Application of isotopic information for estimating parameters in Philip infiltration model

To Wng*,Hi-li XuWei-min Bo

aPowerChina Chengdu Engineering Corporation Limited,Chengdu 610072,China

bState Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering,Hohai University,Nanjing 210098,China

Application of isotopic information for estimating parameters in Philip infiltration model

Tao Wanga,*,Hai-li Xua,Wei-min Baob

aPowerChina Chengdu Engineering Corporation Limited,Chengdu 610072,China

bState Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering,Hohai University,Nanjing 210098,China

Minimizing parameter uncertainty is crucialin the application of hydrologic models.Isotopic information in various hydrologic components of the water cycle can expand our knowledge of the dynamics of water fl ow in the system,provide additional information for parameter estimation,and improve parameter identifi ability.This study combined the Philip infiltration model with an isotopic mixing model using an isotopic mass balance approach for estimating parameters in the Philip infiltration model.Two approaches to parameter estimation were compared:(a)using isotopic information to determine the soilwater transmission and then hydrologic information to estimate the soilsorptivity, and(b)using hydrologic information to determine the soil water transmission and the soil sorptivity.Results of parameter estimation were verifi ed through a rainfallinfiltration experimentin a laboratory under rainfallwith constantisotopic compositions and uniform initialsoilwater content conditions.Experimental results showed that approach(a),using isotopic and hydrologic information,estimated the soil water transmission in the Philip infiltration modelin a mannerthatmatched measured values well.The results of parameter estimation of approach(a)were better than those of approach(b).Itwas also found thatthe analytical precision of hydrogen and oxygen stable isotopes had a signifi canteffect on parameter estimation using isotopic information.

Isotopic information;Hydrologic information;Parameter estimation;Philip infiltration model;Rainfall infiltration experiment

1.Introduction

The successful application of a catchment model depends on the accuracy of hydrologic and hydraulic parameters used for the simulations and structures of the model.Model structures are based on the catchment characteristics and conceptualization of a realistic study system(Fenicia et al.,2008). Because some model parameters are difficult or impossible to measure in the natural world,model parameters are often estimated from secondary information sources(Fonseca et al.,2014).In fact,only some available data are used in model calculation because of the limitation of data(Wagener,2003). Abundance of data is the foundation of understanding model structures and parameter estimation.Data mining and other auxiliary data are two important methods,apart from the traditional measurement method,for collecting information in a given catchment.The data mining method primarily extracts useful information from collected data using mathematical techniques(e.g.,the clustering method),while auxiliary data means an increasing quantity of data independent of stream discharge and other hydrologic data.Hydrogen and oxygen isotopes are good auxiliary data tools and often used to trace water movement in the water cycle in order to provide orthogonal information on the catchment behavior(Fenicia et al.,2008).The combination of isotopic information and hydrologic information can provide plenty of availableinformation for model calculation and reduce uncertainty in parameterestimation.The ways isotopic information is used in a hydrologic modelfor parameter estimation should be further developed.Dunn et al.(2008)studied the mixing processes and mean residence time in a set of nested sub-catchments in northeast Scotland as determined from isotopic data,which could reduce parameter uncertainty in a rainfall-runoff model. Sprenger et al.(2015)used the stable isotope composition of the soil pore water depth profile as a single or additional optimization target,and estimated flow and transport parameters in the unsaturated zone.They found that using both the isotope profiles and the soil moisture time series resulted in good simulation results and strong parameter identifiability. When only data from isotope profi les in combination with textural information were available,the results were still satisfactory(Sprenger etal.,2015).Klaus et al.(2015)studied temporal dynamics of catchment transit times from stable isotope data.They extracted information on catchment mixing from the stable isotope time series instead of prior assumptions of mixing or the shape of transit time distribution,and demonstrated proof of the concepts of the approach with artificial data.This indicated that the Nash-Sutcliffe efficiencies in tracer and instantaneous transit times were higher than 0.9.

The complexities of model structures and number of parameters have a significant effect on parameter estimation using isotopic information.The two-parameter Philip infiltration model with a simple model structure has a specific physical foundation and is widely used to simulate rainfall infi ltration.However,because of limitations in observed hydrologic data,parameter estimation in a Philip infiltration model may be diffi cult.The objective of this study was to combine isotopic information with hydrologic information to estimate the parameters of a Philip infiltration modelthrough a rainfall-infi ltration experiment in a laboratory,and compare them with the results of parameter estimation using only hydrologic information.

2.Methods

2.1.Philip infiltration equation

The Philip infiltration equation(Philip,1957a,1957b)was derived from Richard's equation with water vertically infiltrating into the unsaturated and semi-infinite homogeneous soilunder constantinitialwater contentconditions(Prevedello et al.,2009).The infiltration rate with time,i(t)in cm·h-1,is defined as

where t is the infi ltration time(h),S is the soil sorptivity (cm·h-0.5),and A is the soilwater transmission(cm·h-1).The parameters S and A are related to soil diffusivity and moisture retention characteristics(Mishra et al.,2003).In this paper,S is taken into account as the average soil sorptivity,and A equals the saturated hydraulic conductivity Ks,which does not lead to serious errors in model calculation(Swartzendruber and Youngs,1974).The soil sorptivity S appears to be correlated with the soil water transmission A(Wang et al.,2006).

The cumulative infiltration with time,I(t)in cm,can be expressed as

In reality,Eqs.(1)and(2)are applicable to a limited time span(Prevedello et al.,2009).However,the classical Philip infiltration equation is still widely used for a constant head boundary neglecting the effect of a limited time span.

2.2.Model parameter estimation using hydrologic

information

Two parameters of the Philip infi ltration model need to be estimated:the soil sorptivity S and soil water transmission A. There are also two major methods for parameter estimation using hydrologic information,namely,the linear graphic method and the least squares method(Bristow and Savage, 1987).In the linear graphic method,data of cumulative infiltration with time are plotted on a fi gure with t0.5as the abscissa and I(t)t-0.5as the ordinate.Then,the parameters S and A can be respectively obtained from the intercept and slope of the fi gure.The least squares method is used to optimize S and A through fi tting observed data and Eq.(1)or(2).Notwithstanding that the linear graphic method can easily obtain the model parameters,it is highly arbitrary due to t0.5existing on both axes in order to introduce self-correlation and limitation of data at time t=0(Bonell and Williams,1986).The least squares method shows objective characteristics and is widely used to estimate parameters of a model.In this study,S and A were estimated from observed data of cumulative infi ltration calculated with Eq.(2)using the least square method,and the calculated results were regarded as parameters obtained from hydrologic information.Effects of limited time on model calculation were neglected or considered errors in the parameter estimation process using hydrologic information due to the deficiency of available data.

2.3.Modelparameter estimation using isotopic information

Model parameter estimation using isotopic information is implemented with the isotopic mixing modelbased on isotope mass balance.The isotopic mixing model combines isotopic information with hydrologic information,and can be expressed as

where Cj-1and Cjare the isotopic compositions of mixing water in the mixing tank at time tj-1and tj,respectively,and j indicates the time sequence;Cp,j-1is the isotopic composition of input water(e.g.,rainfall)at time tj-1;ΔV is the volume of water infiltrating into soil from time tj-1to tj;and V0represents the initial soil water volume,which is equal to thevolume of the mixing tank.The application of the isotopic mixing modelis based on certain conditions in which isotopic variations of soil water are primarily caused by isotopic mixing of rainfall and soil water in the process of infi ltration. Itis noted thatthis study only examined the rainfallinfi ltration under rainfallwith constantisotopic compositions and uniform initial soilwater contentconditions.The isotopes of soilwater and rainfall reached a balance between 0.5 h and 1 h after the beginning of the mixing process(Wang et al.,2010).

When water fl ows out of the lower boundary of soil layers, the infi ltration rate gradually becomes a constant,equal to the saturated hydraulic conductivity Ks(Mishra et al.,2003). Thus,A can be indirectly obtained through estimation of the saturated hydraulic conductivity from observed data in the lower boundary of the soil column.The total amount of cumulative infiltration is divided into N equal parts and the volume of each partisΔV.When the infi ltration rate reaches a stable value,eachΔV volume of water infi ltrating into soilwill take the same time intervalΔt.Then,the relationship between ΔV and A isΔV=AΔtB,where B is the area of the crosssection of soil layers.As forΔV volume of water infi ltrating into soil,the water movements are described by the Philip infiltration model while isotopic variations are calculated using the isotopic mixing model.In an isotopic mixing model, theΔV volume of infiltrating water with isotopic composition Cp,j-1mixes with V0volume of water in the mixing tank with the isotopic composition Cj-1.As mixing is completed,the isotopic composition of mixing waters becomes Cj.There is ΔV volume of mixing water immediately fl owing out of the mixing tank,resulting in the volume of the mixing tank maintaining the value of V0.An assumption is introduced that lag timeτof the mixing water fl owing out of the lower boundary equals the time of water movement in the soil column.The relationship between the isotopic composition CjandΔV of outfl ow is established using the isotopic mixing model.DifferentΔV values correspond to different results of Cjwith time through trial calculations.Therefore,A can be estimated with the isotopic results of outflow.The time interval of isotopic results calculated using the isotopic mixing modelshould be treated the same as the time interval of water sampling during the experiment.Subsequently,the root mean squared error(RMSE)between calculations and observations of isotopic compositions of outflow,which is the criterion for estimating parameter A,is computed.

In fact,isotopic information can only be used to establish the parameter A.Another parameter,the soil sorptivity S,is obtained by substituting the established parameter A and observed hydrologic data into Eq.(2).The water movements and isotopic variations above the lower boundary of soil layers are nottaken into account due to lack of relevant information.

3.Rainfall infiltration experiment

A rainfall infi ltration experiment was performed from 8:00 am on May 20 to 8:00 am on May 24,2008.The experimental site was set up in a rainfall simulation laboratory.In order to obtain a uniform initial soil water content profi le,the air-dried soils,from the soil surface of a hillside, were sealed in a container for three days.The initial water content measured by the oven-drying method was 53 g/kg. The initialsoilwater was extracted by the vacuum distillation method,with the values ofδD(deuterium)andδ18O(oxygen-18)being-27‰ and-3.5‰,respectively.The maximum extraction errors ofδD andδ18O using the vacuum distillation method in this experiment were-12‰and-0.7‰,respectively(Wang et al.,2009).Soils were packed into a transparent acrylic column 100 cm long and 15 cm in diameter with a bulk density of 1.22 g·cm-3,a total thickness of 84 cm,and weight of 18.116 kg.

A rainfall simulator was placed above the soil surface, which consisted of a sprinkler made of hypodermic needles similar to those described in Liu etal.(2008).A Marriott tube was used to supply water and a graduated ruler was pasted on itfor measuring infiltration water with time(Fig.1).The water used for simulating rainfall was sealed and stored in a large container 65 cm long and 50 cm in diameter to ensure constant isotopic compositions during the experiment.The values ofδD andδ18O of water used for simulating rainfallwere-50‰and -7.2‰,respectively.A total volume of 16.313 L of water infiltrated into the soil during the experiment.The time from water infi ltration to ponded water appearance was less than 20 min.The wetting front was measured with time,which could be indirectly used to calculate the cumulative infi ltration.The interfaces among the Marriott tube,rainfall simulator,and column were well sealed to reduce the effect of the evaporation fractionation.Water fl owed out of a column after 14.8 h of rainfall infiltration,and the rainfall process lasted 59.35 h.The air temperature ranged from 21.3°C to 25.9°C, with a mean value of 23.1°C,and the relative humidity ranged from 48%to 79%,with a mean value of 58%.

Fig.1.Photo of rainfall infiltration experiment.

Water samples were collected at the bottom outlet of the column using 30-mL plastic bottles at predeterminedintervals.The time of collection for each water sample was recorded in order to calculate the soil water transmission. Hydrogen and oxygen isotopic compositions of water samples were measured using a MAT-253 mass spectrometer in the isotopic laboratory of the Ministry of Land and Resources in Beijing,China.The measured results were expressed as δvalues relative to the international standard Vienna Standard Mean Ocean Water(VSMOW).Analytical precisions were±2‰ and±0.2‰ for hydrogen and oxygen isotope analyses,respectively.

4.Results and discussion

4.1.Isotopic mixing of rainfall and soil water

The application of an isotopic mixing modelis based on the condition in which isotopic variations of soil water in infiltration are mainly caused by the isotopic mixing ofrainfalland soil water.In the cases of rainfall with constant isotopic compositions and a slight effect of evaporation fractionation, the isotopic values of mixed rainfall and soil water should lie between the isotopic values of rainfall and soil water as end members(Shanley et al.,1998).Fig.2 shows the isotopic relationships between rainfall,the initial soil water,and outfl ow of the column.In this fi gure,the number represents the order of isotopic variations of outflow with time.δ18O values of outflow ranged from-7.7‰ to-3.7‰ with an average value of-6.6‰,andδD values ranged from-55‰to-28‰with an average value of-47‰.Fig.2 shows that isotopic values of outflow varying with time were located on or beside the mixing line that connected the isotopic values of rainfall and the initial soil water.The results indicated that isotopic variations of outflow water were primarily caused by the mixing of rainfall and soil water.Some data points away from the mixing line,such as point23 at the end of the experiment, might be mainly attributed to isotopic analysis errors of water samples.

4.2.Results of parameter estimation

Fig.2.Relationship betweenδD andδ18O values of outflow.

The parameters were determined using hydrologic information.Observed data of the cumulative infiltration I(t)were fitted using Eq.(2)and the least squares method.The parameter A was derived as 1.35 cm·h-1and the parameter S was 4.00 cm·h-0.5.The value ofΔt was set as 1 h.The value ofΔV was 239 mL,corresponding to the parameter A determined using hydrologic information,while 210 mL ofΔV with a stable infiltration rate of 1.19 cm·h-1were calculated from observed data.Because the time ponded water appeared above the soil surface was less than 20 min in the experiment,the effect of time on infi ltration for modelcalculation was ignored in this study.The parameter A determined using hydrologic information was close to the observed stable infi ltration rate.

Then,the model parameters were estimated by applying isotopic information.Relationships between A and isotopic compositions of outflow were indirectly determined using the isotopic mixing model.Fig.3 shows the relationship between parameter A and the root mean squared error(RMSE)of simulations and observations of isotopic compositions of outflow.The values of A obtained by the minimum values of RSME of hydrogen and oxygen isotopes were different.The values of A estimated with hydrogen isotopic information were smaller than those observed,while oxygen isotopic information showed a contrary result,with estimated values of A larger than those observed.As hydrogen and oxygen isotopes were simultaneously transported in soil profi les experiencing the slight effect of evaporation fractionation,the reason fordifferent values of A estimated using hydrogen and oxygen isotopic information might just be isotopic analysis errors of water samples and extraction errors in the initial soil water using the vacuum distillation method.The arithmetic average value of A determined using hydrogen and oxygen isotopic information was regarded as the fi nal value of the parameter, i.e.,1.15 cm·h-1.The parameter S,with a value of 4.64 cm·h-0.5,was obtained by substituting the estimated A value and observed data into Eq.(2).

Fig.3.Relationship between parameter A and RMSE for hydrogen and oxygen isotopes.

Table 1 shows the results of parameters estimated using hydrologic and isotopic information.As shown in Table 1,the value of A using only hydrogen or oxygen isotopic information is larger or smaller than that of the observed value,while that using hydrogen and oxygen isotopic information(the arithmetic average value)approaches the observed value.Parameters estimated using oxygen isotopic information are almost the same as those estimated using hydrologic information. Therefore,isotopic information could be used to estimate parameters of the Philip infi ltration model well with insufficient available hydrologic data.Furthermore,the combination of isotopic and hydrological information could increase the quantity of available information for modelcalculation,reduce the uncertainty of parameters,and provide a usefulmethod for parameter estimation.

4.3.Simulation results of isotopic mixing model and Philip infiltration model

ΔV with a value of 203 mL,corresponding to the parameters A and S determined using isotopic information with values of 1.15 cm·h-1and 4.64 cm·h-0.5,respectively,was substituted into the isotopic mixing model to calculate the isotopic variations of outfl ow in the soil column with time. Fig.4 shows isotopic variations of outfl ow using the isotopic mixing modelwith the comparison of observed values.It can be seen that the isotopic mixing model could describe isotopic variations of outfl ow well and combine isotopic and hydrologic information to estimate model parameters. Eq.(3)shows that parameter estimation using isotopic information is affected not only by the isotopic analysis errors of rainfall,but also by isotopic extraction errors with use of the vacuum distillation method.The initial soil water was fi rst extracted from soil using the vacuum distillation method,and then measured using a MAT-253 mass spectrometer with rainfall and mixing water.Errors inevitably existed in the extraction and measurement process of water samples,resulting in an increase in uncertainty of parameter estimation.

Table1 Parameters estimated using hydrologic and isotopic information.

Fig.4.Observed and simulated isotopic values of outflow.

The cumulative infi ltration varying with time before water fl owed outof the lower boundary of soilwas calculated from Eq.(2)using parameters estimated by isotopic and hydrologic information with comparison of observations(as shown in Fig.5).The infi ltration rate gradually became constant with water fl owing out of the soil column.The cumulative infiltration calculated using estimated parameters with isotopic and hydrologic information was 34.87 cm and 35.43 cm,respectively,while the measured value was 34.63 cm at the end of the experiment.This indicates that the value of total cumulative infi ltration using parametersestimated with isotopic and hydrologic information was close to the observed value.

Fig.5.Relationship between cumulative infiltration and time.

5.Conclusions

Sufficient available model information is critical to estimating model parameters.Hydrogen and oxygen isotopes are effective auxiliary data tools for providing large amounts of model information due to their tracer characteristics.In this study,an isotopic mixing model,which combined isotopic and hydrologic information,was used to estimate parameters in a Philip infi ltration model.A ponded water rainfall-infiltration experiment was performed under rainfall with constant isotopic compositions and uniform initial soil water content conditions.The experimental results show that the parameter A estimated using isotopic information was close to the observed value,and errors in isotopic analysis of water samples affected the parameter estimation.Therefore,isotopic information can be used to estimate parameters of a model in the absence of hydrologic information.Application of both isotopic and hydrologic information provides a potential method for determining parameters for modelapplications and reduces the uncertainty in parameter estimation.This study only focused on two parameters of the Philip infiltration model using isotopic information through rainfall-infiltration experiments.Further research might be required for the research method to be used in more complex hydrological models.

Bonell,M.,Williams,J.,1986.Two parameters of the Philip infiltration equation:Their properties and spatial and temporalheterogeneity in a red earth of tropical semi-arid Queensland.J.Hydrol.87(1-2),9-31.http:// dx.doi.org/10.1016/0022-1694(86)90112-5.

Bristow,K.L.,Savage,M.J.,1987.Estimation of parameters for the Philip two-term infiltration equation applied to field soil experiments.Aust.J. Soil Res.25(4),369-375.http://dx.doi.org/10.1071/SR9870369.

Dunn,S.M.,Bacon,J.R.,Soulsby,C.,Tetzlaff,D.,Stutter,M.I.,Waldron,S., Malcolm,I.A.,2008.Interpretation of homogeneity inδ18O signatures of stream water in a nested sub-catchment system in north-east Scotland. Hydrol.Process.22(24),4767-4782.http://dx.doi.org/10.1002/hyp.7088.

Fenicia,F.,McDonnell,J.J.,Savenije,H.H.G.,2008.Learning from model improvement:On the contribution of complementary data to process understanding.Water Resour.Res.44(6),W06419.http://dx.doi.org/ 10.1029/2007WR006386.

Fonseca,A.,Ames,D.P.,Ping,Y.,Botelho,C.,Rui,B.,Vilar,V.,2014. Watershed model parameter estimation and uncertainty in data-limited environments.Environ.Model.Softw.51,84-93.http://dx.doi.org/ 10.1016/j.envsoft.2013.09.023.

Klaus,J.,Chun,K.P.,Mcguire,K.J.,Mcdonnell,J.J.,2015.Temporal dynamics of catchment transittimes from stable isotope data.Water Resour. Res.51(6),4208-4223.http://dx.doi.org/10.1002/2014WR016247.

Liu,J.T.,Zhang,J.B.,Feng,J.,2008.Green-Amptmodelforlayered soils with nonuniform initialwater contentunder unsteady infiltration.Soil Sci.Soc. Am.J.72(4),1041-1047.http://dx.doi.org/10.2136/sssaj2007.0119.

Mishra,S.K.,Tyagi,J.V.,Singh,V.P.,2003.Comparison of infiltration models. Hydrol.Process.17(13),2629-2652.http://dx.doi.org/10.1002/hyp.1257.

Philip,J.R.,1957a.The theory of infiltration:1.The infiltration equation and its solution.SoilSci.83(5),345-358.http://dx.doi.org/10.1097/00010694-200606001-00009.

Philip,J.R.,1957b.The theory of infiltration:4.Sorptivity and algebraic infiltration equations.Soil Sci.84(3),257-264.http://dx.doi.org/10.1097/ 00010694-195709000-00010.

Prevedello,C.L.,Loyola,J.M.T.,Reichardt,K.,Nielsen,D.R.,2009.New analytic solution related to the Richards,Philip,and Green-Amptequations for infiltration.Vadose Zone J.8(1),127-135.http://dx.doi.org/10.2136/ vzj2008.0091.

Shanley,J.B.,Pendall,E.,Kendall,C.,Stevens,L.R.,Michel,R.L., Philips,P.J.,Forester,R.M.,Naftz,D.L.,Liu,B.L.,Stem,L.,etal.,1998. Isotopes as indicators of environmental change.In:Kendall,C., McDonnell,J.J.,eds.,Isotope Tracers in Catchment Hydrology.Elsevier Science B.V.,Amsterdam,pp.761-816.http://dx.doi.org/10.1016/B978-0-444-81546-0.50029-X.

Sprenger,M.,Volkmann,T.H.M.,Blume,T.,Weiler,M.,2015.Estimating flow and transport parameters in the unsaturated zone with pore water stable isotopes.Hydrol.Earth Syst.Sci.19(6),2617-2635.http:// dx.doi.org/10.5194/hess-19-2617-2015.

Swartzendruber,D.,Youngs,E.G.,1974.A comparison of physically-based infiltration equations.Soil Sci.117(3),165-167.http://dx.doi.org/ 10.1097/00010694-197403000-00005.

Wagener,T.,2003.Evaluation of catchment models.Hydrol.Process.17(16), 3375-3378.http://dx.doi.org/10.1002/hyp.5158.

Wang,Q.J.,Zhang,J.H.,Fan,J.,2006.An analytical method for relationship between hydraulic diffusivity and soil sorptivity.Pedosphere 16(4), 444-450.http://dx.doi.org/10.1016/S1002-0160(06)60074-X.

Wang,T.,Bao,W.M.,Chen,X.,Shi,Z.,Hu,H.Y.,Qu,S.M.,2009.Soilwater extraction using vacuum distillation technology.J.Hohai Univ.Nat.Sci. 37(6),660-664.http://dx.doi.org/10.3876/j.issn.1000-1980.2009.06.010 (in Chinese).

Wang,T.,Bao,W.M.,Li,L.,2010.Isotopic variations of soiland inputwater mixing.Hydrogeol.Eng.Geol.37(2),104-107.http://dx.doi.org/10.3969/ j.issn.1000-3665.2010.02.023(in Chinese).

Received 30 November 2015;accepted 15 September 2016

Available online 6 January 2017

This work was supported by the National Natural Science Foundation of China(Grant No.51279057).

*Corresponding author.

E-mail address:wangtaogo@163.com(Tao Wang).

Peer review under responsibility of Hohai University.

©2016 Hohai University.Production and hosting by Elsevier B.V.This is an open access article under the CC BY-NC-ND license(http:// creativecommons.org/licenses/by-nc-nd/4.0/).