation and evaporation, surface evaporation, and surface precipitation, which can be obtained either from diagnostic analysis of available observations or from numerical model outputs.
The diagnostic quantification of CWR based on observations (CWRDQ) is as follows. We extract water vapor and wind fields from the reanalysis products (e.g., the NCEP/NCAR reanalysis data), and obtain precipitation directly from surface station and/or satellite observations. For the state quantities and advection terms, acquirement of threedimensional cloud fields (cloud fraction and cloud water content) is difficult, as routine observations of threedimensional, timevarying cloud fields are not available. We solve the problem as follows. First, satellite observations are used to establish cloud profiles based on atmospheric relative humidity and temperature. Second, NCEP/NCAR reanalysis data are then employed to produce threedimensional cloud fields such as cloud detection (1 for existence of cloud and 0 for no cloud), cloud water content (i.e., hydrometeors content), as well as the integrated cloud water content (i.e., cloud water path, which is equivalent to the state quantity M_{h}). Third, the surface evaporation (E_{s}) and the conversion between water vapor and hydrometeors (C_{vh} and C_{hv}) are derived based on Eqs. (2)–(4). Finally, the CWR and related characteristic variables as well as GM_{h}, GM_{v}, GM_{w} are calculated based on Eqs. (5)–(13).
The numerical quantification of CWR is based on cloud resolving model outputs (CWRNQ). There is an assumption that the cloud resolving model is able to describe the cloud microphysical processes completely and precisely, from which fourdimensional distributions of atmospheric water vapor, hydrometeors, and wind fields can be obtained. The data can then be used to quantify the CWR and related terms/quantities, as well as those characteristic variables in any area during the study period. The CWRNQ method relies heavily on accurate description of cloud water distribution. It is also challenging to ensure that the model is stable after longterm integration, while the simulated wind, water vapor, cloud, and precipitation must be basically consistent with observations.
By use of the CWRDQ method or the CWRNQ method, the basic balance equations of water substance in the atmosphere can be solved, and the items such as M_{x}, Q_{x}, C_{vh}, C_{hv}, and E_{s }can be obtained. Furthermore, the variables including GM_{x} and CWR, PE_{x} and CE_{v}, MM_{x} and RT_{x}, can also be quantitatively derived (x refers to hydrometeors, water vapor, or the total atmospheric water substance). Detailed introduction of the two sets of quantification methods for estimates of CWR and associated applications are presented in the companion paper (Cai et al., 2020).
2.1. Budget equations of atmospheric water substance
2.2. The CWR and related characteristic variables
2.2.1. Definition and quantification of CWR
2.2.2. Precipitation efficiency and condensation efficiency
2.2.3. Mean mass and renewal time
2.3. Calculation algorithms

Based on the quantification results of CWR conducted in North China for two typical months, i.e., April and August 2017 from Cai et al. (2020), the basic features of CWR is analyzed in this section. April is within the spring season when North China is prone to drought with a large demand of agricultural water storage. Spring is the main season for stratiform precipitation enhancement by cloud seeding. August is within the summer flood season when the largest precipitation in North China occurs. The study region for both methods covers 34°–44°N, 108°–121°E, the area size of which is about 1.34 × 10^{12 }km^{2}. The CWRDQ method is applied over 13 × 10 grids with a horizontal resolution of 1°, while the CWRNQ method is applied at a 3km resolution.

The monthly quantification results of CWR and related variables over North China in April and August 2017 are listed in Table 1. For both numerical calculations and diagnostic calculations in the two months, the average state quantities and advection terms for hydrometeors are one order smaller in magnitude than the corresponding quantities of water vapor. The values of the four source/sink items, including P_{s}, are in the same order of magnitude. Therefore, GM_{h} is also one order of magnitude lower than GM_{v}. Except for C_{hv} estimated by the diagnostic method, the value of each physical quantity in August 2017 is higher than that of April 2017. That is, in summer, the contents of cloud water and water vapor in the atmosphere in North China are higher, with more advection and more intense conversion of water vapor to hydrometeors, resulting in more abundant surface precipitation, compared with the situations in Spring.
Variables April 2017 August 2017 CWRNQ CWRDQ CWRNQ CWRDQ MM_{h} 2.19 4.33 3.10 7.41 MM_{v} 82.61 150.46 430.19 478.34 Q_{hi} 151.79 220.44 216.92 253.83 Q_{ho} 126.16 180.44 231.46 263.69 Q_{vi} 3545.30 4403.10 9393.09 10037.83 Q_{vo} 3512.93 4362.06 9266.05 9933.74 C_{vh} 380.71 362.11 2513.62 2164.60 C_{hv} 178.14 99.21 703.43 58.71 P_{s} 228.73 301.30 1814.93 2094.30 E_{s} 237.00 335.54 1595.30 1760.78 CWR 304.83 282.06 944.66 331.37 GM_{h} 533.56 583.36 2733.78 2425.67 GM_{v} 4079.97 4907.00 12107.41 12425.41 Table 1. The CWR and related variables (unit: 10^{11 }kg, equivalent to 0.77 mm for the study region) quantified by the CWRNQ and CWRDQ methods over North China (with an area size of 1.34 × 10^{12} km^{2}) in April and August 2017
According to Table 1, the importance of hydrometeors related quantities and water vapor related variables to the monthly regional CWR calculation can be seen. Take the numerical results as examples. The state quantity MM_{h} is the smallest among the CWR contributors, with its values two orders of magnitude lower than others. The CWR in April (August) is about 30.48 (94.47) billion ton, equivalent to water depth of 22.7 (70.5) mm in the study region. Only approximately 1% of CWR is from MM_{h}. Among the source/sink terms, C_{vh} has the largest value, and P_{s} is slightly smaller than C_{vh}, followed by C_{hv}. For the advection terms, the values of Q_{hi} and Q_{ho} are close, but both are smaller than C_{hv}. To sum up, the contributions of the above five contributors to the CWR in descending order are as follows: C_{vh}, P_{s}, C_{hv}, Q_{hi}, and Q_{ho}. Their percentage contributions are 125.06%, 75.14%, 58.52%, 49.86%, and 41.44% in April, and the values are 266.19%, 192.13%, 74.46%, 22.96%, and 24.5% in August. As P_{s} is a negative term in Eq. (5) for calculation of CWR, the contributions of C_{vh} and P_{s} to CWR could be greater than 100%.
For the diagnostic results, MM_{h} is still the smallest quantity with less than 1% contribution to regional CWR. Among the other five contributors, C_{vh} is the largest, followed by P_{s}, and Q_{hi} and Q_{ho}. These features are similar to the numerical quantification results. However, due to the underestimation of C_{hv} by the diagnostic method, C_{hv} contributes less to the CWR. The contributions of C_{vh}, P_{s}, Q_{hi}, Q_{ho}, and C_{hv} to CWR are descending in order. For example, their contributions to CWR in April 2017 are 128.2%, 106.45%, 77.96%, 63.81%, and 35.08%, respectively.
As for the quantities related to water vapor, the advection terms Q_{vi} and Q_{vo} have the largest values, one order of magnitude higher than the sink/source terms, among which E_{s} is between C_{vh} and C_{hv}. The state quantity MM_{v}, remains the smallest. Different from GM_{h} and CWR, the advection terms make greater contribution to GM_{v}. For instance, in April 2017, the numerical quantified Q_{vi} and Q_{vo} account for 91.18% and 89.75% of GM_{v}, respectively.

Theoretically, since the integral dimensions of the CWR contributors are different, they cannot be accumulated temporally or spatially. The state quantities (M_{x}) are volume integrated. As the study period extends, only M_{x2} changes, resulting in a decreasing importance of state quantities with extension of time. The advection terms (Q_{x}) are integrals along the boundaries of the study region, which increase with time. However, as the study region expands, the advection terms within the area offset and only those along the boundaries reserve, leading to a decreasing importance with expansion of the area. The sink and source terms are accumulative, which increase with both time and space. In this section, we analyze the quantification results over different time periods and various areas in North China by the numerical and diagnostic methods to probe the impact of time and space scales on the CWR calculation.

Table 2 presents the numerical calculation results of the CWR and its contributors in various 10day periods of April and August 2017. The state variables (M_{h1 }and M_{h2}) change with the chosen time period. For each 10day period in April and August 2017, the values of the state terms at the initial and final time moments are different. Note that M_{h2} of the first (second) period is actually the M_{h1} of the second (third) period. When the quantification results of the first period are compared with the entire month, M_{h1} remains unchanged and M_{h2} becomes M_{h} at the end of the month. As a consequence, M_{h} cannot be accumulated with time.
Quantification period M_{h1} M_{h2} Q_{hi} Q_{ho} C_{vh} C_{hv} P_{s} CWR April 1–10 0.11 0.06 5.94 5.02 12.11 8.67 4.38 13.75 11–20 0.06 0.05 5.93 4.95 18.77 6.21 13.56 11.20 21–30 0.05 0.01 3.32 2.65 7.19 2.93 4.94 5.59 1–30 0.11 0.01 15.18 12.62 38.07 17.81 22.87 30.44 August 1–10 0.32 0.67 4.63 6.75 83.94 22.66 59.48 30.08 11–20 0.67 1.33 4.47 4.11 72.05 21.35 50.32 26.78 21–31 1.33 0.98 12.59 12.29 95.38 26.33 71.69 39.60 1–31 0.32 0.98 21.69 23.15 251.36 70.34 181.49 94.47 Table 2. Results of numerically quantified CWR and its contributors (unit: 10^{12 }kg, i.e., billion ton) in different calculation periods
Both the advection terms (Q_{hi} and Q_{ho}) and the source/sink terms (C_{vh}, C_{hv}, and P_{s}) can be accumulated over time; that is, the 10day quantification result is the accumulation of daily quantification value during the 10day period, and the monthly quantification result is the accumulation of all daily quantification in this month or all 10day quantification during the month. Taking C_{hv} as an example: it is 17.81 billion ton during 1–30 April, which is equal to the sum of quantification results of the three 10day periods in April (8.67 + 6.21 + 2.93 = 17.81 billion ton).
Since the state quantities cannot be accumulated with time, the CWR cannot be accumulated with time, either. The CWR in April is 30.44 billion ton, which is not equal to the sum of CWR of the three 10day periods in April (30.55 billion ton). The difference between them is 0.11 billion ton. This is because when the results of individual 10day periods are accumulated to calculate the monthly CWR, M_{h2} of the first and second 10day periods are repeatedly counted. As a result, the sum of 10day CWR values in April is larger than the monthly CWR, and the difference between the two is equal to the sum of M_{h2 }of the first and second 10day periods (0.06 + 0.05 = 0.11 billion ton). Similarly, the sum of 10day CWR values in August is 96.46 billion ton, which is larger than the monthly CWR of August (94.47 billion ton) by 1.99 billion ton. The extra part is the sum of M_{h2} in the first and second 10day periods of August.
Similarly, GM_{x} (x refers to hydrometeors, water vapor, or atmospheric water substance) cannot be accumulated with time, neither PE_{x} nor RT_{x} can be accumulated with time either.
For CWR quantification in a short period (such as daily), the percentage of CWR accounted for by the state quantity (M_{h}) increases significantly, while that accounted for by advection items (Q_{hi }and Q_{ho}) decreases. The sink/source terms (C_{vh}, C_{hv}, and P_{s}) still contribute the most to CWR. Taking April as an example (Fig. 1), it is found that during the 30 days of April, the contributions to CWR of C_{vh}, P_{s}, C_{hv}, Q_{ho}, and M_{h2 }are within the ranges of 5.19%–278.88% (with an average value of 77.64%), 0.11%–240.23% (46.93%), 9.2%–77.81% (40.58%), 1.49%–78.07% (33.11%), and 0.8%–34.88% (16.7%), respectively. Among the CWR contributors, C_{vh} in rainy days is generally larger than the others.
Figure 1. Daily variations of numerical quantification results of the CWR contributing terms (unit: 10^{12 }kg, i.e., billion ton) in North China in April 2017. (a) M_{h1}, C_{vh}, and Q_{hi}, and (b) M_{h2}, C_{hv}, P_{s}, and Q_{ho}.
For the daily CWR quantification, the contribution of the state items cannot be ignored. With the extension of the calculation period, the importance of the state terms gradually decreases, while that of advection and source/sink terms gradually increases. Similar features can be found for water vapor and atmospheric water substance. However, for the quantification of a large area, despite the varying calculation period, the value of C_{vh} remains higher than that of other CWR contributors, i.e., C_{vh} makes the most important contribution to CWR. Therefore, the CWR in a large area mainly depend on cloud physical processes, followed by advection along the boundary.

Figure 2 presents the diagnostic results on the 1° × 1° grids in North China in April 2017. By comparing the grid results with the overall regional quantification results, the impact of spatial scale on CWR can be revealed. To facilitate comparison, the calculation results have been converted into the equivalent water depth per unit area (unit: mm).
Figure 2. Spatial distributions of (a) MM_{h}, (b) Q_{hi}, (c) C_{vh}, (d) P_{s}, (e) GM_{h}, and (f) CWR in North China for April 2017 (unit: mm). The thick black curves denote the provincial boundaries and the red curve denotes the Yellow River.
For the quantification results on 1° grids, M_{h} is the smallest contributor to CWR, with a value below 0.45 mm. The magnitude and spatial distribution of C_{vh} are consistent with those of P_{s}, with the maximum value of about 65 mm. Q_{hi} has the largest value, three times higher than the source/sink terms, and can exceed 210 mm. Therefore, the grid values of GM_{h} and CWR are mainly affected by Q_{hi}, and their spatial distributions are also similar to Q_{hi}. This feature is quite different from the quantification results of the whole region, which is characterized by more contribution to CWR of C_{vh }than Q_{hi.}
For the calculation results of North China, C_{vh} and P_{s} are spatially accumulative, equal to the sum of gridded value within the region. The advection terms (Q_{hi} and Q_{ho}) are integrals along the boundary of the study region, while those at interior grid points have been completely cancelled out. Consequently, advection terms of large region have a much smaller value than the sum of gridded values. For instance, regional Q_{hi} (22.04 billion ton) through the boundary of North China in April are much smaller than the sum of Q_{hi} (229.76 billion ton) at all grid points inside the area. Therefore, when the evaluation area is larger, the effect of Q_{hi} is weakened. Furthermore, CWR cannot be simply spatially accumulated, either. The regional CWR is much smaller than the sum of CWR at all grids as well, and the difference between them should be equal to the sum of gridded Q_{ho} within the area. Similarly, GM_{x} cannot be accumulated with expanded area; consequently, neither PE_{x} nor RT_{x} can accumulate or average with the area.
In summary, the CWR, its contributors, and related variables have unique spatial and temporal characteristics, and cannot be simply accumulated with time or area.

Precipitation is formed by cloud physical processes. Therefore, P_{s} should be closely related to cloud physical parameters such as C_{vh}. According to Fig. 1, there occurred several precipitation processes in April 2017, on 4, 6–8, 13–16, 18–20, and 24–25 April, respectively. For each precipitation event, Q_{hi} increased before the occurrence of precipitation, and C_{vh} increased accordingly. When P_{s} reached its maximum, C_{vh} also reached the maximum, well correlated to P_{s}. Once the precipitation ended, C_{vh} decreased rapidly, evaporation prevailed in the atmosphere, leading to smaller C_{vh} than C_{hv}. Among the CWR contributors, P_{s} and C_{vh} have a very good consistency, with a similar evolution curve over time. In this section, we use the daily quantification results to further analyze the relationship between each of the CWR contributors and the precipitation.
Table 3 lists the correlation coefficient (R) of daily precipitation with CWR contributors in North China from the two methods. The results indicate that C_{vh} and C_{vh }− C_{hv} produced by cloud microphysics processes have the best correlation with P_{s}, showing an obvious positive correlation. The correlation coefficient (R) exceeds 0.9 in both months. MM_{h} also has a certain correlation with P_{s}, which is higher in August than in April. The correlation between Q_{hi} and P_{s} is not as obvious as that between P_{s} and C_{vh} or MM_{h}. The reason might be that the state terms and the advection terms change very little in the hydrometeors budget equation, and P_{s} is mainly determined by C_{vh}.
MM_{h} Q_{hi} C_{vh} C_{vh }− C_{hv} MM_{v} Q_{vi} Q_{vi }− Q_{vo} April R_{NQ} 0.62 0.55 0.95 0.97 0.32 0.14 0.14 R_{DQ} 0.37 0.46 0.99 0.99 0.46 0.54 0.13 August R_{NQ} 0.71 0.44 0.90 0.90 0.33 0.59 0.54 R_{DQ} 0.63 0.53 0.99 0.99 0.30 0.75 0.69 Table 3. Correlation coefficients (R) of daily P_{s} with the CWR contributors in April and August 2017 for numerical (NQ) and diagnostic (DQ) quantification results
Table 3 also lists the correlation coefficients between P_{s} and water vapor related variables commonly used by other studies. For shortterm (such as daily) precipitation, the correlation of MM_{v} with P_{s} is relatively low. The surface precipitation in April is weak, and the correlation of water vapor convergence (Q_{vi }− Q_{vo}) with P_{s} is poor from both quantification methods, although R between the diagnostic Q_{vi} and P_{s} is 0.54. As the precipitation increases in August, correlation between the advection terms and precipitation is significantly enhanced, with values of R of Q_{vi} and Q_{vi }− Q_{vo} with P_{s} exceeding 0.5.

Figure 3 displays scatter plots and fitting curves of gridded daily P_{s} versus C_{vh} (unit: mm) in the study area for April and August 2017 from diagnostic quantification results. P_{s} is highly correlated with C_{vh} with R of 0.98 (April) and 0.99 (August), respectively. The values of P_{s} and C_{vh} demonstrate consistent variation characteristics, i.e., the larger the C_{vh}, the larger the P_{s }is.
Figure 3. Scatter plots and fitting curves of observed daily P_{s} versus C_{vh} (unit: mm) at each grid in North China for (a) April and (b) August 2017.
The correlations between the 1° × 1° gridded P_{s} and other diagnostic contributors to CWR during different periods such as 1, 5, 10day, and 1month are further analyzed and the results are listed in Table 4. In general, among the CWR contributors, C_{vh} has the best correlation with P_{s}. Regardless the length of the period for quantification, R of C_{vh} with P_{s} always exceeds 0.98. The correlations of P_{s} with the advection terms Q_{hi} and Q_{vi} increase with the extension of the study period. For the 1month quantification, R of the advection terms with P_{s} is close to 0.7. The state quantities MM_{h} and MM_{v} are poorly correlated to P_{s}, compared with the other variables.
Period MM_{h} Q_{hi} C_{vh} MM_{v} Q_{vi} April 1day 0.28 0.39 0.98 0.36 0.35 5day 0.30 0.53 0.98 0.46 0.57 10day 0.22 0.58 0.98 0.52 0.69 1month 0.22 0.58 0.98 0.52 0.69 August 1day 0.49 0.51 0.99 0.28 0.41 5day 0.46 0.49 0.99 0.30 0.48 10day 0.51 0.52 0.99 0.39 0.54 1month 0.53 0.64 0.99 0.54 0.62 Table 4. Correlation coefficients (R) of diagnosed CWR contributors over each 1° grid with P_{s} in April and August 2017
The above results indicate that during the processes of water substance variation and precipitation formation in the atmosphere, the relationship between hydrometeors and precipitation is quite close. Compared with state quantities and advection transport, the condensation process from water vapor to hydrometeors has more important impacts on the distribution and magnitude of precipitation. Despite the varying size of the study area and varying length of calculation, P_{s} keeps strongly and positively correlated with C_{vh}. Consequently, it is not sufficient to study the CWR and precipitation by only investigating on the state quantities and/or advection terms, as the hydrometeors converted from water vapor have played a more significant role.

Figure 4 shows the daily evolution of CWR, P_{s}, GM_{h}, and PE_{h} in April 2017 obtained from the CWRNQ method. In general, GM_{h} and CWR accumulate prior to rainfall, and decrease after rainfall occurs. The CWR on days without rainfall is generally smaller than that on rainy days. GM_{h} reaches its peak on 16 April, and is larger than that during 7–8 April. However, P_{s} is also the largest on 16 April, which is significantly larger than that during 7–8 April. Therefore, the CWR during 7–8 April is only a bit larger than that on 16 April. Abundant CWR appears when GM_{h} is very high and PE_{h }(the efficiency of cloud microphysical processes in transforming hydrometeors in the atmosphere into surface precipitation) is low, such as that during 3–10 April (see the two peaks before 12 April 2017). When both GM_{h} and PE_{h} are high, P_{s} is large and CWR is low, as indicated by the three peaks of the curves in Fig. 4 after 12 April 2017.
Figure 4. Daily variations of CWR, P_{s}, and GM_{h} (unit: 10^{12 }kg, i.e., billion ton), as well as PE_{h} (unit: %) in North China in April 2017.
The above results indicate that precipitation is mainly produced by the cloud physical process. It is the hydrometeors that are the truly source of P_{s}. Compared with water vapor, it is more reasonable to use CWR and C_{vh} formed through cloud microphysical processes to investigate precipitation and its variations. When GM_{h} is large but PE_{h} is small (that is, precipitation formation through natural cloud microphysical processes is inefficient), abundant CWR will remain in the atmosphere. Such understandings are important for improving not only the accuracy of precipitation forecast, but also the effects of weather modification and CWR exploitation.

The quantification results in North China show that PE_{h} is much higher than PE_{v}. Taking the numerical quantification result as an example, PE_{h} values in April and August 2017 are 42.87% and 51.65%, while PE_{v} values are only 5.61% and 14.99% respectively. Based on MM5 simulations, Zhou et al. (2010) estimated the PE in a stratiform cloud system that occurred in Henan Province of central China. They found that PE_{h} was about 69.7% and PE_{v} was about 31.1%. Tao et al. (2015) used the CAMS (Chinese Academy of Meteorological Sciences) mesoscale cloudresolving model to calculate the PE in a stratocumulus precipitation process in Beijing. According to their study, PE_{h} and PE_{v} were about 44.9% and 5.6%, respectively. The previous results are in good agreement with those obtained from the monthly results by the CWRNQ method in this study. Similar features can also be derived from the monthly diagnostic results, where PE_{h} of North China is 51.65% in April and 86.34% in August, while PE_{v} is 6.14% in April and 16.85% in August.
It is found that RT_{h} is significantly shorter than RT_{v}. With regard to the diagnostic quantification results, RT_{h} is 10.34 h in April and 2.56 h in August, while RT_{v} is 14.98 (6.92) days in April (August). According to Encyclopedia of China (China Encyclopedia General Committee, 1987), RT_{v} is about 8 days, while RT_{h} is about 2 h (Zhang, 2002). Results of the present study are of the same magnitude as those of previous studies.