A new algorithm for seasonal precipitation forecast based on global atmospheric hydrological water budget

Applied Mathematics and Computation 268 (2015) 478–488

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

A new algorithm for seasonal precipitation forecast based on global atmospheric hydrological water budget Yong-Ping Wu a,b, Guo-Lin Feng a,c,∗ a

College of Physics Science and Technology, Yangzhou University, Yangzhou, Jiangsu Prov., CN-225002, China Ecological Complexity and Modeling Laboratory, University of California, Riverside, CA 92521–0124, USA c Laboratory for Climate Studies of China Meteorological Administration, National Climate Center, Beijing 100081, China b

a r t i c l e

i n f o

Keywords: Seasonal precipitation Precipitation source equation Conversion equation

a b s t r a c t Precipitation forecast has been identified as one of the central issues in climate research. However, the underlying mechanisms of precipitation are far from being understood. In this paper, a new algorithm of forecasting precipitation based on law of conservation of mass in hydrological cycle is proposed and its feasibility is verified. The algorithm mainly include three steps: in the first step, the area we employ is divided into a number of sub-areas, the precipitation source and evaporation whereabouts equations for sub-regions are established, and the rationality of them can be verified by checking whether the precipitation source and evaporation equations meet a self-consistent relationship or not; in the second step, a conversion equation for sub-regional precipitation prediction will be established, which characterize the relationship between precipitation and evaporation in the sub-areas; in the last step, if the regional evaporation, precipitation and moisture divergence (convergence) function keep stable in a certain time scale, then precipitation forecast is achieved by evaporation anomalies and moisture divergence function, which can be predicted according to the prophase sea surface temperature and atmospheric circulation. Finally, the northern and southern hemispheres seasonal precipitation, evaporation and moisture divergence (convergence) weighting coefficients are calculated using this algorithm based on European centre for medium-range weather forecasts (ECMWF) interim re-analysis (ERA-Interim) dataset, which well verifies the feasibility of the algorithm. The obtained results may provide new insights for precipitation forecast in the future. © 2015 Elsevier Inc. All rights reserved.

1. Introduction The seasonal precipitation prediction is very important but difficult in short-term climate prediction, which is also urgent to the needs of society. The elapsed prediction methods are mostly designed from the points of view of statistics and dynamics and always been evaluated according to forecast accuracy and forecast period. Previous studies developed some techniques and algorithms to improve the forecast accuracy. Shamseldin [15], Ju et al. [4] and Jain and Srinivasulu [5] brought forward the season rainfall-runoff prediction method based on BP neural network. Yang et al. [22] raise a data mining approach for heavy rainfall forecasting based on satellite image sequence analysis. Schellart et al. [14], Gu et al. [1] put forward an objective prediction method of monthly precipitation. Wang et al. [20] proposed an objective and quantifiable forecasting method based on optimal factors combinations in precipitation. Wang and Fan [19] provided the three-factor integrated forecasting methods including

∗

Corresponding author at:. College of Physics Science and Technology, Yangzhou University, Yangzhou, Jiangsu Prov., CN-225002, China. Tel.: +8613567344438. E-mail address: [email protected] (G.-L. Feng).

http://dx.doi.org/10.1016/j.amc.2015.06.059 0096-3003/© 2015 Elsevier Inc. All rights reserved.

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eastern equatorial Pacific sea surface temperature (SST), Kuroshio-westerly drift SST in January and the Tibetan plateau snow in spring and winter. Xiong et al. [21] proposed analogue-dynamical prediction in northeast China based on dynamic and optimal configuration of multiple predictors. Yu et al. developed the analogue-dynamical method for error correction of numerical forecasts [23]. Li et al. [6,7] put forward computational uncertainty principle in nonlinear ordinary differential equations and the general explicit difference formulas for numerical differentiation [8]. These methods, to some extent, improved the accuracy of forecasting precipitation. Other researchers developed the coupling and integration models to prolong the forecasting period. However, a longer forecasting period usually could supply valuable time to prevent the flood, while reduce the forecasting accuracy at the same time [2,13,18]. It is a tendency to couple the hydrology forecasting model with the weather forecasting. Most coupled models are based on runoff-precipitation coupled model [3,11] and improved the timeliness of the flood forecasting to a certain extent. However, these runoff-precipitation coupled models could not fundamentally improve the timeliness of precipitation forecast for the atmospheric water cycle and the sources of precipitation have not been considered yet. Atmospheric water cycle including evaporation, water vapor transportation, precipitation and the changes in water vapor content of regional air column usually abide by the principle of atmospheric water balance and energy balance [9]. The regional precipitation is mainly affected by the change of the water vapor content in the regional air column, the regional evaporation, the amount of water vapor transport between this and other regions, and the water vapor transport path related with atmospheric circulation. On the global scale, it will take an average of 10 days for all of the global atmospheric water vapor to change into precipitation [10,12], which represents that the air water will be gradually depleted and there is no precipitation any longer without evaporation. In fact, the annual observed precipitation is much larger than the global atmospheric water vapor content, which indicates that the observed precipitation is primarily the result of the moisture continuously circulating in the atmosphere. Therefore, it is a fundamentally flawed that the atmospheric water cycle was not considered in precipitation forecast. Atmospheric water cycle equation has been established, which masterly links up the different locations evaporation and precipitation using the principle of conservation of mass [24]. However, how to calculate the divergence (convergence) function is still an open question. In this paper, the calculating methods of the divergence (convergence) function was researched and developed. Moreover, the method of hydrological cycle-based forecasting rainfall (HCFR) has been proposed, and the preliminary feasibility verification has been implemented. The article is organized as follows. In Sections 2, we present the basic idea of HCFR algorithm and obtain equations for precipitation source and evaporation whereabouts and precipitation prediction. And in Section 3, we give the validation of the HCFR algorithm. In Section 4, we give conclusion and discussion about the feasibility, limitation and the applicable conditions.

2. Theoretical basis of HCFR algorithm 2.1. The basic idea of HCFR algorithm Due to the effect of atmospheric circulation, the precipitation in any place comes likely from the evaporation in any place on earth; On the contrary, evaporation from any place is likely transported to any place of the earth and become the local precipitation [25]. This reflects the complex relationship between evaporation and precipitation in the process of water cycle. According to the law of conservation of mass, the spatial-temporal distribution of the total precipitation or evaporation is stable in a certain time-space range (the time range belongs to climate-scale and the time-space range should have a minimum), although the proportion of evaporation and precipitation is uneven in respective sub-regions and sub-periods. Therefore, according to this stable distribution and balance in climate-scale, there were relatively stable source and transport path of water vapor for the regional precipitation, and an equation between one regional precipitation and any regions evaporation can be established.

2.2. Equations for precipitation source and evaporation whereabouts Generally, precipitation over land regions is derived from two sources: (1) evaporation and transpiration from the land surface within the region, and (2) water vapor adverted into the region by air motion. Given global is divided into n regions, it is considered that precipitation in any place (Pi ) may come from evaporation in anywhere (Ej ) on the earth. Therefore, given the world is divided into n regions, the precipitation in the ith area can be expressed as:

Pi =

n 

Di j E j

i = 1, 2, 3, . . . , n,

(1)

j=1

where Pi and Ej are precipitation in ith region and evaporation in jth area respectively. Dij is defined as the divergence weighting coefficient of Ej evaporation in jth area which contributes to the precipitation in ith area. This equation was established only in the climatic significance, which implies precipitation in an area during relatively long time scales equals to the sum of precipitation contributed by evaporation from all regions on the earth. Therefore, this is the precipitation source equation. Since expression

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(1) has been established for all areas, it can be expressed as a matrix:

⎛ ⎞

⎛ P1 D11 P 2 ⎜ ⎟ ⎜D21 ⎜.⎟ ⎜ ⎜ .. ⎟ ⎜ .. ⎜ ⎟ ⎜ . ⎜ Pi ⎟ ⎜ ⎜ ⎟ ⎜ Di1 ⎜ .. ⎟ = ⎜ . ⎜.⎟ ⎜ . ⎜ ⎟ ⎜ . ⎜ Pj ⎟ ⎜D ⎜ . ⎟ ⎝ j1 ⎝.⎠ . Dn1 Pn

D12 D22 .. . Di2 .. . D j2 Dn2

··· ··· .. . ··· ··· ··· ··· ···

D1i D2i .. . Dii .. . D ji Dni

··· ··· ··· ··· .. . ··· ··· ···

⎞⎛ ⎞

E1 D1 j · · · D1n D2 j · · · D1n ⎟⎜E2 ⎟ .⎟ ⎟⎜ ⎟⎜ .. ⎟

⎟ ⎟⎜ ⎜ ⎟

Ei ⎟ Di j · · · Din ⎟ ⎟⎜ .⎟ ⎟⎜ ⎜ .. ⎟

⎟⎜ ⎟ ⎟ Ej⎟ D j j · · · D jn ⎟⎜ ⎟ ⎠⎜ ⎝ .. ⎠

(2)

. En

Dn j · · · Dnn

Obviously, Dij satisfies the normalization condition as following: n 

Di j = 1

j = 1, 2, 3, . . . , n.

(3)

i=1

Similarly, the evaporation district can also be expressed as:

Ei =

n 

i = 1, 2, 3, . . . , n,

Ci j Pj

(4)

j=1

or

⎛ ⎞

⎛ E1 C11 ⎜E2 ⎟ ⎜C21 ⎜.⎟ ⎜ ⎜ .. ⎟ ⎜ .. ⎜ ⎟ ⎜ . ⎜ Ei ⎟ ⎜ ⎜ ⎟ ⎜ Ci1 ⎜ .. ⎟ = ⎜ . ⎜.⎟ ⎜ . ⎜ ⎟ ⎜ . ⎜ E j ⎟ ⎜C ⎜ . ⎟ ⎝ j1 ⎝.⎠ . Cn1 En

C12 C22 .. . Ci2 .. . C j2 Cn2

··· ··· .. . ··· ··· ··· ··· ···

C1i C2i .. . Cii .. . C ji C ni

··· ··· ··· ··· .. . ··· ··· ···

⎛ ⎞

⎞ P1 C1 j · · · C1n C2 j · · · C1n ⎟⎜P2 ⎟ .⎟ ⎟⎜ ⎟⎜ .. ⎟ ⎟ ⎟⎜ ⎜ ⎟

Pi ⎟ Ci j · · · Cin ⎟ ⎟⎜ . ⎟, ⎟⎜ ⎜ .. ⎟

⎟⎜ ⎟ ⎟ Pj ⎟ C j j · · · C jn ⎟⎜ ⎟ ⎠⎜ ⎝ .. ⎠ Cn j · · · Cnn

(5)

. Pn

where Cij is the convergence weighting coefficient of the precipitation in jth area which is come from the evaporation in ith area. Cij satisfies the normalization condition as following: n 

Ci j = 1

j = 1, 2, 3, . . . , n.

(6)

i=1

According to (2) and (5), the relationship between the precipitation matrices and evaporation matrices is expressed as:

P = DE = DCP,

(7)

thus,

DC = ,

(8)

where  is an unit matrix. Additionally, according (1), the contribution rate of evaporation in jth region to precipitation in ith region is expressed as (Di j E j )/Pi , it is equal to Cji and Dij is equal to (C ji Ei )/Pj on the contrary. It is indicated that D and C meet the self-consistent relationship, which entails that in a certain time scale, the total amount of water vapor evaporating from the ground, assigned to the different regions and fall back to the ground in the form of precipitation through atmospheric water cycle; likewise, the total precipitation coming from evaporation in different areas are eventually equal to total evaporation from the ground. This reflects the completeness of water cycle maintaining mass conservation at a global scale. Precipitation source equation gives the quantitative relationships between a certain area precipitation and various regions evaporation on Earth. The main source areas of water vapor for ith area precipitation can be found based on the sequence of Cji (j = 1, 2, 3,…), thus the precipitation in ith area can be calculated according to the evaporation come from main sources. The sources of water vapor for most areas are on the ocean, while the ocean evaporation depends on the sea surface temperature. Since ocean evaporation will need a period of t to be transported to an area (especially in inland areas) and form the precipitation there, precipitation in a particular region at time t will be forecasted according to the evaporation and sea surface temperature in water vapor source areas at time t–t. Thereby, the timeliness of precipitation forecast is improved.

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2.3. Conversion equation of precipitation prediction Precipitation Eq. (1) is established only in the climate sense, so the actual precipitation Pi in ith region, the actual evaporation Ej in jth region and divergence weighting coefficient Dij are written as the average of a certain time scale (stable items) plus anomalies (abnormal items), which are expressed as:

⎧  ¯ ⎪ ⎨Pi = Pi + P i

(i, j = 1, 2, 3, . . . , n),

E j = E¯ j + E  j

⎪ ⎩

(9)

Di j = D¯ i j + D i j

where P¯i , E¯ j and D¯ ji are stabilize items, which can be replaced with the corresponding multi-year averages, if the decadal changes is not evident. Pi , E j and Dji are abnormal items. Substituting Eq. (9) into precipitation sources Eq. (1), the precipitation in ith region can be expressed as:

Pi = P¯i + Pi =

n 

D¯ i j E¯ j +

j=1

n 

[D¯ i j E  j + D i j E¯ j + D i j E  j ].

(10)

j=1

The first term is stable and second items are abnormal, and if P¯i , E¯ j and D¯ ji can be regarded as constants at a certain time and space scales (That is, precipitation, evaporation and the divergence weighting coefficient satisfy the constraints), the precipitation forecast in ith region will be converted to the forecast of rainfall anomalies:

Pi =

n 

[E  j D¯ ji + E¯ j D ji + E  j D ji ]

(11)

j=1

Eq. (11) expresses the quantitative relationship between rainfall anomalies in ith region and evaporation anomalies on water vapor sources and the corresponding divergence weighting coefficient. When the impact of evaporation anomalies on ith region rainfall anomalies in some regions is small enough that the corresponding items can be ignored, the rainfall anomalies in ith area will be calculated based on the evaporation anomalies and divergence weighting coefficient anomalies in some main water vapor sources. Here Eq. (11) is called precipitation predicted conversion equation. The evaporation anomalies of water vapor source, due to being associated with early sea surface temperature (SST), are called SST factors, which can be forecasted according to the early SST. The divergence weighting coefficient anomalies are often associated with pre-atmospheric circulation anomalies resulting water vapor transport path abnormalities, called circulation factors, which can be predicted according to the path of the water vapor transport. The key of the HCFR algorithm is the design and calculation of divergence weighting coefficient (Dij ), which is depending on the spatial and temporal scales. In fact, any seasonal rainfall forecast hydrological model must consider the problems of space and time scales [16]. And that in how much time and space scales P¯i , E¯ j and D¯ ji can be seen as constants need hypothesis and verification from the global scale, longer time scales to downscaling gradually. The specific algorithms of design and calculation of divergence weighting coefficient will be given in the following experiment. 3. Feasibility validation of the HCFR algorithm In order to discuss how to use HCFR algorithm to predict the seasonal precipitation of northern hemisphere, the global is divided into two regions that are northern and southern hemispheres. The precipitation anomalies, evaporation anomalies and the divergence weighting coefficient anomalies are calculated based on ERA-interim reanalysis data, for the ERA-interim reanalysis data is consistent with other reanalysis data best through comparative analysis and the standard deviation of annual average precipitation and evaporation are the least. 3.1. Verification of water balance conditions and constraints Global averages of precipitation and evaporation must remain equal to each other on climate time scales [17]. Water balance conditions connote the global seasonal precipitation is equal to the global seasonal evaporation as fallowing:

{PT }G = {ET }G ,

(12)

where P is precipitation, E is evaporation, T is a season and G represents global. For the global atmospheric column, the water balance equation in arbitrary time scale is expressed as PG = EG − WG , where, WG is the incremental amount of global atmospheric water vapor content. Since WG is only about 0.135‰ of PG and EG in seasonal scale by calculating the ERA-interim reanalysis data, it is considered that precipitation and evaporation from ERA-interim reanalysis data satisfy the water balance conditions. Table 1 shows the global average annual precipitation, evaporation and their absolute deviation calculated by using ERA-interim reanalysis data. The absolute deviation percentages of global seasonal precipitation and evaporation varied from 1.27% to 2.29%. It is indicated that precipitation and evaporation is not fully satisfy the constraints, which will bring about systematic error in later calculations and have a certain influence on the verification of HCFR algorithm feasibility. However, since

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Y.-P. Wu, G.-L. Feng / Applied Mathematics and Computation 268 (2015) 478–488 Table 1 1979–2011 global average annual precipitation, evaporation and their absolute deviation and t test value.

Spring Summer Autumn Winter Annual

P E P E P E P E P E

Annual average (1015 kg)

Absolute deviation (1015 kg)

Deviation percentage (%)

t

135.53 133.55 140.02 139.31 135.00 130.63 132.74 129.80 543.29 516.90

2.82 2.68 3.04 2.99 2.92 2.52 3.04 2.76 11.05 6.56

2.08 2.01 2.17 2.15 2.16 1.93 2.29 2.13 2.03 1.27

0.01 0.40 0.00 0.00 0.00

the maximum deviation is not more than 2.29%, the seasonal and inter-annual precipitation and evaporation can be used as constants at global scale, which denotes precipitation and evaporation data basically satisfy the constraints. In addition, the differences between precipitation and evaporation at global annual and seasonal scales was tested by the t test algorithm, the results show that the maximum t value is in summer with 0.40, which does not pass the 0.5 degree (t > 0.68) test. It is indicated that there is no significant difference between the global season precipitation and evaporation as well as that between annual precipitation and annual evaporation, and it is considered that precipitation and evaporation meet water balance conditions at seasonal scales. Precipitation predicted conversion Eq. (11) shows that to achieve the northern hemisphere seasonal precipitation forecasts is based on that the seasonal northern hemisphere and southern hemisphere precipitation and evaporation satisfy the constraints as following: {PT }N = C1 ; {ET }N = C2 ; {ET }s = C3 , where, {PT }N and {ET }N are northern hemisphere precipitation and evaporation, respectively, {ET }s is southern hemisphere evaporation and T represents season. Fig. 1 shows that all seasonal precipitation and evaporation in global, northern and southern hemisphere essentially unchanged. What’s more, the maximum deviation percentage of the northern hemisphere seasonal precipitation is in spring with 4.23%, which is less than 5% (Table 2). Thus, it is considered that the northern hemisphere precipitation in all season satisfy the constraints. 3.2. Hemispheres precipitation sources and evaporation whereabouts equations The global is divided into southern and northern hemisphere along the equator as that in Fig. 2. PS , PN , ES and EN denote southern and northern hemispheres precipitation and southern hemispheres evaporation, QSN , QNS , RSN and RNS denote the amount of water vapor transported and runoff from the southern hemisphere to the northern hemisphere and that from the northern to the southern hemisphere. WS and WN are the increments of the atmospheric water vapor content for northern and southern hemispheres. According to the mass conservation law, water balance equations for southern and northern hemispheres atmospheric column are expressed as follows:

(EN + QSN ) − (PN + QNS ) = WN , (ES + QNS ) − (PS + QSN ) = WS ,

(13)

the sum of the two equations is:

(EN +ES ) − (PN +PS ) =WN + WS .

(14)

That is:

EG − PG = WG ,

(15)

where EG , PG and WG are global evaporation, precipitation and the increment of the atmospheric water vapor content. SinceWG tends to 0, (13) can be simplified as:

EN + QSN = PN + QNS , ES + QNS = PS + QSN .

(16)

According to Eq. (1), precipitation source equations for southern and northern hemispheres are expressed as:

⎧ ⎨PN = DNN EN + DNS ES , ⎩

(17)

PS = DSN EN + DSS ES ,

where Di j (i, j = N, S) is divergence weighting coefficient, which satisfy the normalization condition of DNN + DSN = 1, DNS + DSS = 1. Evaporation whereabouts equations are expressed as following according to (4):

EN = CNN PN + CNS PS , ES = CSN PN + CSS PS ,

(18)

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Fig. 1. The time series of hemispheres precipitation and evaporation in different seasons (P, E, Ps, Es, Pn, En are global, southern and northern hemisphere precipitation and evaporation, 1, 2, 3, 4 represent spring, summer, autumn and winter). Table 2 1979–2011 northern hemisphere annual average precipitation, its absolute deviation and deviation percentage in different seasons. Seasons

Annual average (1015 kg)

Absolute deviation (1015 kg)

Deviation percentage (%)

Spring Summer Autumn Winter

58.99 83.64 80.08 57.57

2.50 2.65 2.11 1.99

4.23 3.17 2.64 3.46

where Ci j (i, j = N, S) are convergences weighting coefficients, which satisfy the normalization condition of CNN + CSN = 1, CNS + CSS = 1. In addition, Di j and Ci j (i, j = N, S) also satisfy the self-consistent relationship:



DNN

DNS

DSN

DSS



CNN

CNS

CSN

CSS





=

1 0



0 . 1

(19)

3.3. Design and verification of divergence (convergence) weighting coefficients There are two methods to design divergence (convergence) weighting coefficients: (1) according to the relationship of the earth surface water cycle elements such as precipitation, evaporation and runoff; (2) according to the relationship of the atmospheric water cycle elements such as precipitation, evaporation and water vapor transportation. The latter is used to design and calculation the divergence (convergence) weighting coefficients, because this study pays more attention to the impact of water vapor source evaporation on regional precipitation and focus on predicting precipitation anomalies by evaporation anomalies from water vapor sources. Eq. (3) shows that the air moisture in northern hemisphere mainly comes from northern hemisphere

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Y.-P. Wu, G.-L. Feng / Applied Mathematics and Computation 268 (2015) 478–488

Fig. 2. Schema of the water cycle relationship between southern and northern hemisphere.

Fig. 3. Schematic of calculating water vapor divergence weighting coefficient (Dij ).

evaporation (EN ) and the water vapor transportation from southern hemisphere (QSN ), while the whereabouts of the air moisture in northern hemisphere includes precipitation in northern hemisphere (PN ) and the amount of water vapor transportation to the southern hemisphere (QNS ). The two sources of water vapor for the air in northern hemisphere can be considered full equality (symmetrical) and be reassigned to PN and QNS in the same proportion. According to the physical meaning of divergence weighting coefficient, DNN refers the ratio of the part EN which is contributed to PN and the total EN . Fig. 3 shows the two ways for northern hemisphere evaporation to convert to the northern hemisphere precipitation: (1) the evaporation from the northern hemisphere directly convert into the northern hemisphere precipitation in the ratio of PN /(PN + QNS ) (the solid line corresponding to  1 ); (2) the evaporation in the northern hemisphere is transported to the southern hemisphere in the ratio of 2 ), in which, there is QSN /(PS + QSN ) ratio being transported back to the northern QNS /(PN + QNS ) (dotted line corresponding to  hemisphere from southern hemisphere (dotted line corresponding to  3 ) and contributing to the northern hemisphere precipiQNS QSN PN PN 4 ). Thus DNN = P +Q + P +Q · P +Q · P +Q is obtained as tation in the ratio of PN /(PN + QNS ) (dotted line corresponding to  N NS N NS S SN N NS well as DSN , DSS and DNS as following:

⎧ PN QNS QSN PN ⎪ DNN = + · · ⎪ ⎪ PN + QNS PN + QNS PS + QSN PN + QNS ⎪ ⎪ ⎪  P  ⎪ QSN QNS QSN ⎪ N ⎪ ⎨DNS = P + Q · P + Q + P + Q · P + Q S SN N NS N NS S SN  P  Q Q Q ⎪ NS S SN NS ⎪ D = · + · ⎪ SN ⎪ PN + QNS PS + QSN PS + QSN PN + QNS ⎪ ⎪ ⎪ ⎪ ⎪ PS QSN QNS PS ⎩D = + · · SS

PS + QSN

PS + QSN

PN + QNS

(20)

PS + QSN

Similarly, the convergence weighting coefficients are obtained as following:

⎧ EN QSN QNS EN ⎪ CNN = + · · ⎪ ⎪ EN + QSN EN + QSN ES + QNS EN + QSN ⎪ ⎪ ⎪  E  ⎪ QNS QSN QNS ⎪ N ⎪ ⎨CNS = E + Q · E + Q + E + Q · E + Q S NS N SN N SN S NS  E  QSN QNS QSN ⎪ S ⎪ C = · + · ⎪ SN ⎪ EN + QSN ES + QNS ES + QNS EN + QSN ⎪ ⎪ ⎪ ⎪ ⎪ ES QNS QSN ES ⎩ CSS =

ES + QNS

+

ES + QNS

·

EN + QSN

·

(21)

ES + QNS

It is clearly that the moisture divergence (convergence) weighting coefficients satisfy the normalization condition and the water balance equation, shown in (20) and (21). But, whether do they meet the self-consistent relationship requires further calculation and verification. In addition, the moisture divergence weighting coefficient (Dij ) is calculated by the amount of precipitation and moisture transport in Eq. (20), this calculation scheme is called P program. And the calculation scheme, in (21), to calculate convergence weighting coefficient (Cij ) by the amount of evaporation and moisture transport is called E program.

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Table 3 The annual average Dij (Cij) for northern hemisphere by E and P program in 4 reasons. Spring



DNN DSN

DNS DSS

CNN CSN

CNS CSS







 

Summer









0.94 0.06

0.06 0.94

0.93 0.07

0.05 0.95

Autumn









1.00 0.00

0.25 0.75

0.77 0.23

0.00 1.00

Winter

0.99 0.01

0.16 0.84

0.86 0.14

0.01 0.99











0.83 0.17

0.03 0.97

0.97 0.03

0.15 0.85



Table 4 Precipitation from data and calculated by E and P programs, their standard deviation (percentage) and t.

E_P1 P_P1 E_P2 P_P2 E_P3 P_P3 E_P4 P_P4

Data (1015 kg/a)

Calculation (1015 kg/a)

Standard deviation (1015 kg/a)

Standard deviation/data (%)

t

58.99 60.43 83.64 82.47 80.08 75.09 57.57 56.28

60.75 58.99 82.32 83.64 74.90 80.08 56.09 57.57

1.39 1.29 1.86 2.09 3.66 3.53 1.34 1.17

2.40 2.23 2.20 2.48 4.54 4.37 2.31 2.01

0.00 0.10 0.00 0.00

Notes: E_P and P_P represent precipitation calculated by E and P programs for northern hemisphere respectively; 1, 2, 3 and 4 represent spring, summer, autumn and winter respectively.

According to water balance equation, Eq. (16), the amount of precipitation in Eq. (20) can replaced with amount of evaporation and water vapor transportation, and thus the moisture divergence weighting coefficient (Dij ) can also be calculated by E program. The moisture divergence weighting coefficient (Dij ) and convergence weighting coefficient (Cij ) are calculated by both P program and E program. The results show in Table 3 that: (1) In the case of keeping two decimal places, Dij calculated by E and P program is exactly the same as well as Cij (there is no listed respectively here); (2) The moisture divergence weighting coefficient (Dij ) and convergence weighting coefficient (Cij ) satisfy the normalization condition in seasonal scale; (3) The multiplication of moisture divergence and convergence weighting coefficient is not equal to the unit matrix, that is DC = , with the average deviation of 15.9%. There are two possible reasons for this error: (1) The moisture divergence (convergence) weighting coefficient calculation is based on the approximate water balance equation, see Eq. (16); (2) The data used in this study have some errors. The average precipitation in northern hemisphere for 1979–2011 and its absolute deviation and absolute deviation percentage are calculated using E and P program respectively (see Table 4). The results show in Table 4 that the maximum rate of deviation between the calculated value and data value is 4.54% and the minimum is 2.01%. This indicates that it is reliable to use HCFR algorithm to predict precipitation. Additionally, the difference of the two programs accepted the t-test and significance tests and the results show that the t value does not pass the 95% level of confidence. This indicates that there is no significant difference between the two programs. So E program, as the unified program is used to predict the precipitation anomalies for northern hemisphere in following studies. 3.4. Development and validation of seasonal precipitation predicted conversion equation for northern hemisphere According to precipitation predicted conversion Eq. (11), the precipitation predicted conversion equation for northern hemisphere can be obtained as following:

PN = (D¯ NN E  N + D NN E¯N + D NN E  N ) + (D¯ NS E  S + D NS E¯S + D NS E  S ),

(22)

where PN is the precipitation anomalies for northern hemisphere, E¯N (E¯S ) and EN (ES ) are stable term and anomalies of evaporation respectively for northern (southern) hemispheres, D¯ (D¯ ) and D (D ) are stable term and anomalies of the moisture diverNN

NS

NN

NS

gence weighting coefficient between evaporation of northern (southern) hemisphere and precipitation of northern hemisphere, respectively. Among them, the stable term can be regarded as a known quantity, because of satisfying the constraints. DNN EN and DNS ES are the high-order infinitesimal, which account for about 6% of the amount of PN in this case and could be ignored. But, in fact, DNN EN and DNS ES , due to less divided areas, are retained in this study. The northern hemisphere seasonal precipitation anomalies PN are calculated by E program using ERA-interim reanalysis data, shown in Fig. 4. The results show that the inter-annual variability of calculated precipitation is consistent with that of data well in four seasons. But the decadal trends in four seasons are not very good consistency, especially in summer and autumn, the decadal trends are quite different, shown in Fig. 4 (b) and (c). The possible cause of this difference is the use of precipitation predicted conversion equation, in which E¯ j and D¯ i j (i, j = N, S) is instead of the average for many years and this may eliminate or weaken the original decadal trend. To reduce the error, the calculated northern hemisphere precipitation anomalies by E program, EP’, are revised using polynomial fitting curve method. And the cubic polynomial trend, due to no too large amount of calculation and the best fitting, was chosen to revise the error. The

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Fig. 4. The anomalies time series of observed and calculated northern hemisphere precipitation by E program in spring (a), summer (b), autumn (c) and winter (d) for 1979–2011 (P’ is observed precipitation anomalies, EP’ is calculated precipitation anomalies by E program) .

results showed that the corrected precipitation anomalies (E&3PolP’) are very consistent with the actual precipitation anomalies in four seasons, shown in Fig. 5. Additionally, the correlation coefficients of revised calculations with the data are calculated and tested, which are 0.91, 0.89, 0.92 and 0.90 for spring, summer, autumn and winter, respectively, and all pass the 0.001 confidence level. These above results indicate that the moisture divergence (convergence) weighting coefficient can be designed and calculated to establish the precipitation predicted conversion equation aid the water vapor transportation of atmospheric water cycle between different regions. This demonstrates that HCFR algorithm is feasible to predict precipitation in a certain spatial and temporal scales. 4. Conclusion and discussion In this paper, an algorithm of hydrological cycle based forecasting rainfall (HCFR) was proposed based on that an amount of water in atmospheric water cycle maintains the balance in a certain spatial and temporal scales. In this algorithm, we establish the precipitation source equation and the evaporation whereabouts equation, design and calculate the weighting coefficients of moisture divergence (convergence) through the water vapor transportation between regional precipitation and evaporation from the areas which support water vapor source. On this basis, the quantitative relationship between the amount of regional precipitation and evaporation and the regional precipitation predicted conversion equation are established, which converts the precipitation anomalies forecasting into the forecasting of both evaporation anomalies of water vapor source and moisture divergence (convergence) weighting coefficients anomalies skillfully. Since the evaporation of water vapor source is mainly

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Fig. 5. The time series of the revised and the observed precipitation anomalies of northern hemisphere in spring (a), summer (b), autumn (c) and winter (d) for 1979–2011 years.

affected by the impact of pre-SST and the moisture divergence weighting coefficient mainly by the impact of pre-atmospheric circulation, which needs a period of t, a regional precipitation anomalies at time t can be predicted according to vapor source SST anomalies and atmospheric circulation anomalies at t–t moment, and thus improve the timeliness of precipitation forecast. In this paper, the applicable conditions and basic ideas of HCFR algorithm are also analyzed, and the feasibility on HCFR has been demonstrated using ERA-interim reanalysis data. The results show that: HCFR algorithm is feasible in season scale and the scope of the southern, northern hemisphere as well as global, which also proved the basic idea of HCFR algorithm is correct and provide a new perspective for future to further improve the timeliness of regional precipitation. HCFR algorithm makes use of coupled hydrology-weather to forecast rainfall firstly, which not only consider the fundamental sources of precipitation, but also consider the moisture balance in the global distribution and quantitative relationship, and so has broad application prospects. However, can this algorithm be applied in smaller spatial and temporal scales? How much accuracy can be achieved? How to describe the relationship between vapor source SST and the amount of evaporation quantitatively? How to characterize quantitative indicators of the path of water vapor transportation and its quantitative relationship with moisture divergence weighting coefficient? These issues need further exploration.

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Acknowledgments This research was in part supported by the Major National Scientific Research Programs of China (grant nos. 2012CB955902, 2013CB430204) and National Natural Science Foundation of China (grant nos. 41375078, 41375079, 41201062, 41271083). References [1] W. Gu, L. Chen, W. Li, et al., Development of a downscaling method in China regional summer precipitation prediction, Acta Meteorol. Sin 25 (2011) 303–315. [2] G.E. Halkos, K.D. Tsilika, Analyzing and visualizing the synergistic impact mechanisms of climate change related costs, Appl. Math. Comput. 246 (2014) 586–596. [3] R. Hostache, P. Matgen, A. Montanar, et al., Propagation of uncertainties in coupled hydro -meteorological forecasting systems: A stochastic approach for the assessment of the total predictive uncertainty, Atmos. Res. 100 (2011) 263–274. [4] Q. Ju, Z. Yu, Z. Hao, et al., Division-based rainfall-runoff simulations with BP neural networks and Xinanjiang model, Neurocomputing 72 (2009) 2873–2883. [5] A. Jain, S. Srinivasulu, Development of effective and efficient rainfall-runoff models using integration of deterministic, real-coded genetic methods and artificial neural network techniques, Water. Resour. Res. 40 (2004) W04302. [6] J.P. Li, Q. Zeng, J. Chou, Computational uncertainty principle in nonlinear ordinary differential equations I, Numerical results, Sci. China (E) 43 (2000) 449– 460. [7] J.P. Li, Q. Zeng, J. Chou, Computational uncertainty principle in nonlinear ordinary differential equations II, Theoretical analysis, Sci. China (E) 44 (2001) 55–74. [8] J.P. Li, General explicit difference formulas for numerical differentiation, J. Comput. Appl. Math 183 (2005) 29–52. [9] G.W. Liu, Atmosphere Process in Hydrologic Cycle (in Chinese), Science Press, Beijing, 1997. [10] I. Matyasovszky, I. Bogardi, J. Ganoulis, Impact of global climate change on temperature and precipitation in Greece, Appl. Math. Comput. 71 (1995) 119–150. [11] F. Pappenberger, K.J. Beven, N.M. Hunter, et al., Cascading model uncertainty from medium range weather forecasts (10 days) through a rainfall-runoff model to flood inundation predictions within the European Flood Forecasting System (EFFS), Hydrol. Earth Syst. Sci. 9 (2005) 381–393. [12] J.P. Peixoto, A.H. Oort, Physics of Climate, American Institute of Physics Press, New York, 1991. [13] A. Rossa, K. Liechti, M. Zappa, et al., The COST 731 action: A review on uncertainty propagation in advanced hydro-meteorological forecast systems, Atmos. Res 100 (2011) 150–167. [14] A. Schellart, S. Liguori, S. Krämer, et al., Comparing quantitative precipitation forecast methods for prediction of sewer flows in a small urban area, Hydrol. Sci. J. 59 (2014) 1418–1436. [15] A.Y. Shamseldin, Application of a neural network technique to rainfall-runoff modelling, J. Hydrol 199 (1997) 272–294. [16] Q. Shao, M. Li, An improved statistical analogue downscaling procedure for seasonal precipitation forecast, Stoch. Env. Res. Risk. A 27 (2013) 819–830. [17] S. Sherwood, Q. Fu, A drier future, Science 343 (2014) 737–739. [18] A. Ting, Comparison of different aggregation methods in coupling of the numerical precipitation forecasting and hydrological forecasting, Procedia Eng. 28 (2012) 786–790. [19] H.J. Wang, K. Fan, A new scheme for improving the seasonal prediction of summer precipitation anomalies, Weather Forecast 24 (2009) 548–554. [20] Q.G. Wang, G.L. Feng, Z.H. Zheng, R. Zhi, A study of the objective and quantifiable forecasting based on optimal factors combinations in precipitation in the middle and lower reaches of the Yangtze river in summer, Chinese J. Atmos. Sci. 35 (2011) 287–297. [21] K.G. Xiong, J.H. Zhao, G.L. Feng, et al., A new method of analogue–dynamical prediction of monsoon precipitation based on analogue prediction principal components of model errors, Acta. Physica. Sinica 61 (2012) 149204. [22] Y.B. Yang, H. Lin, Z.Y. Guo, et al., A data mining approach for heavy rainfall forecasting based on satellite image sequence analysis, Comput Geosci 33 (2007) 20–30. [23] H. Yu, J.P. Huang, W.J. Li, G.L. Feng, Development of the analogue-dynamical method for error correction of numerical forecasts, J. Meteorol. Research. ACTA. PHYS. SIN-CH ED 28 (2014) 934–947. [24] X.W Zhang, Atmospheric water circulation equation, Plateau Meteorol. 25 (2006) 190–194. [25] X.W. Zhang, S.X. Zhou, Preliminary Exploration in Air Hydrology (in Chinese), China Meteorological Press, Beijing, 2010.