skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: On the probabilistic structure of water age: Probabilistic Water Age

Abstract

We report the age distribution of water in hydrologic systems has received renewed interest recently, especially in relation to watershed response to rainfall inputs. The purpose of this contribution is first to draw attention to existing theories of age distributions in population dynamics, fluid mechanics and stochastic groundwater, and in particular to the McKendrick-von Foerster equation and its generalizations and solutions. A second and more important goal is to clarify that, when hydrologic fluxes are modeled by means of time-varying stochastic processes, the age distributions must themselves be treated as random functions. Once their probabilistic structure is obtained, it can be used to characterize the variability of age distributions in real systems and thus help quantify the inherent uncertainty in the field determination of water age. Finally, we illustrate these concepts with reference to a stochastic storage model, which has been used as a minimalist model of soil moisture and streamflow dynamics.

Authors:
 [1];  [1]
  1. Duke Univ., Durham, NC (United States). Department of Civil and Environmental Engineering
Publication Date:
Research Org.:
Duke Univ., Durham, NC (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Biological and Environmental Research (BER) (SC-23)
OSTI Identifier:
1454928
Grant/Contract Number:  
SC0006967
Resource Type:
Accepted Manuscript
Journal Name:
Water Resources Research
Additional Journal Information:
Journal Volume: 51; Journal Issue: 5; Journal ID: ISSN 0043-1397
Publisher:
American Geophysical Union (AGU)
Country of Publication:
United States
Language:
English
Subject:
54 ENVIRONMENTAL SCIENCES; 58 GEOSCIENCES; age distribution; residence time; stochastic soil moisture; streamflow; Von Foerster equation; harmonic mean

Citation Formats

Porporato, Amilcare, and Calabrese, Salvatore. On the probabilistic structure of water age: Probabilistic Water Age. United States: N. p., 2015. Web. doi:10.1002/2015WR017027.
Porporato, Amilcare, & Calabrese, Salvatore. On the probabilistic structure of water age: Probabilistic Water Age. United States. doi:10.1002/2015WR017027.
Porporato, Amilcare, and Calabrese, Salvatore. Thu . "On the probabilistic structure of water age: Probabilistic Water Age". United States. doi:10.1002/2015WR017027. https://www.osti.gov/servlets/purl/1454928.
@article{osti_1454928,
title = {On the probabilistic structure of water age: Probabilistic Water Age},
author = {Porporato, Amilcare and Calabrese, Salvatore},
abstractNote = {We report the age distribution of water in hydrologic systems has received renewed interest recently, especially in relation to watershed response to rainfall inputs. The purpose of this contribution is first to draw attention to existing theories of age distributions in population dynamics, fluid mechanics and stochastic groundwater, and in particular to the McKendrick-von Foerster equation and its generalizations and solutions. A second and more important goal is to clarify that, when hydrologic fluxes are modeled by means of time-varying stochastic processes, the age distributions must themselves be treated as random functions. Once their probabilistic structure is obtained, it can be used to characterize the variability of age distributions in real systems and thus help quantify the inherent uncertainty in the field determination of water age. Finally, we illustrate these concepts with reference to a stochastic storage model, which has been used as a minimalist model of soil moisture and streamflow dynamics.},
doi = {10.1002/2015WR017027},
journal = {Water Resources Research},
number = 5,
volume = 51,
place = {United States},
year = {2015},
month = {4}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Citation Metrics:
Cited by: 15 works
Citation information provided by
Web of Science

Save / Share:

Works referenced in this record:

Mathematical models for cellular systems the von Foerster equation. Part I
journal, September 1965

  • Trucco, E.
  • The Bulletin of Mathematical Biophysics, Vol. 27, Issue 3
  • DOI: 10.1007/BF02478406

Some simple models for nonlinear age-dependent population dynamics
journal, April 1979


The McKendrick partial differential equation and its uses in epidemiology and population study
journal, September 1997


Probabilistic dynamics of soil nitrate: Coupling of ecohydrological and biogeochemical processes: STOCHASTIC DESCRIPTION OF SOIL NUTRIENTS
journal, March 2008

  • Botter, G.; Daly, E.; Porporato, A.
  • Water Resources Research, Vol. 44, Issue 3
  • DOI: 10.1029/2007WR006108

Residence Time Theory
journal, May 2008

  • Nauman, E. Bruce
  • Industrial & Engineering Chemistry Research, Vol. 47, Issue 10
  • DOI: 10.1021/ie071635a

Theory of population transport
journal, November 1972


Quantifying catchment-scale mixing and its effect on time-varying travel time distributions: QUANTIFYING CATCHMENT-SCALE MIXING
journal, June 2012

  • van der Velde, Y.; Torfs, P. J. J. F.; van der Zee, S. E. A. T. M.
  • Water Resources Research, Vol. 48, Issue 6
  • DOI: 10.1029/2011WR011310

Superstatistics
journal, May 2003


Superstatistics of hydro-climatic fluctuations and interannual ecosystem productivity
journal, January 2006

  • Porporato, Amilcare; Vico, Giulia; Fay, Philip A.
  • Geophysical Research Letters, Vol. 33, Issue 15
  • DOI: 10.1029/2006GL026412

Transit Times for Water in a Small Till Catchment from a Step Shift in the Oxygen 18 Content of the Water Input
journal, December 1996

  • Rodhe, A.; Nyberg, L.; Bishop, K.
  • Water Resources Research, Vol. 32, Issue 12
  • DOI: 10.1029/95WR01806

Applications of Mathematics to Medical Problems
journal, February 1925


Residence time distribution theory for unsteady stirred tank reactors
journal, September 1969


Simulation of groundwater age distributions
journal, December 1998

  • Varni, Marcelo; Carrera, Jesús
  • Water Resources Research, Vol. 34, Issue 12
  • DOI: 10.1029/98WR02536

Analysis of soil carbon transit times and age distributions using network theories
journal, January 2009

  • Manzoni, Stefano; Katul, Gabriel G.; Porporato, Amilcare
  • Journal of Geophysical Research, Vol. 114, Issue G4
  • DOI: 10.1029/2009JG001070

Dose-structured population dynamics
journal, July 2007


Direct Simulation of Groundwater Age
journal, February 1996


Representation of space–time variability of soil moisture
journal, September 2005

  • Isham, V.; Cox, D. R.; Rodríguez-Iturbe, I.
  • Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 461, Issue 2064
  • DOI: 10.1098/rspa.2005.1568

On a first-passage problem for a cumulative process with exponential decay
journal, January 1976


Shot noise models for the generation of synthetic streamflow data
journal, February 1977


State-dependent fire models and related renewal processes
journal, October 2006


Effect of different jump distributions on the dynamics of jump processes
journal, June 2010


Toward a general theory of the age in ocean modelling
journal, January 1999


Catchment residence and travel time distributions: The master equation: CATCHMENT RESIDENCE TIMES
journal, June 2011

  • Botter, Gianluca; Bertuzzo, Enrico; Rinaldo, Andrea
  • Geophysical Research Letters, Vol. 38, Issue 11
  • DOI: 10.1029/2011GL047666

Impact of hydroclimatic fluctuations on the soil water balance: HYDROCLIMATIC FLUCTUATIONS AND SOIL WATER BALANCE
journal, June 2006


Prescription-induced jump distributions in multiplicative Poisson processes
journal, June 2011


Transient soil-moisture dynamics and climate change in Mediterranean ecosystems: TRANSIENT SOIL-MOISTURE DYNAMICS
journal, November 2008

  • Viola, F.; Daly, E.; Vico, G.
  • Water Resources Research, Vol. 44, Issue 11
  • DOI: 10.1029/2007WR006371

Kinematics of age mixing in advection-dispersion models: AGE MIXING IN ADVECTION-DISPERSION MODELS
journal, December 2013

  • Benettin, Paolo; Rinaldo, Andrea; Botter, Gianluca
  • Water Resources Research, Vol. 49, Issue 12
  • DOI: 10.1002/2013WR014708

A review and evaluation of catchment transit time modeling
journal, November 2006


Non-linear age-dependent population dynamics
journal, September 1974

  • Gurtin, Morton E.; Maccamy, Richard C.
  • Archive for Rational Mechanics and Analysis, Vol. 54, Issue 3
  • DOI: 10.1007/BF00250793

Compartment Models and Reservoir Theory
journal, November 1971


Residence time distributions of variable flow processes
journal, October 1977


Stochastic modeling of soil salinity: STOCHASTIC SOIL SALINITY
journal, April 2010

  • Suweis, S.; Rinaldo, A.; Van der Zee, S. E. A. T. M.
  • Geophysical Research Letters, Vol. 37, Issue 7
  • DOI: 10.1029/2010GL042495

Dynamical modelling of concentration-age-discharge in watersheds
journal, May 2010

  • Duffy, Christopher J.
  • Hydrological Processes, Vol. 24, Issue 12
  • DOI: 10.1002/hyp.7691

A note on the concepts of age distribution and transit time in natural reservoirs
journal, January 1973


Transport of kinetically sorbing solute by steady random velocity in heterogeneous porous formations
journal, April 1994


Fractal stream chemistry and its implications for contaminant transport in catchments
journal, February 2000

  • Kirchner, James W.; Feng, Xiahong; Neal, Colin
  • Nature, Vol. 403, Issue 6769
  • DOI: 10.1038/35000537