Accepted Manuscript Application of isotopic information for estimating parameters in Philip infiltration model Tao Wang, Hai-li Xu, Wei-min Bao PII:
S1674-2370(17)30006-6
DOI:
10.1016/j.wse.2017.01.006
Reference:
WSE 81
To appear in:
Water Science and Engineering
Received Date: 30 November 2015 Accepted Date: 15 September 2016
Please cite this article as: Wang, T., Xu, H.-l., Bao, W.-m., Application of isotopic information for estimating parameters in Philip infiltration model, Water Science and Engineering (2017), doi: 10.1016/ j.wse.2017.01.006. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Application of isotopic information for estimating parameters in Philip infiltration model Tao Wang a, *, Hai-li Xu a, Wei-min Bao b a
PowerChina Chengdu Engineering Corporation Limited, Chengdu 610072, China State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, China Received 30 November 2015; accepted 15 September 2016 Available online ***
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b
Abstract
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Minimizing parameter uncertainty is crucial for the application of hydrologic models. Isotopic information in various hydrologic components of the water cycle can expand our knowledge of the dynamics of water flow in the system, provide additional information for parameter estimation, and improve parameter identifiability. 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 soil water transmission and then hydrologic information to estimate the soil sorptivity, and (b) using hydrologic information to determine the soil water transmission and the soil sorptivity. Results of parameter estimation were verified through a rainfall infiltration experiment in a laboratory under rainfall with constant isotopic compositions and uniform initial soil water content conditions. Experimental results show that approach (a), using isotopic and hydrologic information, estimated the soil water transmission in the Philip infiltration model in a manner that matched measured values well. The results of parameter estimation of approach (a) were better than those of approach (b). It was also found that the analytical precision of hydrogen and oxygen stable isotopes had a large effect on parameter estimation using isotopic information. Keywords: Isotopic information; Hydrologic information; Parameter estimation; Philip infiltration model
1. Introduction
The successful application of a catchment model depends on the accuracy of hydrologic and hydraulic parameters
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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
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(Wagener, 2003). Abundance of data is the foundation of understanding model structures and parameter estimation. Data mining and other auxiliary data are two important major methods for collecting information in a given catchment, apart from the traditional measurement method. The data mining method primarily extracts useful information from
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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 available information for model calculation and reduce uncertainty in parameter estimation. The ways isotopic information is used in a hydrologic model for 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 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 good parameter ————————————— This work was supported by the National Natural Science Foundation of China (Grant No. 51279057). * Corresponding author. E-mail address:
[email protected] (Tao Wang).
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identifiability. If data from only isotope profiles in combination with textural information were available, the results were still satisfactory (Sprenger et al., 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
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physical foundation and is widely used to simulate rainfall infiltration. However, because of limitations in observed hydrologic data, parameter estimation in a Philip infiltration model may become difficult. The objective of this study was to combine isotopic information with hydrologic information to estimate the parameters of a Philip infiltration model through the rainfall-infiltration experiment in a laboratory, and compare them with the results of parameter estimation only using hydrologic information.
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2. Methods 2.1. Philip infiltration equation
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The Philip infiltration equation (Philip, 1957a, b) was derived from Richard’s equation with water vertically infiltrating into the unsaturated and semi-infinite homogeneous soil under constant initial water content conditions (Prevedello et al., 2009). The infiltration rate with time, i ( t ) in cm·h–1 , is defined as
i ( t ) = 0.5St −0.5 + A
(1)
where t is the infiltration time (h), S is the soil sorptivity (cm·h–0.5), and A is the soil water transmission (cm·h–1). The parameters S and A are related to soil diffusivity and moisture retention characteristic (Mishra et al., 2003). In this paper,
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parameter S is taken into account as the average soil sorptivity, and parameter 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 (2)
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I ( t ) = St 0.5 + At
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 limited time span.
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2.2. Model parameter estimation using hydrologic information Two parameters of the Philip infiltration 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 figure with t 0.5 as the abscissa and I ( t ) t −0.5 as the vertical coordinate. Then, the parameters S and A can be respectively obtained from the intercept and slope of the figure. The least squares method is applied to optimize S and A through fitting 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 t 0.5 existing 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 infiltration 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 2
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calculation were neglected or considered errors in the parameter estimation process using hydrologic information due to the deficiency of available data.
2.3. Model parameter estimation using isotopic information Model parameter estimation using isotopic information is implemented with the isotopic mixing model based on isotope mass balance. The isotopic mixing model combines isotopic information with hydrologic information, and can be expressed as
∆VCp + V0C j −1
(3)
∆V + V0
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Cj =
where C j −1 and C j are the isotopic compositions of mixing water in the mixing tank at time t j −1 and t j , respectively, and j indicates the time sequence; Cp is the isotopic compositions of input water (e.g., rainfall) at time t j −1 ; ∆V is the volume of water infiltrating into soil from time t j −1 to t j ; and V0 represents the initial soil water volume, which is
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equal to the volume of the mixing tank. The application of the isotopic mixing model is based on certain conditions in which isotopic variations of soil water are primarily caused by isotopic mixing between rainfall and soil water in the process of infiltration. It is noted that this study only examined the rainfall infiltration under rainfall with constant
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isotopic compositions and uniform initial soil water content conditions. The isotopes of soil water and rainfall reached a balance between 0. 5 h and 1 h after the beginning of mixing process (Wang et al., 2010). When water flows out of the lower boundary of soil layers, the infiltration 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 soil column. The total amount of cumulative infiltration is divided into N equal parts and the volume of each part is ∆V . When the infiltration rate reaches a stable value, each ∆V volume of water infiltrating into soil will take the same time interval ∆t . Then, the relationship
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between ∆V and A is ∆V = A∆tB , where B is the area of the cross-section of soil layers. As for ∆V volume of water infiltrating 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, the ∆V volume of infiltrating water with isotopic compositions Cp mixes with V0 volume of water in the mixing tank with the isotopic composition C j −1 . As mixing is completed, the isotopic composition of mixing waters becomes C j . There is ∆V volume of mixing water
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immediately flowing out of the mixing tank, resulting in the volume of the mixing tank maintaining the value of V0 . An assumption is introduced that lag time τ of the mixing water flowing out of the lower boundary equals the time of water movement in the soil column. The relationship between isotopic composition C j and ∆V of outflow is
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established using the isotopic mixing model. Different ∆V values correspond to different results of C j with 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 model should 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 be only 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 not taken into account because of lack of relevant information.
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ACCEPTED MANUSCRIPT 3. Rainfall infiltration experiment A rainfall infiltration 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 an initial uniform soil moisture content profile, 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 initial soil water was extracted by the vacuum distillation method, with the values of δD (deuterium) and δ18O (oxygen-18) being –27 ‰ and –3.5‰, respectively. The maximum extraction errors of δD and δ18O using the vacuum distillation method in this experiment were –12 ‰ and –0.7 ‰,
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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 et al. (2008). A Marriott tube was used to supply water and had a graduated ruler pasted on it for 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
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values of δD and δ18O of water used for simulating rainfall were –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 infiltration to ponded water
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appearance was less than 20 min. The wetting front was measured with time, which could be indirectly used to calculate the cumulative infiltration. The interfaces among the Marriott tube, rainfall simulator, and column were well sealed to reduce the effect of the evaporation fractionation. Water flowed 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
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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 predetermined intervals. 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 δ values relative to the international standard VSMOW (Vienna Standard Mean Ocean Water). Analytical precisions were ±2‰ and ±0.2‰ for hydrogen and oxygen isotope analyses, respectively. 4
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4. Results and discussion
4.1. Isotopic mixing between rainfall and soil water The application of an isotopic mixing model is based on a certain condition that isotopic variations of soil water in infiltration are mainly caused by the isotopic mixing between rainfall and soil water. In the cases of rainfall with constant isotopic compositions and slight effect of evaporation fractionation, the isotopic values of mixing water of 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 outflow of the column. In this figure, the
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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‰, and δD 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 between rainfall and soil water. Some data points away from the mixing line, such as
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point 23 at the end of experiment, might be mainly attributed to isotopic analysis errors of water samples.
Fig. 2. Relationship between δD and δ18O values of outflow.
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4.2. Results of parameter estimation
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–1 and the parameter S was 4.00 cm·h–0.5. The value of ∆t was set as 1 h. The value of ∆V was 239 mL, corresponding to the parameter A
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determined using hydrologic information, while 210 mL of ∆V with a stable infiltration rate of 1.19 cm·h–1 were calculated from observed data. Because the time that ponded water appeared above the soil surface was less than 20 min in the experiment, the effect of time on infiltration for model calculation was ignored in this study. The parameter A
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determined using hydrologic information was close to the observed stable infiltration 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 was 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 profiles with a slight effect of evaporation fractionation, the reason for different 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 final value of the parameter,
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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
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observed data into Eq. (2).
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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 parameter A using only hydrogen or oxygen was larger or smaller than that of the observation value, while the observed value was approached with use of hydrogen and oxygen isotopic information (the arithmetic average value). Parameters estimated using oxygen isotopic information were almost the same as those estimated using hydrologic
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information. Both approached the measured data. Therefore, isotopic information could be used to estimate parameters of the Philip infiltration model well with insufficient available hydrologic data. Furthermore, the combination of isotopic and hydrological information could increase the quantity of available information for model calculation, reduce the Table 1
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uncertainty of parameters, and provide a useful method for parameter estimation. Parameters estimated using hydrologic and isotopic information Parameter estimation method
A (cm·h–1)
S (cm·h–0.5)
Using hydrogen isotopic information
0.93
5.32
Using oxygen isotopic information
1.36
3.97
Using hydrogen and oxygen isotopic information
1.15
4.64
Using hydrologic information
1.35
4.00
Observation
1.19
–
6
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ACCEPTED MANUSCRIPT 4.3. Simulation results of isotopic mixing model and Philip infiltration model ∆V with a value of 203 mL, corresponding to the parameters A and S determined using isotopic information with values of 1.15 cm·h–1 and 4.64 cm·h–0.5, respectively, was substituted into the isotopic mixing model to calculate the isotopic variations of outflow in the soil column with time. Fig. 4 shows isotopic variations of outflow using the isotopic mixing model with the comparison of observed values. It can be seen that the isotopic mixing model could describe isotopic variations of outflow 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 first extracted
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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
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increase in uncertainty of parameter estimation.
Fig. 4. Observed and simulated isotopic values of outflow.
The cumulative infiltration varying with time before water flowed out of the lower boundary of soil was calculated from Eq. (2) using parameters estimated by isotopic and hydrologic information with comparison of observations (as shown in Fig. 5). The infiltration rate gradually became constant with water flowing 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 infiltration using parameters estimated with isotopic and hydrologic information was close to the observed value.
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Fig. 5. Relationship between cumulative infiltration and time.
5. Conclusions
Sufficient available model information is critical for 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
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this study, an isotopic mixing model, which combined isotopic and hydrologic information, was used to estimate parameters in a Philip infiltration 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
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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 model applications 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
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