tracers: rt 1 mcguire, osu isotope hydrology shortcourse prof. jeff mcdonnell dept. of forest...

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1McGuire, OSU

Isotope Hydrology Shortcourse

Prof. Jeff McDonnell

Dept. of Forest EngineeringOregon State University

Residence Time Approaches using

Isotope Tracers

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2McGuire, OSU

© Oregon State University

Outline

Day 1 Morning: Introduction, Isotope Geochemistry Basics Afternoon: Isotope Geochemistry Basics ‘cont, Examples

Day 2 Morning: Groundwater Surface Water Interaction, Hydrograph

separation basics, time source separations, geographic source separations, practical issues

Afternoon: Processes explaining isotope evidence, groundwater ridging, transmissivity feedback, subsurface stormflow, saturation overland flow

Day 3 Morning: Mean residence time computation Afternoon: Stable isotopes in watershed models, mean residence

time and model strcutures, two-box models with isotope time series, 3-box models and use of isotope tracers as soft data

Day 4 Field Trip to Hydrohill or nearby research site

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How these time and space scales relate to what we have discussed so far

Bloschel et al., 1995

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This section will examine how we make use of isotopic variability

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Outline

What is residence time?

How is it determined? modeling background

Subsurface transport basics

Stable isotope dating (18O and 2H)

Models: transfer functions

Tritium (3H)

CFCs, 3H/3He, and 85Kr

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Residence Time

Mean Water Residence Time (aka: turnover time, age of water leaving a system, exit age, mean transit time, travel time, hydraulic age, flushing time, or kinematic age)

tw=Vm/Q

For 1D flow pattern: tw=x/vpw

where vpw =q/

Mean Tracer Residence Time

0

0

)(

)(

dttC

dtttC

t

I

I

t MQC

dttC

tCtg I

I

I /

)(

)()(

0

Residence time distribution

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Why is Residence Time of Interest?

It tells us something fundamental about the hydrology of a watershed

Because chemical weathering, denitrification, and many biogeochemical processes are kinetically controlled, residence time can be a basis for comparisons of water chemistry

Vitvar & Burns, 2001

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Tracers and Age Ranges

Environmental tracers: added (injected) by natural processes, typically

conservative (no losses, e.g., decay, sorption), or ideal (behaves exactly like traced material)

0.01 0.1 1 10 100 1000 10000 100000

Age (years)

18O

D

85Kr

3He-3H

Tritium

CFCs

14C

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Modeling Approach

Lumped-parameter models (black-box models): System is treated as a whole & flow pattern

is assumed constant over modeling period Used to interpret tracer observations in system outflow (e.g.

GW well, stream, lysimeter)

Inverse procedure; Mathematical tool: The convolution integral

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Convolution

A convolution is an integral which expresses the amount of overlap of one function h as it is shifted over another function x. It therefore "blends" one function with another

It’s frequency filter, i.e., it attenuates specific frequencies of the input to produce the result

Calculation methods: Fourier transformations, power spectra Numerical Integration

y t h t x t d( ) ( ) ( )

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The Convolution Theorem

{ ( )* ( )} ( ) ( – ) exp( )

( ) ( ) exp(– )

( ){ ( exp(– )}

( ) exp(– ) ( ( (

f t g t f x g t x dx i t dt

f x g t x i t dt dx

f x G i x dx

f x i x dxG F G

Proof:

Trebino, 2002

Y()=F()G() and

|Y()|2=|F()| 2 |G()| 2

)()()(*)( GFtgtf

We will not go through this!!

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x()

g() = e -a

Folding

g(-)

e -(-a

Displacement

g(t-)

e -a(t-

t

Multiplication

x()g(t-)

t

Integrationy(t)

tt

Shaded area

1

2

3

4

Step

Convolution: Illustration of how it works

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Example: Delta Function

Convolution with a delta function simply centers the function on the delta-function.

This convolution does not smear out f(t).

Thus, it can physically represent piston-flow processes.

( ) ) ( ) ( – )

( )

f t t a f t u u a du

f t a

Modified from Trebino, 2002

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Matrix Set-up for Convolution

= [length(x)+length(h)]-1=

leng

th(x

)

=

= x(t)*h y(t)

= 0

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Similar to the Unit Hydrograph

Time

Precipitation

Infiltration Capacity

Excess Precipitation

Excess Precipitation

Hydrographs for Event

0

500

1000

1500

2000

2500

0 1 2 3Time(hrs)

Flo

w

Tarboton

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Instantaneous Response Function

Excess Precipitation P(t)

0

500

1000

1500

2000

2500

0 1 2 3 4

0

500

1000

1500

2000

2500

0 1 2 3

Unit Response Function U(t)

Event Response Q(t)

d)t(U)(P)t(Q

Tarboton

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Subsurface Transport Basics

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Subsurface Transport Processes

Advection Dispersion Sorption Transformations

Modified from Neupauer

& Wilson, 2001

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Advection

t=t1 t2>t

1

t3>t

2

FLOWFLOW

Solute movement with bulk water flow


& Wilson, 2001

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& Wilson, 2001

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Dispersion

FLOWFLOW

Solute spreading due to flowpath heterogeneity


& Wilson, 2001

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& Wilson, 2001

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Sorption

t=t1 t2>t1

FLOWFLOW

Solute interactions with rock matrix


& Wilson, 2001

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& Wilson, 2001

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Transformations

t=t1 t2>t1

MICROBE

CO2

Solute decay due to chemical and biological reactions


& Wilson, 2001

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Stable Isotope Methods

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Stable Isotope Methods

Seasonal variation of 18O and 2H in precipitation at temperate latitudes

Variation becomes progressively more muted as residence time increases

These variations generally fit a model that incorporates assumptions about subsurface water flow

Vitvar & Burns, 2001

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- 2 0

- 1 5

- 1 0

- 5

0

18

O (

pe

r m

il S

MO

W)

Jan-93 Jan-94 Jan-95 Jan-96-10

0

10

20

air

te

mp

era

ture

(°C

)

Neversink watershed, 1993 - 1996

Vitvar, 2000

Seasonal Variation in 18O of Precipitation

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Seasonality in Stream Water

Deines et al. 1990

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1200 1300 1400 1500 1600 1700 1800-14

-12

-10

-8

-6

-4

-2

0

Time [days]

Oxy

gen-

18 (

per

mil) Precipitation or recharge signal

Streamflow signal

Example: Sine-wave

0 1000 20000

2

4

6x 10-3

Time [days]

g(t

)

Mean = 235 d

Cin(t)=A sin(t)

Cout(t)=B sin(t+)

T=-1[(B/A)2 –1)1/2

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Convolution Movie

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Transfer Functions Used for Residence Time Distributions

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Common Residence Time Models

0 0.5 1 1.5 20

0.002

0.004

0.006

0.008

0.01

0.012

Normalized time (t/T)

g(t

)

0 0.5 1 1.5 20

1

2

3

4

5

6x 10

-3

Normalized time (t/T)

eta=1eta=1.25eta=1.5eta=1.75eta=2eta=2.25eta=2.5eta=2.75eta=3

D/vx=5D/vx=3.5D/vx=2D/vx=1D/vx=0.1D/vx=0.2D/vx=0.5D/vx=0.01D/vx=0.05D/vx=0.005

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Piston Flow (PFM)

Assumes all flow paths have transit time All water moves with advection

Represented by a Dirac delta function:

0 1 2 3 40

0.2

0.4

0.6

0.8

1

t/T

g(t

)

)()( Tttg

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Exponential (EM)

Assumes contribution from all flow paths lengths and heavy weighting of young portion.

Similar to the concept of a “well-mixed” system in a linear reservoir model

0 2 4 6 8 10 120

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

g(t

)

t/T

)/exp()( 1 TtTtg

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Dispersion (DM)

Assumes that flow paths are effected by hydrodynamic dispersion or geomorphological dispersion

Arises from a solution of the 1-D advection-dispersion equation:

0 2 4 6 8 100

0.002

0.004

0.006

0.008

0.01

t/T

g(t

)

x

Cv

x

CD

t

C

2

2

tD

T

T

tt

T

tDtg

p

p

41exp

4)(

21

2/1

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Exponential-piston Flow (EPM)

Combination of exponential and piston flow to allow for a delay of shortest flow paths

for t (1-andg(t)=0 for t< (1--1)

1exp)( T

t

Ttg

0 2 4 6 8 10 120

0.05

0.1

0.15

0.2

t/T

g(t

)

Piston flow = 1

1

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Heavy-tailed Models

Gamma

Exponentials in series

/

1

exp)(

)( tttg

2211

exp1

exp)(T

t

TT

t

Ttg

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Exit-age distribution (system response function)

Unconfined aquiferEM: g(t’) = 1/T exp(-t‘/T)

Maloszewski and Zuber

Confined aquifer PFM: g(t’) = (t'-T)

Kendall, 2001

PFM PFMEMEM

EPMEM

DM

DM

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Exit-age distribution (system response function) cont…

Partly Confined Aquifer:

EPM: g(t’) = /T exp(-t'/T + -1) for t‘≥T (1 - 1/)g(t’) = 0 for t'< T (1-1/ )

Maloszewski and Zuber

Kendall, 2001

DM

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Dispersion Model Examples

-20

-15

-10

-5

0O

-18

(per

mil

)

0

0.05

0.1

0.15

0.2

g(t

)

-20

-15

-10

-5

0

O-1

8 (p

er m

il)

0

0.05

0.1

0.15

0.2

g(t

)

0 20 40 60 80-20

-15

-10

-5

0

Time (months)

O-1

8 (p

er m

il)

0 20 40 60 800

0.05

0.1

0.15

0.2

Time (months)

g(t

)

MRT = 6 monthsD/vx = 0.3



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0.00

0.02

0.04

0.06

0.08

0 500 1000 1500Time (d)

g(t) DM: Dp = 0.27MRT = 10.5 mon.

EPM: 21% PistonMRT = 10.5 mon.

0 500 1000 1500Time (d)

DM: Dp = 0.36MRT = 8.5 mon.

EPM12% PistonMRT = 9.5 mon.

Residence Time Distributions can be Similar

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0 0.5 1 1.5 2 2.5 3 3.5 40

0.05

0.1

0.15

0.2

FittedTransfer Functions

Normalized time [t/T]

Fu

nct

ion

we

igh

ting

[g(t

)]

40 45 50 55 60 65 70 75-11

-10

-9

-8

Simulation Results with Optimized Parameters

time

tra

cer

con

ten

t

ConvolutionOutput Obs

Uncertainty

10 12 14 160

10

20

30

Piston Flow %

Fre

q

140 160 1800

10

20

30

MRT

Fre

q

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Identifiable Parameters?

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Review: Calculation of Residence Time

Simulation of the isotope input – output relation:

Calibrate the function g(t) by assuming various distributions of the residence time:

1. Exponential Model

2. Piston Flow Model

3. Dispersion Model

t

in dtCtgtC0

)()()(

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Input Functions

Must represent tracer flux in recharge Weighting functions are used to “amount-weight” the tracer

values according recharge: mass balance!!

Methods: Winter/summer weighting:

Lysimeter outflow

General equation:

t

in dtCtgtC0

)()()(

ininiN

iii

iiin CC

P

PNt

1

)(

0

0

)()(

)()()(

)(

dtwg

dttwg

tCin where w(t) = recharge

weighting function

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Models of Hydrologic Systems

Cin Cout

Model 1

1-

CoutCin 1-

Model 3

Upper reservoir

Lower reservoirCoutCin 1-

Model 2

Direct runoff

Maloszewski et al., 1983

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Stewart & McDonnell, 2001

Soil Water Residence Time

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Example from Rietholzbach

1994 1995 1996 1997

Mean baseflow residence time = 12.5 mo

-14

-12

-10

-8

-13

-11

-9

18O

(p

er m

il S

MO

W)

Rietholzbach watershed, Switzerland

dispersion model

exp/piston-flow model

measured values

Vitvar, 1998

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Model 3…

Uhlenbrook et al., 2002

Stable deep signal

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New Zealand

42 degrees S1

72

de

gre

es

E

Figure 1

How residence time scales with basin area

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Figure 2

Digital elevation model and stream network

Contour interval 10 meters

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1 10 1000.0

0.5

1.0

K catchment (17 ha)

M15 catchment (2.6 ha)

Sub-catchment size ha

1 10 1000.0

0.5

1.0

Median sub-catchment size = 8.2 ha




Fre

qu

en

cy

1 10 1000.0

0.5

1.0

1 10 1000.0

0.5

1.0

Bedload catchment (280 ha)

PL14 catchment (80 ha)

Figure 3

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0 2 4 6 8

1

2M

ean t

ritium

ag

e y

ears

Median sub-catchment size ha

M15 (2.6 ha)

K (17 ha)

Bedload (280 ha)

PL14 (17 ha)

Figure 4

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500 m

Scale

-7

0

-3.5

Low

High

RIF

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Determining Residence Time of Old(er) Waters

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What’s Old?

No seasonal variation of stable isotope concentrations: >4 to 50 years

Methods:

Tritium (3H)

3H/3He

CFCs

85Kr

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Tritium

Historical tracer: 1963 bomb peak of 3H in atmosphere

1 TU: 1 3H per 1018 hydrogen atoms Slug-like input 36Cl is a similar tracer

Similar methods to stable isotope models Half-life () = 12.43

Tritium Input

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Tritium (con’t)

Piston flow (decay only):tt=-17.93[ln(C(t)/C0)]

Other flow conditions:')'()'exp()()(

0dtttgttCtC

t

in

Manga, 1999

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Zeit [Jahre]

1950 1960 1970 1980 1990 2000

3 H-K

onze

ntra

tione

n [T

.U.]

10

100

1000 3H-Input im Bruggagebiet

1992 1993 1994 1995 1996 1997 1998 1999

3H

[T.U

.]

10

15

20

25

sim. 3H-Konzentrationen3H-Messungen mit analyt. Fehler

Spring: Stollen t0 = 8.6 a, PD = 0.22

3H-Input-Bruggagebiet3H-Input

lumped parameter models

Time [yr.]

3 H [

TU

]

Deep Groundwater Residence Time

Time [yr.]

3 H [

TU

]


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3He/3H

As 3H enters groundwater and radioactively decays, the noble gas 3He is produced

Once in GW, concentrations of 3He increase as GW gets older

If 3H and 3He are determined together, an apparent age can be determined:

1

H

Heln

3

*31tt

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Determination of Tritiogenic He

Other sources of 3He: Atmospheric solubility (temp dependent) Trapped air during recharge Radiogenic production ( decay of U/Th-

series elements)

Determined by measuring 4He and other noble gases

3H

e/3H

age

(ye

ars)

20

0

30

10

1 5 10 50

Tage (years)

20.5 years

Modified from Manga, 1999

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Chlorofluorocarbons (CFCs)

CFC-11 (CFCL3), CFC-12 (CF2Cl2), & CFC-13 (C2F3Cl3) long atm residence time (44, 180, 85 yrs)

Concentrations are uniform over large areas and atm concentration are steadily increasing

Apparent age = CFC conc in GW to equivalent atm conc at recharge time using solubility relationships

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85Kr

Radioactive inert gas, present is atm from fission reaction (reactors)

Concentrations are increasing world-wide

Half-life = 10.76; useful for young dating too

Groundwater ages are obtained by correcting the measured 85Kr activity in GW for radioactive decay until a point on the atm input curve is reached

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85Kr (con’t)

Independent of recharge temp and trapped air

Little source/sink in subsurface Requires large volumes of water

sampled by vacuum extraction (~100 L)

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Model 3…


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Large-scale Basins

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Notes on Residence Time Estimation

• 18O and 2H variations show mean residence times up to ~4 years only; older waters dated through other tracers (CFC, 85Kr, 4He/3H, etc.)

• Need at least 1 year sampling record of isotopes in the input (precip) and output (stream, borehole, lysimeter, etc.)

• Isotope record in precipitation must be adjusted to groundwater recharge if groundwater age is estimated

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Class exerciseftp://ftp.fsl.orst.edu/pub/mcguirek/rt_lecture

Hydrograph separation Convolution FLOWPC

Show your results graphically (one or several models) and provide a short write-up that includes:

– Parameter identifiability/uncertainty– Interpretation of your residence time distribution in

terms of the flow system

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References

Cook, P.G. and Solomon, D.K., 1997. Recent advances in dating young groundwater: chlorofluorocarbons, 3H/3He and 85Kr. Journal of Hydrology, 191:245-265.

Duffy, C.J. and Gelhar, L.W., 1985. Frequency Domain Approach to Water Quality Modeling in Groundwater: Theory. Water Resources Research, 21(8): 1175-1184.

Kirchner, J.W., Feng, X. and Neal, C., 2000. Fractal stream chemistry and its implications for contaminant transport in catchments. Nature, 403(6769): 524-527.

Maloszewski, P. and Zuber, A., 1982. Determining the turnover time of groundwater systems with the aid of environmental tracers. 1. models and their applicability. Journal of Hydrology, 57: 207-231.

Maloszewski, P. and Zuber, A., 1993. Principles and practice of calibration and validation of mathematical models for the interpretation of environmental tracer data. Advances in Water Resources, 16: 173-190.

Turner, J.V. and Barnes, C.J., 1998. Modeling of isotopes and hydrochemical responses in catchment hydrology. In: C. Kendall and J.J. McDonnell (Editors), Isotope tracers in catchment hydrology. Elsevier, Amsterdam, pp. 723-760.

Zuber, A. and Maloszewski, P., 2000. Lumped parameter models. In: W.G. Mook (Editor), Environmental Isotopes in the Hydrological Cycle Principles and Applications. IAEA and UNESCO, Vienna, pp. 5-35. Available: http://www.iaea.or.at/programmes/ripc/ih/volumes/vol_six/chvi_02.pdf

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Outline

Day 1 Morning: Introduction, Isotope Geochemistry Basics Afternoon: Isotope Geochemistry Basics ‘cont, Examples

Day 2 Morning: Groundwater Surface Water Interaction, Hydrograph

separation basics, time source separations, geographic source separations, practical issues

Afternoon: Processes explaining isotope evidence, groundwater ridging, transmissivity feedback, subsurface stormflow, saturation overland flow

Day 3 Morning: Mean residence time computation Afternoon: Stable isotopes in watershed models, mean residence

time and model strcutures, two-box models with isotope time series, 3-box models and use of isotope tracers as soft data

Day 4 Field Trip to Hydrohill or nearby research site

tracers: rt 1 mcguire, osu isotope hydrology shortcourse prof. jeff mcdonnell dept. of forest...

Documents

rt mcguire

osu oregon state university

isotope tracers

kr slide

convolution integral

travel time

turnover time

isotope time series