The document describes a component that computes the forward probability density functions (pdfs) of residence time, travel time, and evapotranspiration time given injection time, using backward pdfs as inputs. The component outputs are travel time and evapotranspiration time forward pdfs in tridimensional matrices, as well as the partitioning coefficient. Examples of using the component in an OMS simulation file are also provided.

1. Bancheri
LINKERS
JGrass-NewAge: ProbabilitiesForward component
Marialaura Bancheri
Correspondence:
marialaura.bancheri@unitn.it
Dipartimento di Ingegneria Civile
Ambientale e Meccanica, Trento,
Mesiano di Povo, Trento, IT
Full list of author information is
available at the end of the article
Abstract
These pages teach how to run the ProbabilitiesForward component inside the OMS 3
console. Some preliminary knowledge and installation of OMS is mandatory (see @Also
useful). This component deals with the computation of the forward probability density
functions (pdfs) of the residence time, travel time and evapotranspiration time
conditional on the injection time, according to (). Once computed the backward pdfs, it
is possible to obtain the residence time, the travel times and evapotranspiration times
forward pdfs, thanks to the definition of the partitioning coefficient (Θ) and the
Niemi’s relationship ((1)) . The results of the component are given in tridimensional
matrices, in which two dimensions are the injection and current times, and the third is
the value of the variable.
@Version:
0.1
@License:
GPL v. 3
@Inputs:
• Rainfall (mm);
• Evapotranspiration (mm);
• Discharge (m3
/m2
);
• Travel times backward pdfs [T−1
];
• Evapotranspiration times backward pdfs [T−1
];
• Station ID (-);
• Start date (String);
• Start Date Water Budget (String);
• End date (String).
@Outputs:
• Travel times forward pdfs [T−1
];
• Evapotranspiration forward times pdfs [T−1
];
• Partitioning coefficient [-].
@Doc Author: Marialaura Bancheri
@References:
• See References section below
Keywords: OMS; JGrass-NewAGE Component Description; Travel times

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Code Information
Executables
This link points to the jar file that, once downloaded can be used in the OMS console:
https://github.com/Mariolina88/OMS_Project_TT/tree/master/lib
Developer Info
This link points to useful information for the developers, i.e. information about the code
internals, algorithms and the source code
https://github.com/geoframecomponents
Also useful
To run JGrass-NewAGE it is necessary to know how to use the OMS console. Information
at: ”How to install and run the OMS console”,
https://alm.engr.colostate.edu/cb/project/oms).
JGrasstools are required for preparing some input data (information at:
http://abouthydrology.blogspot.it/2012/11/udig-jgrasstools-resources-in-italian.
html
To visualize results you need a GIS. Use your preferred GIS, following its installation
instructions. To make statistics on the results, you can probably get benefits from R:
http://www.r-project.org/ and follow its installation instruction.
To whom address questions
marialaura.bancheri@unitn.it
Authors of documentation
Marialaura Bancheri (marialaura.bancheri@unitn.it)
This documentation is released under Creative Commons 4.0 Attribution International

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Component Description
The forward travel time pdfs are defined as:
pQ(t − τ|τ) :=
q(t, τ)
Θ(τ)J(τ)
(1)
and
pET
(t − τ|τ) :=
aeT (t, τ)
(1 − Θ(τ))J(τ)
(2)
where where q(t, τ) [L3
T−2
] and aeT (t, τ) [L3
T−2
] are the part of the discharge Q(t)
and evapotranspiration ET (t) exiting the control volume at time t but composed by water
molecules entered at instant τ. Θ(τ) is the partition coefficient between the two outflows
considered and is defined as:
Θ(τ) := lim
t→∞
Θ(t, τ) := lim
t→∞
VQ(t, τ)
VQ(t, τ) + VET
(t, τ)
(3)
where:
VQ(t, τ) :=
t
0
q(t, τ)dt (4)
and
VAET
(t, τ) =
t
0
aeT (t, τ)dt (5)
Θ(τ) has been introduced to complete the algebra of probabilities, in presence of more
that one outflow. However studying it is important by itself, because it summarizes the
relevant element of hydrologic fluxes partition.
Thanks to Niemi’s relationship, (1), it is possible to relate the backward and forward
travel time pdfs, once computed Θ(τ):
Q(t)pQ(t − τ|t) = Θ(τ)pQ(t − τ|τ)J(τ) (6)
and:
AET (t)pET
(t − τ|t) = [1 − Θ(τ)]pET
(t − τ|τ)J(τ) (7)
Detailed Inputs description
The input file is a .csv file containing a header and one or more time series of input data,
depending on the number of stations involved. Each column of the file is associated to a
different station.
The file must have the following header:
• The first 3 rows with general information such as the date of the creation of the file
and the author;

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• the fourth and fifth rows contain the IDs of the stations (e.g. station number 8:
value 8, ID, ,8);
• the sixth row contains the information about the type of the input data (in this
case, one column with the date and one column with double values);
• the seventh row specifies the date format (YYYY-MM-dd HH:mm).
All the previous information shown in the figure 1.
Figure 1 Heading of the .csv input file
Rainfall
The rainfall is given in time series of (mm) for the investigated station .
Evapotranpiration
The evapotranpiration is given in time series of (mm) for the investigated station.
Discharge
The discharge is given in time series for the investigated station in (m3
/m2
).
Station ID
Station ID is the ID of the investigate station.
Start date
Start date is a string containing the first day of the simulation.
Start Date Water Budget
Start Date Water Budget is a string containing the first day of the simulation of the water
budget time series, if it is different from the start date of the actual simulation.
End date
End date is a string containing the last day of the simulation
Detailed Outputs description
The output are given in tridimensional matrices, t × ti × variable value, which must be
integrated in the injection time dimension to obtain the time series of the mean travel
times and the mean evapotranspiration times. Fig. 2 shows an example of a pdfs matrix.
The injection times are in the first column, whose length is l. In this case l = 5. The
actual time is t = ti −i, with i varying in the interval [0, l−1], in this case [0, 4]. Each row
contains a number of values equals to t. So for example for the first row i = 0 and there

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are 5 elements, the second 4 and so on. If, at a given injection time ti, the input rainfall
is zero, the rows has one zero value. This structures take into account two important
aspect: the first is that when t < ti, the variables are zero; the second is the speed of the
computation, which, in this case, is the minimum possible.
Figure 2 : Example of output pdfs matrix: for each time step, the length of the row decrease of one
value, till the end of the time series
Travel times forward pdfs
Fig. 3 shows an example of a forward travel time pdf for a given injection time, which
properly normalize to 1 when integrated over t.
0 500 1000 1500 2000 2500
0.00000.00100.00200.0030
Forward travel time pdf
Time [h]
ForwardTTpdf[1/T]
Figure 3 : Evolution in time of the forward travel time pdf for a given injection time
Evapotranspiration times forward pdfs
Fig. 4 shows an example of a forward evapotranspiration time pdf for a given injection
time, which properly normalize to 1 when integrated over t.

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0 500 1000 1500 2000 2500
0.0000.0020.0040.006
Forward evapotranspiration time pdf
Time [h]
ForwardETpdf[1/T]
Figure 4 : Evolution in time of the forward evapotranspiration time pdf for a given injection time
Partitioning coefficient Θ(t)
Fig. 5 shows an example of a time series of Θ(t), for a given injection time, obtained
using data from the Posina River. At the beginning Θ(t, τ) shows large oscillations due to
hourly and daily oscillations, especially in evapotranspiration. Because Θ(t, τ) is defined
through integrals, these oscillation are progressively dumped and become irrelevant after
a couple of weeks.
0 500 1000 1500 2000 2500
0.00.20.40.60.81.0
Partitioning coefficient
Time [h]
Theta[-]
Figure 5 : Evolution in time of the residence time pdf for a given injection time
Examples
The following .sim file is customized for the use of the forward probabilities component.
The .sim file can be downloaded from here:
https://github.com/GEOframeOMSProjects/OMS_Project_TT/tree/master/ForwardPdfs/
simulation
import static oms3.SimBuilder.instance as OMS3
def home = oms_prj
def startDate= "1994 -01 -01 00:00"
def endDate= "1995 -01 -01 00:00"
OMS3.sim {
model(while:" reader_data_J .doProcess") {
components {

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" reader_data_J " "org.jgrasstools.gears.io. timedependent .
OmsTimeSeriesIteratorReader "
" reader_data_pQ " "org. jgrasstools.gears.io. timedependent .
OmsTimeSeriesIteratorReader "
" reader_data_pET " "org. jgrasstools .gears.io. timedependent .
OmsTimeSeriesIteratorReader "
" reader_data_Qback " "org. jgrasstools .gears.io. timedependent
. OmsTimeSeriesIteratorReader "
" reader_data_ETback " "org.jgrasstools.gears.io. timedependent
. OmsTimeSeriesIteratorReader "
"pdfs" " travelTimesFor . ProbabilitiesForward "
" writer_theta " "org.jgrasstools.gears.io. timedependent
. OmsTimeSeriesIteratorWriter "
"writer_pQ" "org.jgrasstools .gears.io. timedependent .
OmsTimeSeriesIteratorWriter "
"writer_pET" "org.jgrasstools.gears.io. timedependent .
OmsTimeSeriesIteratorWriter "
"writer_Q" "org.jgrasstools.gears.io. timedependent .
OmsTimeSeriesIteratorWriter "
"writer_ET" "org.jgrasstools .gears.io. timedependent .
OmsTimeSeriesIteratorWriter "
}
parameter{
" reader_data_J .file" "${home }/ data/rainfall.csv"
" reader_data_J .idfield" "ID"
" reader_data_J .tStart" "${startDate}"
" reader_data_J .tEnd" "${endDate}"
" reader_data_J .tTimestep" 60
" reader_data_J . fileNovalue " " -9999"
" reader_data_pQ .file" "${home }/ data/pQ.csv"
" reader_data_pQ .idfield" "ID"
" reader_data_pQ .tStart" "${startDate}"
" reader_data_pQ .tEnd" "${endDate}"
" reader_data_pQ .tTimestep" 60
" reader_data_pQ . fileNovalue " " -9999"
" reader_data_pET .file" "${home }/ data/pET.csv"
" reader_data_pET .idfield" "ID"
" reader_data_pET .tStart" "${startDate}"
" reader_data_pET .tEnd" "${endDate}"
" reader_data_pET .tTimestep" 60
" reader_data_pET . fileNovalue" " -9999"
" reader_data_Qback .file" "${home }/ data/Qback.csv"
" reader_data_Qback .idfield" "ID"
" reader_data_Qback .tStart" "${startDate}"
" reader_data_Qback .tEnd" "${endDate}"
" reader_data_Qback .tTimestep" 60
" reader_data_Qback .fileNovalue" " -9999"
" reader_data_ETback .file" "${home }/ data/ETback.csv"
" reader_data_ETback .idfield" "ID"
" reader_data_ETback .tStart" "${startDate}"
" reader_data_ETback .tEnd" "${endDate}"
" reader_data_ETback .tTimestep" 60
" reader_data_ETback . fileNovalue " " -9999"
"pdfs.ID" 209

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"pdfs.tStartDate" "${startDate}"
"pdfs. tStartDateWaterBudget " "${startDate}"
"pdfs.tEndDate" "${endDate}"
"pdfs. inPathToDischarge " "${home }/ data/Q.csv"
"pdfs.inPathToET" "${home }/ data/ET.csv"
"writer_pQ.file" "${home }/ output/pQ_for.csv"
"writer_pQ.tStart" "${startDate}"
"writer_pQ.tTimestep" 60
"writer_pQ. fileNovalue " " -9999"
"writer_pET.file" "${home }/ output/pET_for.csv"
"writer_pET.tStart" "${startDate}"
"writer_pET.tTimestep" 60
"writer_pET. fileNovalue " " -9999"
" writer_theta .file" "${home }/ output/theta.csv"
" writer_theta .tStart" "${startDate}"
" writer_theta .tTimestep" 60
" writer_theta .fileNovalue" " -9999"
"writer_Q.file" "${home }/ output/Qfor.csv"
"writer_Q.tStart" "${startDate}"
"writer_Q.tTimestep" 60
"writer_Q.fileNovalue " " -9999"
"writer_ET.file" "${home }/ output/ETfor.csv"
"writer_ET.tStart" "${startDate}"
"writer_ET.tTimestep" 60
"writer_ET. fileNovalue " " -9999"
}
connect {
" reader_data_J .outData" "pdfs. inPrecipvalues "
" reader_data_pQ .outData" "pdfs. inPQ_backValues
"
" reader_data_pET .outData" "pdfs. inPET_backValues "
" reader_data_Qback .outData" "pdfs. inQ_tiValues "
" reader_data_ETback .outData" "pdfs. inET_tiValues "
"pdfs.outHMPQfor" "writer_pQ.inData"
"pdfs. outHMPETfor " "writer_pET.inData"
"pdfs.outHMtheta" " writer_theta .inData"
"pdfs.outHMQfor" "writer_Q.inData"
"pdfs.outHMETfor" "writer_ET.inData"
}
}
}
Data and Project
The following link is for the download of the input data necessaries to execute the com-
ponents (as shown in the .sim file in the previous section ) :
https://github.com/GEOframeOMSProjects/OMS_Project_TT/tree/master/ForwardPdfs/
data
The following link is for the download of the OMS project for the TT component:
https://github.com/GEOframeOMSProjects/OMS_Project_TT
%

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References
1. Niemi, A.J.: Residence time distributions of variable flow processes. The International Journal of Applied Radiation and
Isotopes 28(10), 855–860 (1977)