Change of Annual Extreme Water Levels and Correlation with River Discharges in the Middle-lower Yangtze River: Characteristics and Possible Affecting Factors

As one of the fastest developing regions in China, the middle-lower Yangtze River （MLYR） is vulnerable to floods and droughts. With obtained time series of annual highest water level （HWL）, annual lowest water level （LWL） and the corresponding fiver discharges from three gauging stations in MLYR that covering the period 1987-2011, the current study evaluated the change character- istics of annual extreme water levels and the correlation with fiver discharges by using the methods of Vend test, Mann-Whitney-Pettitt （MWP） test and double mass analysis. Major result indicated a decreasing/increasing trend for annual HWL/LWL of all stations in MLYR during the study period. A change point in 1999 was identified for annual HWL at the Hankou and Datong stations. The year 2006 was found to be the critical year that the relationship between annual extreme water levels and fiver discharges changed in the MLYR. With contrast to annual LWL in MLYR, further investigation revealed that the change characteristics of annual HWL were highly consistent with regional precipitation in the Yangtze River Basin, while the linkage with Three Gorges Dam （TGD） operation is not strong. Our observation also pointed out that the effect of serious down cutting of the riverbed and the enlargement of the cross-section area during the initial period of TGD operation caused the downward trend of the relationship between annual LWL and river discharge. Whereas, the relatively raised river water level before the flood season due to TGD regulation since 2006 explained for the changing upward trend of the relationship between annual HWL and river discharge.

Effective groundwater level recovery from mining reduction: Case study of Baoding and Shijiazhuang Plain area

The effective recovery of water level is a crucial measure of the success of comprehensive groundwater over-exploitation management actions in North China.However,traditional evaluation method do not directly capture the relationship between mining and other equilibrium elements.This study presents an innovative evaluation method to assess the water level recovery resulting from mining reduction based on the relationship between variation in exploitation and recharge.Firstly,the recharge variability of source and sink terms for both the base year and evaluation year is calculated and the coefficient of recharge variationβis introduced,which is then used to calculate the effective mining reduction and solve the water level recovery value caused by the effective mining reduction,and finally the water level recovery contribution by mining reduction is calculated by combining with the actual volume of mining reduction in the evaluation area.This research focuses on Baoding and Shijiazhuang Plain area,which share similar hydrogeological conditions but vary in groundwater exploitation and utilization.As the effect of groundwater level recovery with mining reduction was evaluated in these two areas as case study.In 2018,the results showed an effective water level recovery of 0.17 m and 0.13 m in the shallow groundwater of Shijiazhuang and Baoding Plain areas,respectively.The contributions of recovery from mining reduction were 76%and 57.98%for these two areas,respectively.It was notable that the water level recovery was most prominent in the foothill plain regions.From the evaluation results,it is evident that water level recovery depends not only on the intensity of groundwater mining reduction,but also on its effectiveness.The value of water level recovery alone cannot accurately indicate the intensity of mining reduction,as recharge variation significantly influences water level changes.Therefore,in practice,it is crucial to comprehensively assess the impact of mining reduction on water level recovery by combining the coefficient of recharge variation with the contribution of water level recovery from mining reduction.This integrated approach provide a more reasonable and scientifically supported basis,offering essential data support for groundwater management and conservation.To improve the accuracy and reliability of evaluation results,future work will focus on the standardizing and normalizing raw data processing.

Study on the Lowest Navigable Water Level

The design lowest navigable water level is a primary index to the channel grade division.But differ- ent method can get different calculated result,and that even brings contradiction to decide the navigation grade.In this paper the differences between guaranteed rate method and guaranteed rate-frequency method on the fundamental concept of guaranteed rate and the calculated result are carried out.According to the theoreti- cal expression forms of the two methods,the reason leading to the difference is an...

Characteristics of coseismic water level changes at Tangshan well for the Wenchuan MS_8.0 earthquake and its larger aftershocks

Coseismic water level changes which may have been induced by the Wenchuan Ms8.0 earthquake and its 15 larger aftershocks （Ms〉5.4） have been observed at Tangshan well. We analyze the correlation between coseismic parameters （maximum amplitude, duration, coseismic step and the time when the coseismic reach its maximum amplitude） and earthquake parameters （magnitude, well-epicenter distance and depth）, and then compare the time when the coseismic oscillation reaches its maximum amplitude with the seismogram from Douhe seismic station which is about 16.3 km away from Tangshan well. The analysis indicates that magnitude is the main factor influencing the induced coseismic water level changes, and that the well-epicenter distance and depth have less influence. Ms magnitude has the strongest correlation with the coseismic water level changes comparing to Mw and ML magnitudes. There exists strong correlation between the maximum amplitude, step size and the oscillation duration. The water level oscillation and step are both caused by dynamic strain sourcing from seismic waves. Most of the times when the oscillations reach their maximum amplitudes are between S and Rayleigh waves. The coseismic water level changes are due to the co-effect of seismic waves and hydro-geological environments.

Using the concept of ecological groundwater level to evaluate shallow groundwater resources in hyperarid desert regions

This paper, based on the analysis and calculation of the groundwater resources in an arid region from 1980 to 2001, put forward the concept of ecological groundwater level threshold for either salinity control or the determination of ecological warning. The surveys suggest that soil moisture and soil salinity are the most important environmental factors in determining the distribution and changes in vegetation. The groundwater level threshold of ecological warning can be determined by using a network of groundwater depth observation sites that monitor the environmental moisture gradient as reflected by plant physiological characteristics. According to long-term field observations within the Ejin oases, the groundwater level threshold for salinity control varied between 0.5 m and 1.5 m, and the ecological warning threshold varied between 3.5 m and 4.0 m. The quantity of groundwater re- sources （renewable water resources, ecological water resources, and exploitable water resources） in arid areas can be calculated from regional groundwater level information, without localized hydrogeological data. The concept of groundwater level threshold of ecological warning was established according to water development and water re- sources supply, and available groundwater resources were calculated. The concept not only enriches and broadens the content of groundwater studies, but also helps in predicting the prospects for water resources development.

Mean water level setup/setdown in the inlet-lagoon system induced by tidal action—a case study of Xincun Inlet,Hainan Island in China

With the tides propagating from the open sea to the lagoon, the mean water level （MWL） in the inlet and lagoon becomes different from that at the open sea, and a setup/setdown is generated. The change of MWL （setup/setdown） in the system imposes a great impact on regulating the development of tidal marshes, on determining the long-term water level for harbor maintenance, on the planning for the water front development with the flood control for the possible inundation, and on the interpretation of the historical sea level change when using tidal marsh peat deposits in the lagoon as the indicator for open sea＇ s sea level. In this case study on the mechanisms which control the setup/setdown in Xincun Inlet, Hainan in China, the 2-D barotropic mode of Eulerian - Lagrangian CIRCulation （ELCIRC） model was utilized. After model calibration and verification, a series of numerical experiments were conducted to examine the effects of bottom friction and advection terms, wetting and drying of intertidal areas, bathymetry and boundary conditions on the setup/setdown in the system. The modeling results show that setup occurs over the inlet and lagoon areas with an order of one tenth of the tide range at the entrance. The larger the bottom friction is, a larger setup is generated. Without the advection term, the setup is reduced clue to a decrease of water level gradient to compensate for the disappearance of the advection term. Even without overtides, a setup can still be developed in the system. Sea level rise and dredging in the inlet and tidal channel can cause a decrease of setup in the system, whereas shoaling of the system can increase the setup. The uniqueness of the Xincun Inlet with respect to MWL change is that there is no evident setdown in the inlet, which can be attributed to the complex geometry and bathymetry associated with the inlet system.

Impact assessment of climate change and human activities on annual highest water level of Taihu Lake

The annual highest water level of Taihu Lake （Zm） is very significant for flood management in the Taihu Basin. This paper first describes the inter-annual and intra-annual traits of Zm from 1956 to 2000. Then, using the Mann-Kenall （MK） and Spearman （SP） nonparametric tests, the long-term change trends of area precipitation and pan evaporation in the Taihu Basin are determined. Meanwhile, using the Morlet wavelet transformation, the fluctuation patterns and change points of precipitation and pan evaporation are analyzed. Also, human activities in the Taihu Basin are described, including land use change and hydraulic project construction. Finally, the relationship between Zm, the water level of Taihu Lake 30 days prior to the day of Zm （Z0）, and the 30-day total precipitation and pan evaporation prior to the day of Zm （P and E0, respectively） is described based on multi-linear regression equations. The relative influence of climate change and human activities on the change of Zm is quantitatively ascertained. The results demonstrate that： （1） Zm was distinctly higher during the 1980-2000 period than during the 1956-1979 period, and the 30 days prior to the day of Zm are the key phase influencing Zm every year; （2） P increased significantly at a confidence level of 95% during the 1956-2000 period, while the reverse was true for E0; （3） The relationship between Zm, P and E0 distinctly changed after 1980; （4） Climate change and human activities together caused frequent occurrences of high Zm after 1980; （5） Climate change caused a substantially greater Zm difference between the 1956-1979 and 1980-2000 periods than human activities. Climate change, as represented by P and E0, was the dominant factor raising Zm, with a relative influence ratio of 83.6%, while human activities had a smaller influence ratio of 16.4%.

Effect of Water Level Fluctuations on Wintering Goose Abundance in Poyang Lake Wetlands of China

The Poyang Lake is a Ramsar site and is the important over-wintering site for migratory waterbirds along the East Asian-Australasian Fly way. Examining the effects of water level fluctuations on waterbird abundance and analyzing the influencing mechanism is critical to waterbird protection in the context of hydrological alteration. In this study, the effect of water level regime on wintering goose abundance was examined and the influencing mechanism was interpreted. Synchronous waterbirds survey data, hydro- logical data, Moderate Resolution Imaging Spectroradiometer-Normalized Difference Vegetation Index （MODIS-NDVI） data and habi- tat data derived from Landsat TNUETM data and HJ/CCD data were combined. The satellite-derived Green Wave Index （GWI） based on MODIS-NDVI dataset was applied to detect changes in goose food resources. It was found that habitat size and vegetation conditions are key factors determining goose abundance. Geese numbers were positively correlated with habitat area, while intermediate range of vegetation productivity might benefit the goose abundance. Water level affects goose abundance by changing available habitat areas and vegetation conditions. We suggested that matching hydrological regime and exposed meadows time to wintering geese dynamics was crucial in the Poyang Lake wetlands. Our study could provide sound scientific information for hydrological management in the context of waterbird conservation.

Centrifugal model test on a riverine landslide in the Three Gorges Reservoir induced by rainfall and water level fluctuation

Frequent soil landslide events are recorded in the Three Gorges Reservoir area,China,making it necessary to investigate the failure mode of such riverside landslides.Geotechnical centrifugal test is considered to be the most realistic laboratory model,which can reconstruct the required geo-stress.In this study,the Liangshuijing landslide in the Three Gorgers Reservoir area is selected for a scaled centrifugal model experiment,and a water pump system is employed to retain the rainfall condition.Using the techniques of digital photography and pore water pressure transducers,water level fluctuation is controlled,and multi-physical data are thus obtained,including the pore water pressure,earth pressure,surface displacement and deep displacement.The analysis results indicate that:Three stages were set in the test(waterflooding stage,rainfall stage and drainage stage).Seven transverse cracks with wide of 1–5 mm appeared during the model test,of which 3 cracks at the toe landslide were caused by reservoir water fluctuation,and the cracks at the middle and rear part were caused by rainfall.During rainfall process,the maximum displacement of landslide model reaches 3 cm.And the maximum deformation of the model exceeds 12 cm at the drainage stage.The failure process of the slope model can be divided into four stages:microcracks appearance and propagation stage,thrust-type failure stage,retrogressive failure stage,and holistic failure stage.When the thrust-type zone caused by rainfall was connected or even overlapped with the retrogressive failure zone caused by the drainage,the landslide would start,which displayed a typical composite failure pattern.The failure mode and deformation mechanism under the coupling actions of water level fluctuation and rainfall are revealed in the model test,which could appropriately guide for the analysis and evaluation of riverside landslides.

Relationship Between Changes of River-lake Networks and Water Levels in Typical Regions of Taihu Lake Basin,China

The typical regions of the Taihu Lake Basin,China,were selected to analyze the variation characteristics of river-lake networks under intensive human activities.The characteristics of the fractal dimension of river networks and lakes for different periods were investigated and the influences of river system evolution on water level changes were further explored through the comparison of their fractal characters.The results are as follows:1) River network development of the study area is becoming more monotonous and more simple;the number of lakes is reducing significantly,and the water surface ratio has dropped significantly since the 1980s.2) The box dimension of the river networks in all the cities of the study area decreased slowly from the 1960s to the 1980s,while the decrease was significant from the 1980s to the 2000s.The variations of lake correlation dimension are similar to those of the river network box dimensions.This is unfavorable for the storage capacity of the river networks and lakes.3) The Hurst exponents of water levels were all between 0.5 and 1.0 from the 1960s to the 1980s,while decreased in the 2000s,indicating the decline in persistence and increase in the complexity of water level series.The paper draws a conclusion that the relationship between the fractal dimension of river-lake networks and the Hurst exponents of the water level series can reveal the impacts of river system changes on flood disasters to some extent:the disappearance of river networks and lakes will increase the possibility of flood occurrence.

Model test of the influence of cyclic water level fluctuations on a landslide

Many landslides in reservoir areas continuously deform under cyclic water level fluctuations due to reservoir operations. In this paper,a landslide model, developed for a typical colluvial landslide in the Three Gorges Reservoir area, is used to study the effect of cyclic water level fluctuations on the landslide. Five cyclic water level fluctuations were implemented in the test, and the fluctuation rate in the last two fluctuations doubled over the first three fluctuations. The pore water pressure and lateral landslide profiles were obtained during the test. A measurement of the landslide soil loss was proposed to quantitatively evaluate the influence of water level fluctuations. The test results show that the first water level rising is most negative to the landslide among the five cycles. The fourth drawdown with a higher drawdown rate caused further large landslide deformation. An increase of the water level drawdown rate is much more unfavorable to the landslide than an increase of the water level rising rate. In addition, the landslide was found to have an adaptive ability to resist subsequent water level fluctuations after undergoing large deformation during a water level fluctuation. The landslide deformation and observations in the field were found to support the test results well.

Response of Regeneration Diversity of Carex Lasiocarpa Community to Different Water Levels in Sanjiang Plain,China

The species diversity at the regeneration stage, inflenced by different water levels, is important for community composition in the later growing season.Regeneration diversity of Carex lasiocarpa community under different water levels was studied at two stages, recruitment and adult, in the Sanjiang Plain, Heilongjiang Province, China.The results showed that, at the two growing stages, important value of C.lasiocarpa population and species richness of the community decreased with the increasing water level, while the Simpson and Shannon-Wiener diversity indexes and Pielou evenness index increased.Under different water levels, community diversities were higher at the recruitment stage, while population important values of C.lasiocarpa were higher at the adult stage.Indexes in vegetation evaluation must be chosen prudentially for successful restoration and effective management of wetlands, and especially for wetland restoration, the optimal time should be selected according to the restoration objectives and costs.

Study of Interval Type-2 Fuzzy Controller for the Twin-tank Water Level System

For dealing with large static error due to poor immunity of the traditional fuzzy control, a novel interval type-2 fuzzy control system is proposed. By extending the typical membership functions to interval type-2 membership functions, the proposed control system can efficiently reduce the uncertain disturbance from real environment without increasing the design complexity. The simulation results on the water tank level control system showed that the proposed method succeeded in better static and dynamic control with stronger robust performance than the traditional fuzzy control method.

Deformation and failure mechanism of Yanjiao rock slope influenced by rainfall and water level fluctuation of the Xiluodu hydropower station reservoir

With the construction of the Xiluodu hydropower station on the Jinsha River,the reservoir impoundment began in 2013 and the water level fluctuates annually between 540 m and 600 m above sea level.The Yanjiao rock slope which is located on the left bank of the Jinsha River 75 km upstream of the Xiluodu dam site,began to deform in 2014.The potential failure of the slope not only threatens Yanjiao town but also affects the safe operation of the Xiluodu reservoir.This paper is to find the factors influencing the Yanjiao slope deformation through field investigation,geotechnical reconnaissance,and monitoring.Results show that the Yanjiao slope can be divided into a bank collapse area(BCA)and a strong deformation area(SDA)based on the crack distribution characteristics of the slope.The rear area of the slope has been experiencing persistent deformation with a maximum cumulative displacement(GPS monitoring point G4)of 505 mm and 399 mm in the horizontal and vertical directions,respectively.The potential failure surface of the slope is formed 36 m below the surface based on the borehole inclinometer.The bank collapses of the Yanjiao slope are directly caused by the reservoir impoundment while the deformation area of the slope is affected by the combination of the rainfall and reservoir water level fluctuation.Based on mechanism of the Yanjiao slope,prestressed anchor combined with the surface drainage and slope unloading are recommended to prevent potential deformation.

Spatial variations of tidal water level and their impact on the exposure patterns of tidal land on the central Jiangsu coast

The exposed area of intertidal zone varies with tidal water level changes, If intercomparisons of satellite images are adopted as a method to determine geomorphological changes of the intertidal zone in response to accretion or erosion processes, then the effect of water level variations must be evaluated. In this study, two Landsat TM images overpassing the central Jiangsu coastal waters on 2 January and 7 March 2002, respectively, were treated by the changing detection analysis using Image Differencing and Post-classification Comparison. The simultaneous tide level data from four tide gauge stations along the coast were used for displaying the spatial variations of water levels and determining the elevations of waterlines. The results show that the spatial variations of water levels are highly significant in the central Jiangsu coastal waters. The huge differences of tidal land exposure patterns between the two imaging times are related mainly to the spatial variations of tidal water levels, which are controlled by the differences in tidal phases for different imaging times and the spatial variations of water level over the study area at each imaging time. Under complex tidal conditions, e.g., those of the central Jiangsu coastal waters, the tide-surge model should be used to eliminate effectively the effects of water level variations on remote sensing interpretation of geomorphological changes in the intertidal zone.

Effect of Sea Level Variation on Calculation of Design Water Level

The long-term variation and seasonal variation of sea level have a notable effect on the calculation of engineering water level. Stich an effect is first analyzed in this paper. The maximal amplitude of inter-annual anomaly of monthly mean sea level along the China coast is larger than 60 cm. Both the storm surge disaster and cold wave disaster are seasonal disasters in various regions, so the water level corresponding to the 1% of the cumulative frequency in the cumulative frequency curve of hourly water level data for different seasons in various sea areas is different from design water level., for example, the difference between them reaches maximum in June, July and August for northern sea area, and maximum in September, October and November for Southern China Sea, The hourly water level data of 19 gauge stations along the China coast are analyzed. Firstly, the annual mean sea level for every station is obtained; secondly, linear changing rates of annual mean sea level are obtained with the stochastic dynamic method; thirdly, the astronomical tide and storm surge tide are obtained by subtracting the linear fitting part from the original hourly data, finally, two distributions corresponding to the astronomical tide and wind tide are obtain ed according to whether the astronomical tide and storm tide are correlative or not. So the two check water levels are obtained with the joint probability method, The maximal difference between the two water levels of 100 years' recurrence is more than 30 cm. Both of the two check water levels have disadvantages in the use of observation data, so the mean value is suggested to be taken as the final check water level. A comparison between the two check-water levels indicates that the effect of sea level variation upon design water level and check water level is larger than 80 cm at some stations.

Effects of water level fluctuation on sedimentary characteristics and reservoir architecture of a lake, river dominated delta

The hydrodynamic conditions present in a river delta's formation are a highly important factor in the variation between its sedimentary regulation and characteristics. In the case of the lacustrine basin river-dominated delta, water level fluctuations and fluviation, are both important controlling factors of the sedimentary characteristics and reservoir architecture. To discuss the effects of water level fluctuation on sediment characteristics and reservoir architecture of this delta, the Fangniugou section in the east of the Songliao Basin was selected for study. Based on an outcrop investigation of the lacustrine basin river-dominated delta, combining with an analysis of the major and trace chemical elements in the sediments to determine the relative water depth, through architecture bounding surfaces and lithofacies division, sedimentary microfacies recognition and architectural element research, this work illustrated the effects of water level fluctuation on the reservoir architecture and established sedimentary models for the lacustrine basin river-dominated delta under various water level conditions. The results show that there are 8 lithofacies in the Fangniugou section. The fan delta front, which is the main object of this study, develops four sedimentary microfacies that include the underwater distributary channel, river mouth bar, sheet sand and interdistributary bay. The effects of water level fluctuation on different orders geographic architecture elements are respectively reflected in the vertical combination of the composite sand bodies, the plane combination of the single sand bodies, the particle size changes in the vertical of hyperplasia in the single sand body, the coset and lamina. In the case of the sand body development of the petroliferous basin, varying water level conditions and research locations resulted in significant variation in the distribution and combination of the sand bodies in the lacustrine basin.

Image-based Water Level Measurement Method under Stained Ruler

This paper propoes the water level measuring method based on the image, while the ruler used to indicate the water level is stained. The contamination of the ruler weakens or eliminates many features which are required for the image processing. However, the feature of the color difference between the ruler and the water surface are firmer on the environmental change compare to the other features. As the color differeaces are embossed, only the region of the ruler is limited to eliminate the noise, and the average image is produced by using several continuous frames. A histogram is then produced on the height axis of the produced intensity average image. Local peaks and local valleys are detected, and the section between the peak and valley which have the greatest change is looked for. The valley point at this very moment is used to detect the water level. The detected water level is then converted to the actual water level by using the mapping table. The proposed method is compared to the ultrasonic based method to evaluate its accuracy and efficiency on the various contaminated environments.

Soil anti-scourability enhanced by herbaceous species roots in a reservoir water level fluctuation zone

Revegetation is one of the successful approaches to soil consolidation and streambank protection in reservoir water level fluctuation zones(WLFZs).However,little research has been conducted to explore the impact of herbaceous species roots on soil anti-scourability during different growth stages and under different degrees of inundation in this zone.This study sampled two typical grasslands(Hemarthria compressa grassland and Xanthium sibiricum grassland)at two elevations(172 and 165 m a.s.l.)in the water level fluctuation zone(WLFZ)in the Three Gorges Reservoir(TGR)of China to quantify the changes in soil and root properties and their effects on soil anti-scourability.A simulated scouring experiment was conducted to test the soil anti-scourability in April and August of 2018.The results showed that the discrepancy in inundation duration and predominant herbaceous species was associated with a difference in root biomass between the two grasslands.The root weight density(RWD)values in the topsoil(0-10 cm)ranged from 7.31 to 13 mg cm^(-3) for the Hemarthria compressa grassland,while smaller values ranging from 0.48 to 8.61 mg cm^(-3) were observed for the Xanthium sibiricum grassland.In addition,the root biomass of the two herbs was significantly greater at 172 m a.s.l.than that at 165 m a.s.l.in the early recovery growth period(April).Both herbs can effectively improve the soil properties;the organic matter contents of the grasslands were 128.06%to 191.99%higher than that in the bare land(CK),while the increase in the water-stable aggregate ranged from 8.21%to 18.56%.Similarly,the topsoil antiscourability indices in both the herbaceous grasslands were larger than those in the CK.The correlation coefficients between the root length density(RLD),root surface area density(RSAD)and root volume density(RVD)of fine roots and the soil antiscourability index were 0.501,0.776 and 0.936,respectively.Moreover,the change in the soil antiscourability index was more sensitive to alternations in the RLD with diameters less than 0.5 mm.Overall,the present study showed that the perennial herbaceous(H.compressa)has great potential as a countermeasure to reduce or mitigate the impact of erosion in the WLFZ of the Three Gorges Reservoir.

The Application of a Grey Markov Model to Forecasting Annual Maximum Water Levels at Hydrological Stations

Compared with traditional real-time forecasting,this paper proposes a Grey Markov Model(GMM) to forecast the maximum water levels at hydrological stations in the estuary area.The GMM combines the Grey System and Markov theory into a higher precision model.The GMM takes advantage of the Grey System to predict the trend values and uses the Markov theory to forecast fluctuation values,and thus gives forecast results involving two aspects of information.The procedure for forecasting annul maximum water levels with the GMM contains five main steps:1) establish the GM(1,1) model based on the data series;2) estimate the trend values;3) establish a Markov Model based on relative error series;4) modify the relative errors caused in step 2,and then obtain the relative errors of the second order estimation;5) compare the results with measured data and estimate the accuracy.The historical water level records(from 1960 to 1992) at Yuqiao Hydrological Station in the estuary area of the Haihe River near Tianjin,China are utilized to calibrate and verify the proposed model according to the above steps.Every 25 years' data are regarded as a hydro-sequence.Eight groups of simulated results show reasonable agreement between the predicted values and the measured data.The GMM is also applied to the 10 other hydrological stations in the same estuary.The forecast results for all of the hydrological stations are good or acceptable.The feasibility and effectiveness of this new forecasting model have been proved in this paper.