Discussion of Present Isotopic Hydrograph Separation (IHS) Method
Discussion of Present Isotopic Hydrograph Separation (IHS) Method
Weimin Bao
State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering
Hohai University
Nanjing,P.R. China
wmbao@https://www.360docs.net/doc/c015044243.html,
Tao Wang, Haiying Hu, Simin Qu
State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering
Hohai University
Nanjing,P.R. China
wangtaogo@https://www.360docs.net/doc/c015044243.html,
Abstract—Runoff sources and dominant flow paths are still poorly understood in most catchments. Consequently, isotope hydrograph separation methods are widely used to improve our understanding about runoff generation. Isotope hydrograph separation (IHS) is based on the mass balance equation of water flux and isotope tracer concentration. The isotope analysis machine (for example, mass spectrum) available can only get the δ-value of the isotope tracer. An analysis is presented for quantifying the error in two- and three- component tracer-based hydrograph separation. This analysis compared the error between ratio balance method and mass balance method in computing mixed fractions to determine which method is more accurate. In the ratio balance method, the ratio of isotope tracer in used in concentration mass balance equation; comparatively, in the mass balance method, the concentration of isotope tracer is used in the same equation. Many studies about two- and three-component tracer-based hydrograph separation are used to illustrate the application of the analysis. Because of the abundance of rare isotopes are so negligible, the difference between two methods is small in the isotopic hydrograph separation studies. In scientific research, it is not appropriate to substitute ratio for concentration in the isotope tracer concentration mass balance equation. Therefore, we recommend the use of mass balance method in the application of isotope hydrograph separation research to improve the degree of accuracy.
Keywords- isotope hydrograph separation (IHS); two component; three component; ratio balance method; mass balance method
I.I NTRODUCTION
The process whereby rainfall becomes runoff is still poorly understood[1,2]. Hydrographs are an enticing focus for hydrologic research: they integrate the variety of upstream routing and watershed flow pathways especially the soil water movement. So it is difficult to separate the water sources of hydrograph. Early hydrograph-oriented analysis focused on arbitrary graphical separations of streamflow components such as the inflection point on the recession limb of hydrograph [3,4]. Since the advent of the graphical hydrograph separation method, work in the past two decades has focused on the use of tracers as a more objective means to separate the storm hydrograph. Stable IHS[5] has developed as a common technique in small watershed hydrology [6]. The tracer-based separation approaches have the advantage of providing more processed-based information about water sources of runoff.
For stable isotope D and 18O of the water molecule in the water cycle, only mixing and physical processes such as evaporation, condensation and infiltration can fractionate or change the relative proportions of different isotopes of the same element in various compounds. The most important fractionation governing the occurrence of D and 18O is caused by the mixing of different water sources. As a result of fractionation processes, water often develops unique isotopic compositions (ratio of heavy to light isotopes) that may be indicative of their source or the processed that formed them. The isotope hydrograph separation is based on the mass balance equations of different isotopes to separate the storm hydrograph into different water sources. The isotope hydrograph separation is a physically-based hydrograph separation method.
Because of the minor difference of variations of stable isotope composition in the nature, stable isotope compositions are normally reported as δ values. δ values are reported in units of parts per thousand (denoted as ‰ or permil) relative to a standard of known composition. δvalues are calculated by:
δ=(R/R SMOW -1)*1000 ‰ (1) where R denotes the ratio of the heavy to light isotopes (e.g., R(D)=[D]/[H], R(18O)=[18O]/[16O]), and R and R SMOW are the ratios in the sample and standard, respectively. For a sample, a different standard will result in different δ values.δD and δ18O values are normally reported relative to the SMOW standard[7].
In the isotope hydrograph separation, there are two mass balance equations describing the fluxes of water and the isotope tracer concentration in the stream. By substituting δ value or the ratio of isotope tracer for the concentration of it in the mass balance equation and then combining with the water flux mass balance equation to calculate the proportion of event water and pre-event water. Strictly speaking, if we substitute δ value or ratio of isotope tracer for the concentration of it in the tracer isotope concentration mass balance equation, the concentration mass balance equation cannot exist any more. As a result of this simplification, the specific objective of this study is to quantify the error in the results of isotope hydrograph separation. The examples of two- and three-component hydrograph separation are given below.
2009 International Conference on Environmental Science and Information Application Technology
II.Q UESTION ON PRESENT ISOTOPE HYDROGRAPH
SEPARATION
A.Two-Component Hydrograph Separation
The two-component hydrograph separation method divide runoff hydrograph into event water (surface runoff derived from precipitation, also referred to as new water) and pre-event water (groundwater runoff, water stored in the catchment prior to a storm event, also referred to as old water). The two-component hydrograph separation is based on the simultaneous solution of the steady-state mass balance equations describing the fluxes of water and the isotope tracer in the stream. Examples which use deuterium as tracer in the two-component hydrograph separation are given below. These equations are of the form:
Q t=Q s+Q g (2)
Q tδt=Q sδs+Q gδg (3) where Q is the discharge (m3/s), δrepresents the δ value of isotope tracer, and subscript t, s, g refer to the total discharge, pre-event component, and event component, respectively.
Given the conversion of δ and ratio of isotope tracer, (3) can be rewritten as:
Q t RD t=Q s RD s+Q g RD g (4) where RD t=[D]t/[H]t, RD s=[D]s/[H]s,RD g=[D]g/[H]g, RD is the isotopic ratio of deuterium in the water, [D] is the abundance of deuterium in the water, [H] is the abundance of hydrogen in the corresponding water and subscript t, s, g refer to the total discharge, event component, and pre-event component, respectively.
Equations (2) and (4) are combined to calculate the contribution of the event water and the pre-event water in the total streamflow.
Q s=Q t(RD t-RD g)/(RD s-RD g) (5)
Q g=Q t(RD t-RD s)/(RD g-RDs g) (6) Equations (5) and (6) are commonly used in the two-component hydrograph separation method in which we substitute isotopic ratio for isotopic concentration to construct the tracer isotope mass balance equation (referred to as Ratio Balance Method). A new method which is based on strictly tracer isotope mass conservation is proposed (referred to as Mass Balance Method). According to the new method, (3) is rewritten as:
Q t C t=Q s C s+Q g C g (7) where C t, C s, C g are the isotope tracer concentration of total water, event water and pre-event water, respectively.
The isotope tracer concentration can be converted to the isotopic ratio. The formula of this conversion relationship can be deduced using deuterium tracer as an example.
Given RD=[D]/[H], RT=[T]/[H], it can be concluded that:
RD/(1+RD+RT)=([D]/[H])/(1+[D]/[H]+[T]/[H]) (8) where RD is the isotopic ratio of deuterium in the water, RT is the isotopic ratio of tritium in the water, and [H], [D] and [T] is the abundance of hydrogen, deuterium and tritium in the water respectively.
CD=[D]/([H]+[D]+[T])=RD/(1+RD+RT) (9) Equation (9) is the conversion formula between the isotopic concentration and the isotopic ratio. CD is the isotopic concentration of deuterium. From the above equation, it is evident that the isotopic concentration is different from the isotopic ratio. Therefore, it is not appropriate to substitute the isotopic ratio for the isotopic concentration. Equation (9) has different expressions for different water sources. The formulas are:
C t=R
D t/(1+RD t+RT t)
C s=R
D s/(1+RD s+RT s)
C g=R
D g/(1+RD g+RT g)
Given the above conversion formula, (7) can be rewritten as:
111
g
t s
t s g
t t s s g g
RD
RD RD
Q Q Q
RD RT RD RT RD RT
=+
++++++
(10)
Equations (2) and (10) are combined to solve the contribution of the event water and prevent water in the total streamflow again.
()()
()()
111
1
11
t g g t s s s t
t t
s g g s
RD RT RD RT RD RT
Q Q
RD RT
RD RT RD RT
+?+++ =
++
+?+
(11)
1
(1)(1)
(1)(1)1
g g
t s s t
g t
g s s g t t
RD RT
RD RT RD RT
Q Q
RD RT RD RT RD RT
++
+?+
=
+?+++
(12)
We can see the difference if we look at the right hand side of (5) and (6) and (11) and (12). The results from the Ratio Balance Method are different from Mass Balance Method because the former cannot strictly meet tracer mass conservation equation.
Nevertheless, in the tracer mass balance equation, the values of the isotopic concentration and the isotopic abundance are equivalent.
Tracer mass balance equation expressed as isotopic abundance is written as:
Q t[D]t=Q s[D]s+Q g[D]g (13) Let: N=[H]+[D]+[T] (given that hydrogen has three different isotopes, that is, hydrogen, deuterium, and tritium, and N is the total isotope abundance of the unit water body)
According to (9), the isotope abundance of deuterium can be written as:
[D ]= N * RD /(1+RD +RT ) (14)
For different water sources, there are different formulas:
[D ]t =N t *RD t /(1+RD t +RT t ) , [D ]s =N s *RD s /(1+RD s +RT s ) , [D ]g =N g *RD g /(1+RD g +RT g ) .
Given the principle of the total isotope abundance in
the unit water body is always the same, that is N t =N s =N g . Then (13) can also be rewritten as the form of (10). From the analysis of the deduction above, some conclusions can be achieved:
1) The reason for the error of (4) is because of isotope fractionation. If there is no isotope fractionation, the abundance of abundant isotope in the unit water body of total streamflow is the same. That means [H ]t =[H ]s =[H ]g , and then the (4) can be rewritten as (13).
2) The principle of the total isotope abundance of the unit water body is identical was introduced in the Mass Balance Method. As a result, the reduction of abundance of rare isotope in the unit water body corresponds to the increase of the abundance of abundant isotope, and vice versa. Consequently, it is concluded that for every water source, the total isotope abundance in the unit water body is identical. This assumption is reasonable and the Mass Balance Method can strictly meet the mass conservation equation.
3) The results of the Ratio Balance Method cannot strictly meet the mass conservation equation. Mass Balance Method includes all hydrogen isotopes and can be deduced on the basis of strictly mass conservation equation.
B. Three-Component Hydrograph Separation
A three-component isotope hydrograph separation [8] has been used to include soil water as a separate component, that is channel precipitation (generally referred to surface runoff results from precipitation), soil water and groundwater. We will give the details using deuterium and 18O as examples. The basic equations are of the form:
Q t =Q s +Q i +Q g (15) Q t δD t =Q s δD s +Q i δD i +Q g δD g (16) Q t δ18O t =Q s δ18O s +Q i δ18O i +Q g δ18O g (17)
where δD and δ18O are the δ-values of deuterium and 18O tracer, subscript t , s , i , g refer to the total discharge, channel precipitation, soil water and groundwater respectively.
Given the conversion between the δ-value and the isotopic ratio, (16) and (17) can be rewritten as:
Q t RD t =Q s RD s +Q i RD i +Q g RD g (18)
Q t R 18
O t =Q s R 18
O s +Q i R 18
O i +Q g R 18
O g (19) where RD and R 18O are the isotopic ratio of deuterium and 18O tracer (RD =[D ]/[H ], R 18O =[18O ]/[16O ]).
Equations (15), (18) and (19) can be combined to calculate the contribution of every water sources. The results are written in the form of vector as:
1
181818181111s t i t s i g t s i g g t t Q Q Q RD RD RD RD R O R O R O Q Q R O ?????????????
=?????
???
????
?
????
? (20)
Let
'181818111s i g s i g A RD RD RD R O R O R O ??
??
=????
??,'181t t B RD R O ????=??
????,'s t i t g t Q Q X Q Q ????=??
????, then (20) can be rewritten as:
X'=A'-1B' (21)
Equation (20) is commonly used basic equation in the three-component hydrograph separation. The new method that strictly meets mass conservation can be deduced. The basic equations are of the forms:
Q t =Q s +Q i +Q g (22)
Q t CD t =Q s CD s +Q i CD i +Q g CD g (23) Q t C 18O t =Q s C 18O s +Q i C 18O i +Q g C 18O g (24)
where CD and C 18O are isotopic concentration of deuterium and 18O tracer. Given:
CD =RD /(1+RD +RT ) and C 18O =R 18O /(1+ R 18O + R 17O ) (22), (23) and (24) are combined to calculate the proportions of every water sources, the result is written in the form of vector as:
X =A -1B (25)
where 181818
181718171817111111111g s i s s i i g g g s i
s s i i g g R D R D R D A R D R T R D R T R D R T R O R O R O R O R O R O R O R O R O ????
??????=??++++++??????++++++?
?, ())181817
111t t t t t t B R D R
D R D R O R O R O ??
??
??=++??++????, s t i t g t Q Q X Q Q ????=??????,
R 17O is the isotopic ratio of 17O tracer (R 17O =[17O ]/[16O ]). It is apparent that the formulas of (20) and (25) are
different. Therefore, the calculated proportions of three components from Ratio Balance Method and Mass
Balance Method will be different. Three-component Ratio Balance Method cannot meet the mass balance.
III. R ESULTS AND DISCUSSION
From the above analysis we can see that strictly speaking, there are some errors in Ratio Balance Method which cannot meet the mass balance. Table I summarizes the results from the two balance methods based on the examples of international hydrologist’s experiments. From the table I, we can see that there is little difference between the two separation methods. The reason is the proportion of rare isotope in the atom is so small and the proportion of abundant isotope in the atom is large, then the value of R t , R s and R g is small. For example,
R 18O (VSMOW )=(2005.2±0.43) ×10-6 RD (VSMOW )=(155.76±0.10) ×10-6
The difference between isotopic ratios of different water source is small because of the small magnitude value of isotopic ratio. Although there is not large difference between isotopic ratios of different water sources, it cannot be negligible. It is isotope fractionation that leads to the small difference which is the objective to use isotopic hydrograph separation methods. In summary, Mass Balance Method can strictly meet mass conservation but Ratio Balance Method by substituting the isotopic ratio for the isotopic concentration cannot. The results of Ratio Balance Method deviate from the results of Mass Balance Method.
IV. C ONCLUSIONS
Although the calculated proportions of different source water from two different methods are similar. It does not mean that we can substitute the isotopic ratio for the isotopic concentration in the mass balance equation of tracer concentration to calculate the contributions of different water sources in isotopic hydrograph separation.
In scientific research, it is not appropriate to substitute the isotopic ratio for the isotopic concentration in the isotope tracer concentration mass balance equation. Therefore, we recommend the use of Mass Balance Method in the application of isotope hydrograph separation research to improve the degree of accuracy.
A CKNOWLEDGMENT
This work was supported by the National Nature Science Foundation of China (Grant No. 50679024)
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TABLE I.
C OMPARISON OF RESULTS FROM TWO DIFFERENT BALANCE METHODS
Hydrologists Location Number Tracer
Separation method
（two or three component ）
Percentage old water
Ratio Balance Method （%） Mass Balance Metho d （%）
Sklash et al.[9] New Zealand Pit 1 D Two 70 69.6
Pit 2 D Two 62 62.1 Pit 3 D Two 55 55.0 Site A D Two 69 68.8 Pit 5 D Two 94 94.0 Site D D Two 75 75.1
Turner et al.[10] Australia 2 18
O Two 69 69.2
D Two 74 74.2
3 18
O Two 73 72.6 D Two 76 75.6
5 18
O Two 95 95.1 D Two 92 91.5
DeWalle et al.[8] Pennsylvania A 1 18
O Two 55 55.4
B 2 18
O Three 68 68.2
C 3 18
O Two 69 69.4