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WATER RESOURCES RESEARCH, VOL. 43, W03439, doi:10.1029/2004WR003913, 2007
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A simple model for predicting water table fluctuations
in a tidal marsh
Franco A. Montalto,1,2 Jean-Yves Parlange,1 and Tammo S. Steenhuis1
Received 17 December 2004; revised 7 March 2006; accepted 14 August 2006; published 27 March 2007.
[1] Wetland restoration efforts are ongoing in many urban estuaries. In this context the
hydrologic characteristics of restored wetlands are of paramount importance since the
spatially and temporally variable position of the water table and of soil saturation
establishes the oxidation state of the substrate, which, in turn, affects the wetland’s
biogeochemical composition and the biological communities it is capable of supporting. A
relatively simple analytical model developed here describes tidal marsh hydrology from
creek bank to interior, considering transient drainage, net meteorological inputs, and tidal
effects. Given a series of physical and time-dependent inputs, the analytical solution
derived predicts the position of the water table at points along a transect perpendicular to a
tidal creek. Validation of the model using water table time series data collected along three
transects at Piermont Marsh, a tidal wetland on the Hudson River in the New York/
New Jersey Estuary, indicates good general agreement between observations and
predictions, although it may not be precise enough for some kinds of ecological
applications. A sensitivity analysis on the model indicates that a range of pairs of
transmissivity and specific yield values that increase with distance from the creek results
in the same spatial and temporal fluctuations in the water table. This equifinality result is
discussed as it relates to the predictive capacity of the model presented.
Citation: Montalto, F. A., J.-Y. Parlange, and T. S. Steenhuis (2007), A simple model for predicting water table fluctuations in a tidal
marsh, Water Resour. Res., 43, W03439, doi:10.1029/2004WR003913.
1. Introduction
[2] In urban estuaries, filling, dredging, damming, and
bulkheading have historically resulted in dramatic alteration
of estuarine hydrology, with significant impacts to the
functioning of wetlands [Mitsch and Gosselink, 2000]. To
mitigate for wetland functional impairment, restoration,
creation, and enhancement efforts seek to reestablish selfperpetuating ecosystems with hydrologic regimes typical of
the surrounding region [Middleton, 1999]. One way to
improve the success of these efforts, is through study of
reference marshes. In particular, modeling of the hydrological processes occurring in relatively undisturbed reference
marshes can be helpful in assessing the impacts of ongoing
estuarine modifications on existing wetlands, and also in
prioritizing, planning, and designing regional wetland
improvement efforts [Niedowski, 2000; Shisler, 1990;
Zedler, 2001].
[3] In tidal areas, hydrology at least partially controls the
exchange of nutrients, organic matter, and pollutants
between upland watersheds, tidal wetlands and surface
waters [Gardner, 1975; Heinle and Flemer, 1976; Valiela
et al., 1978; Luther et al., 1982; Hemond et al., 1984;
Jordan and Correll, 1985; Yelverton and Hackney, 1986].
1
Department of Environmental and Biological Engineering, Cornell
University, Ithaca, New York, USA.
2
Now at Earth Institute at Columbia University, New York, New York,
USA.
Copyright 2007 by the American Geophysical Union.
0043-1397/07/2004WR003913$09.00
Recent hydrologic research in tidal wetlands has involved
the development of theoretical models suggesting linkages
between tidally and meteorologically driven groundwater
flows and both soil aeration rates [Ursino et al., 2004; Li
et al., 2005], and vegetation patterns [Silvestri and Marani,
2004], emphasizing the importance of groundwater flows
on tidal marsh ecology [Wilson and Gardner, 2005; Marani
et al., 2005]. In particular, the spatially and temporally
variable position of the water table establishes soil saturation patterns, which, in turn, affect the oxidation state and
biogeochemical composition of the substrate, as well as the
biological communities the wetland is capable of supporting. Together these factors determine many aspects of tidal
marsh ecohydrology.
[4] Although a variety of wetland hydrology models have
been attempted, comparison of these attempts is complicated
by intermarsh variability, the application of different modeling methodologies, and inconsistency in the focus of the
modeling efforts. The most basic attempts [Youngs, 1965;
Youngs et al., 1989] involve solution of basic drainage
equations with unsteady water tables assumed to behave
as a continuous succession of steady states. Application of
the classic, nonsteady state drainage models (described by
Ritzema [1994]) to wetland hydrology was not found in the
literature. Some [Parlange et al., 1984; Nielsen, 1990;
Barry et al., 1996; Li et al., 2000a, 2000b, 2002; Li and
Jiao, 2003] have used the well-known Boussinesq equation
to model the transient position of the water table in a
shallow, rigid aquifer in contact with a sinusoidally oscillating reservoir. Jeng et al. [2005] develop an analytical
solution to the Boussinesq equation to address spring-neap
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tide induced fluctuations in a sloping coastal aquifer. More
detailed hydrologic models have been derived to describe
various individual components of tidal marsh hydrology,
such as the vertical flow of groundwater in response to
evapotranspiration demand and the piezometric pressure of
an underlying aquifer [Hemond and Fifield, 1982], surface
infiltration [Hemond et al., 1984], horizontal fluxes [Nuttle
and Hemond, 1988], and the stress and pressure changes
due to tidal loading of the marsh surface [Reeves et al.,
2000]. Attempts to link together different hydrologic processes in wetland environments have been undertaken using
numerical models [Ursino et al., 2004; Wilson and Gardner,
2005; Li et al., 2005; Skags et al., 2005; Thompson et al.,
2004; Twilley and Chen, 1998]. However, few attempts to
validate wetland hydrology models in actual wetland environments have actually been performed.
[5] Validated hydrologic models can be used to better
understand the drivers of particular wetland hydrologic
patterns, or to compare the ecohydrological consequences
of different restoration scenarios in a given wetland. Invoking the Dupuit assumption and assuming uniform soil
properties, Harvey et al. [1987] found good agreement
between a finite difference solution to the Boussinesq
equation, and observations of tidally induced water table
fluctuations made in a narrow and regularly flooded Chesapeake Bay tidal marsh. Input parameters, initial and
boundary conditions, were derived from field measurements. The model was validated with four days of piezometric head observations collected on a 15-min time
interval. Considering horizontal drainage and net atmospheric fluxes, Nuttle [1988] found an analytical solution
to the one-dimensional Boussinesq equation, also assuming
uniform soil properties, but using observations in an irregularly flooded Massachusetts marsh for validation. Initial
and boundary conditions were obtained from curve fitting,
substrate properties were derived from field measurements,
and the fluctuating creek stage was approximated by the
time-averaged creek level and a periodic component reflecting the influence of semidiurnal tides. Good agreement was
reported between model predictions and piezometric head
observations, although the observations used for validation
were recorded only once every other day for a 2-month
period.
[6] In this paper, we develop a simple, yet robust,
analytical groundwater model and use data from Piermont
Marsh (Rockland Co., NY), a brackish tidal marsh located
on the Hudson River in the NY/NJ Estuary [Montalto et al.,
2006] to discuss its validity. The need for research in
reference marshes in this geographic region is explained
in Montalto and Steenhuis, [2004]. A computer program
was devised to run model simulations. Given a series of
physical and time-dependent inputs, the model predicts the
position of the water table at points along a transect
perpendicular to a tidal creek. The model is used for an
equifinality analysis to explore the sensitivity of combinations of input parameters to represent the observed water
table height. The approach differs from both Harvey et al.
[1987] and Nuttle [1988] in terms of the initial condition of
the water table considered. (For example, Nuttle [1988]
used the complimentary error function to represent the
initial condition in his analytical solution.) Additionally,
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the validation process employs a much longer set of
observations, collected at a much finer time step.
2. Site Description and Methodology
[7] Piermont Marsh is an irregularly flooded, brackish
tidal marsh located approximately 40 km north of New York
City along the western bank of the Hudson River (Figure 1).
The vegetation of Piermont Marsh is a combination of
Spartina alterniflora, along the edges of some creeks and
the Hudson River coastline; invasive Phragmites australis,
occupying approximately 75% of the overall marsh surface;
and the typical mix of grasses, sedges, and cattails found in
high marsh environments along the northeastern coast of the
United States. Tidal fluctuations in this span of the Hudson
River are sinusoidal and semidiurnal, with a tidal range of
approximately 1.1 m. Salinity is in the mesohaline range.
[8] Various data were collected at Piermont Marsh to
characterize the site hydrology [Montalto et al., 2006] and
to validate the model. Tide gages were installed in the
creeks and Hudson River. The elevation of mean high water
in this span of the river is 0.83 m in the 1929 National
Geodetic Vertical Datum, (NGVD-29). Topographic surveying, conducted using real-time kinematic GPS and a laser
plane unit, revealed a primarily flat marsh surface, at an
average overall elevation of 0.95 ± 0.05 m NGVD-29. A
soil core extracted from the interior of Piermont Marsh
uncovered peat extending from the surface to a depth of 3 m,
below which an organic silty clay layer extends almost
completely uninterrupted to a depth of 9.5 m below the
surface. Basal sediments were not encountered in this, the
only soil core attempted in the marsh [Wong and Peteet,
1999]. The saturated hydraulic conductivity was measured
using the auger hole method. The average saturated hydraulic conductivity of the creek bank, 6.6 104 cm/s, is lower
than the overall marsh average, 7.75 103 cm/s [see
Montalto et al., 2006, Figure 3].
[9] The total depth of precipitation falling every 10 min
was measured continuously using a rain gauge. Daily
potential evapotranspiration (PET) rates were estimated by
the Northeast Regional Climate Center using MORECS, a
model employing the Penman-Monteith equation, and utilizing data measured at White Plains, NY, approximately
10 km due east of Piermont Marsh.
[10] Pressure transducers were used to measure the position of the water table elevation in a series of wells installed
along three different transects at Piermont Marsh (Figure 1).
Known as transects 2, 3, and 4, they are referred to in this
paper by the names used to describe them in Montalto et al.
[2006]. Wells were positioned along each transect at 6, 12,
18, 24, 36, and 48 m from the creek bank levee. Data
loggers were used to record the position of the water table
simultaneously for approximately one lunar month in
all wells along each of the transects. Data was recorded at
10-min intervals. Measurements were made from 24 April
through 22 May 1999 along transect 2, 25 May through
22 June 1999 along transect 3, and 28 November through
26 December 2000 along transect 4. Transects 2 and 3 were
located perpendicular to an approximately 4 m wide,
unnamed tidal creek located in the northern half of Piermont
Marsh. Transect 4 was situated perpendicular to the slightly
wider Crumkill Creek, located in the marsh center
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Figure 1. Piermont Marsh, located approximately 40 km north of New York City along the western
bank of the Hudson River. Transects 2 and 3 are along an unnamed tidal creek at the northern end of the
marsh. Transect 4 is off Crumkill Creek at the southern end of the marsh.
(Figure 1). Both creeks were less than 2 m deep at mean
tide.
[11] A more complete description of the vegetation,
substrate characteristics, tidal range, and hydrology of the
site is provided elsewhere [Montalto et al., 2006]. A brief
description of its hydrology follows. Figures 2, 3, and 4,
adapted from Montalto et al. [2006], depict several weeks of
water table observations in all wells along transect 2 in
spring, transect 3 in summer, and transect 4 in winter
conditions. The elevations are reported in height above
the 0 m NGVD-29 datum. Different colors are used to
designate the water table elevation at different well locations, with gray and dark blue used for the creek stage and
levee well, respectively. Figures 2, 3, and 4 also display the
zone of 95% confidence, a measure of the error incurred by
pressure transducer calibration, which was in almost all
cases, less than ±1 cm. Uncertainty of elevations surveyed
using RTK GPS techniques accounts for another error of
0.5 to 2 cm. Thus, when pressure transducer noise is not a
factor, the observed water table elevations may, in general,
be considered accurate to within ±3 cm.
[12] As illustrated in Figures 2, 3, and 4, spring tides and
other meteorologically induced high tides periodically inundate and saturate the surface of Piermont Marsh. Inundations along transect 2 occurred, for example on 1 – 3 May
and then not again until 14 –20 May. As described below
and illustrated in Figure 5, immediately after the recession
of inundating tides, a sloped water table is observed in wells
located within 24 m of all tidal creeks. The water table is
nearly flat and close to the surface further inland.
[13] In between inundations, water is lost from the marsh
to both evapotranspiration and horizontal drainage to the
creek. These losses cause the water table to drop throughout
the marsh. Because surface inundation occurs several times
each month, and losses are relatively small, the water table
always remains within 10 cm of the surface, across the
entire marsh interior (defined, for the purpose of this paper,
as all areas located at least 36 m from creek banks), even
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Figure 2. (top) Water table observations (NGVD-29) and (bottom) net meteorological conditions
(P-PET) in centimeters per 10-min interval along transect 2. The average marsh surface elevation is
at 1.02 m NGVD-29 and is indicated with a dotted line.
throughout prolonged dry periods in the summer. The water
table drops up to 30 cm below the marsh surface in portions
of the marsh located closer to creek banks.
[14] Tidal fluctuations in the Hudson River and connected
creek networks cause periodic, semidiurnal, and tidally
induced fluctuations in the subsurface marsh water table.
Fluctuations of up to 30 cm were observed in creek bank
wells. Rarely were water table fluctuations in excess of 5 cm
observed in wells located further than 6 m from the creek
bank. No tidal fluctuations were noted further than 18 m
from creek bank. [Montalto et al., 2006].
[15] More sporadic fluctuation of the water table was
caused by noninundating high tides, such as at the levee
and 6 m locations along transect 2 on 27 April, or at the levee,
6 m, and 12 m locations along transect 3 on 28 May. During
these instances, the water table rose with the incoming tide,
but then subsided much more slowly as the tide ebbed. We
speculate that this hysteresis effect is caused when tidal water
that entered the marsh aquifer through the large fiddler crab
holes and other macropores is forced, subsequently, to drain
out to the creek through the soil matrix. Other potential
explanations include anisotropic hydraulic conductivity and
nonuniform soil properties along the transect.
[16] Rainfall increased the water table elevation most
prominently in the creek bank region, but did not appear
to significantly alter the water table of the marsh interior
(for example, see transect 2, on 8 May). This was likely the
result of a larger specific yield in the organic rich marsh
interior compared with the denser creek banks.
3. Model Development
[17] An analytical model was developed to simulate
spatial and temporal water table fluctuations in an irregularly flooded tidal marsh, bounded by two, parallel creeks.
The solution derived is symmetric about the marsh midpoint. Evapotranspiration and precipitation vary temporally
but were assumed not to vary spatially across the marsh.
The substrate was uniform with an inelastic soil matrix, and
a uniform specific yield. A no flow boundary condition was
located between the upper peat strata and the underlying
organic, silty clay horizon.
[18] On the basis of observations of small differences in
nested piezometer readings and increasing salinity with
depth, groundwater upwelling appeared to be minimal. A
downward gradient of 0.02 m/m was measured during and
after spring tide inundations. A similar gradient in the
opposite direction was noted during long dry periods between
tide inundations. Measured salinity levels were 8 –10 mg/L at
the surface and 14– 16 mg/L at 2.5 m depth. If groundwater
upwelling were important the deeper piezometers would have
demonstrated lower salt concentrations than at the surface.
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Figure 3. (top) Water table observations (NGVD-29) and (bottom) net meteorological conditions
(P-PET) in centimeters per 10-min interval along transect 3. The average marsh surface elevation is
at 1.03 m NGVD-29 and is indicated with a dotted line.
[19] Flow normal to the transects was ignored. Moreover,
since horizontal distances were much larger than the depth
of the aquifer, Dupuit assumptions were valid. Flow could
therefore be modeled in one dimension perpendicular to the
creek bank.
[20] Because of both the small depth of the unsaturated
zone and frequent inundations, we assumed that moisture
content in the unsaturated zone remained constant and equal
to the saturated moisture content minus the specific yield.
The actual value of the saturated moisture content is not
important though, since only specific yield appears in the
equations.
[21] In addition, inundation of the marsh surface as well
as surface runoff are assumed to take place instantaneously.
Since groundwater was only moderately saline (<16 mg/L
NaCl), density flow effects were ignored.
[22] The governing equation is the Boussinesq equation
[Boussinesq, 1877] for unsteady flow in a phreatic aquifer
with accretion. Given the above assumptions and linearizing
(because changes in water table depth are small compared to
the overall depth of groundwater, this equation can be
written as follows:
Sy
@d
@2d
¼wþT 2
@t
@x
ð1Þ
where Sy is the specific yield [L3/L3], t is the time [T], d is
the elevation of the water table [L], T is the aquifer
transmissivity [L2/T], x is the coordinate perpendicular to
the creek [L] and w is the net accretion [L/T]. This equation
is similar in form to the classical, one-dimensional, linear
heat flow equation, for which many solutions have been
derived [see Carslaw and Jaeger, 1959].
[23] To solve equation (1) directly, a solution must be
found that meets the following conditions:
d ð x ¼ 0; t Þ ¼
1
X
Ai cosðwi t ai Þ
ð2Þ
i¼1
@d x ¼ L2 ; t
¼0
@x
ð3Þ
d ð x; t ¼ 0Þ ¼ f ð xÞ:
ð4Þ
[24] The levee water table is used as the creekside flow
boundary (x = 0) to avoid the difficulty of modeling seepage
face phenomena on the creek bank wall, and also to avoid
the need to model radial flow from the creeks since the
creek bed does not rest on the impermeable lower boundary
to flow. Regarding the latter, radial flow would invalidate
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Figure 4. (top) Water table observations (NGVD-29) and (bottom) net meteorological conditions
(P-PET) in centimeters per 10-min interval along transect 4. The average marsh surface elevation is
at 1.01 m NGVD-29 and is indicated with a dotted line.
the Dupuit assumptions. According to Hooghoudtp[1940],
ffiffiffi
radial flow occurs within a distance equal to D/ 2 back
from the middle of a partial penetrating creek. At this site,
the aquifer thickness, D, is approximately equal to 3 m,
which, as noted before, is the distance between the average
water level in the levee well and the boundary of the highly
permeable peat and the organic silty clay layer horizon with
a low conductivity. To ensure validity of the Boussinesq
equation, therefore, the boundary condition needed to be
located at least 2.1 m from the center of the creek, a
condition met by selecting the creek bank levee since both
creeks were wider than 4 m.
[25] Equation (2) is the periodic boundary condition at
the levee well due to the tides with Ai, wi and ai representing
the amplitude, angular velocity, and phase angles, respectively as a Fourier series. Equation (3) is the ‘‘no-flux’’
condition at the marsh midpoint. The function, f(x), in
equation (4) describes the initial water table profile, defined
as the water table profile observed in the marsh immediately
after the ebbing of an inundating tide. Although it is not
general, a linear, sloping segment (Figure 5) has been
shown to represent reasonably and sufficiently the post-
inundation water table profile in the creek bank region
[Montalto et al., 2006]. Accordingly, it is defined as
follows:
f ðxÞ
¼ mx þ b
¼ mx1 þ b
0 < x < x1
x1 < x < L=2
ð5Þ
where L is the distance between two tide creeks, as
illustrated in Figure 6, and x1, b, and m are fitting
parameters.
[26] Because equation (1) is linear, the original problem
can be decomposed into three, individually modeled processes, the solutions of which are added, yielding a time and
space dependent solution for the position of the marsh water
table, d(x, t), vis-à-vis:
d ð x; tÞ ¼ d1 ð x; t Þ þ d2 ð x; t Þ þ d3 ð x; t Þ þ davg
ð6Þ
where d1(x, t), d2(x, t) and d3(x, t) represent the individual
contribution to water table height of respectively the
following processes: horizontal drainage flow to the creek
(process 1 shown in Figure 6, top); the net effect of
precipitation and evapotranspiration on the water table
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model simulation is described separately below. A typical
model simulation started at the beginning of the observations (t = 0), and continued through the end of the
observation period, a period that spanned at least one full
lunar month. Each inundation event during the observation
period reset the initial conditions, as will be described in the
following.
3.1. Process 1: Horizontal Drainage
[28] The water table profile established in the marsh upon
recession of inundating tidal waters is similar after each
Figure 5. Immediate post inundation water table profile
(t = 0) used for the initial conditions for flow process 1
along (top) transect 2, (middle) transect 3, and (bottom)
transect 4. It was derived from well observations made after
several spring tide inundation events and is represented by
two line segments (equation (9)).
(process 2 shown in Figure 6, middle); and the tidally induced
oscillations of the marsh water table (process 3 shown in
Figure 6, bottom). The davg, is used so that the water table can
be given in height above the NGVD-29 datum. The boundary
and initial conditions for individual water table heights are
determined such that d(x,t) in equation (6) meets the
conditions set forth in equations (2) – (5).
[27] Each of these three processes with boundary and
initial conditions and the computer program used to run the
Figure 6. Graphical illustration of the three processes (not
to scale). (top) Process 1, which simulates horizontal
sideways drainage toward the creek, (middle) process 2,
which simulates the effects of meteorological fluxes across
the marsh surface, and (bottom) process 3, which models
the propagation of tidal pulses in the aquifer.
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flooding event, and decays subsequently as a result of
horizontal drainage to the creek. Meteorological fluxes
through the marsh surface are not considered in this process.
Equation (1) is solved by separation of variables with the
following conditions:
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in horizontal flow. Then, separating variables and integrating, we obtain
Zt
d2u ðt Þ ¼ d2u ðt ¼ 0Þ þ
wðt Þ
dt
Sy
ð14Þ
0
d1 ð x ¼ 0; t Þ ¼ 0
ð7Þ
@d1 x ¼ L2 ; t
¼0
@x
ð8Þ
d1 ð x; t ¼ 0Þ ¼ f ð xÞ where f ð xÞ ¼ mx þ b 0 < x < x1
:
f ð xÞ ¼ mx1 þ b x1 < x < L=2 ð9Þ
The solution is written as
1
X
ð2n þ 1Þ2 p2 t T
d1 ðx; t Þ ¼
Cn exp
L2
Sy
n¼0
!
sin
ð2n þ 1Þpx
L
d2c x; t ¼ tj ¼ hoj
ð10aÞ
where; Cn ¼
4
ð2n þ 1Þp
bþ
mL
ð2n þ 1Þpx1
sin
ð2n þ 1Þp
L
where d2u(t) is the cumulative sum of all individual net
meteorological fluxes through time, t, and w(t) is equal to
the precipitation, P(t), minus the total potential evapotranspiration, PET(t), at time, t. The potential, and not actual,
evapotranspiration values were used because the marsh
substrate, even above the water table, was almost always
near saturation during the observation period.
[31] 2. Equation (14) does not meet the boundary condition at x = 0. To adjust the solution so that it meets the
boundary condition imposed by equation (11), the amount
of water gained and lost through the creek bank due to
evaporation and rainfall is calculated by solving equation (1)
by separation of variables, to satisfy equation (11),
equation (12), and the following:
;
ð10bÞ
where hoj equals the total daily precipitation minus the total
daily evapotranspiration on day j. This operation yields the
Glover Dumm equation [Dumm, 1954; Ritzema, 1994]:
1
4hoj X
ð2n þ 1Þpx
1
sin
p n¼0 2n þ 1
L
!
2 2
ð2n þ 1Þ p t tj T
exp
Sy
L2
d2cj ð x; t Þ ¼
x is the distance from the levee well and all other parameters
are as defined previously.
3.2. Process 2: Meteorological Effects
[29] This solution quantifies the net effect of evapotranspiration and precipitation on the marsh water table profile.
The Kraijenhoff van de Leur [1958] solution did not
converge for the small time steps used here and, consequently, a more cumbersome two-step procedure is used.
The solution needed to meet the following conditions:
d2 ð x ¼ 0; t Þ ¼ 0
ð11Þ
@d2 x ¼ L2 ; t
¼0
@x
ð12Þ
d2 ðx; t ¼ 0Þ ¼ 0:
ð13Þ
In the first step of the solution, the net meteorological flux is
initially assumed to act uniformly over the whole transect
water table. The second step involves adjusting the solution
once a day to satisfy equation (11). A daily adjustment is
considered appropriate because the small error introduced
during part of the day and the additional computational
burden when more frequent adjustments are made. The
equations derived for each of these two steps are elaborated
below. Their usage in the computer simulation is described
in the section 3.4.
[30] 1. To compute the uniform effects of precipitation
and evapotranspiration, the second term on the right side of
equation (1) is set equal to zero, because a uniform increase
or decrease in the elevation of the water table does not result
ð15Þ
ð16Þ
where d2c(x, t) represents the position of the transect water
table as a result of all net meteorological fluxes through the
marsh surface and all associated horizontal movement of
water through the creek bank, occurring during a given day;
t is the time elapsed since the beginning of the current
interval of the simulation; and tj is the time elapsed between
the beginning of the current interval of the simulation and
the beginning of day j (for example: tj = 0 when j = 1).
3.3. Process 3: Tidal Effects
[32] This process simulates the propagation of creek stage
fluctuations in the marsh aquifer. The tidal fluctuations
observed in the creek bank levee well are used as boundary
condition.
[33] Many solutions to this problem have been derived
[Steggewentz, 1933; Todd, 1980; Knight, 1981; Parlange
et al., 1984; Nuttle, 1988; Hughes et al., 1998; Sun, 1997;
Li et al., 2000a, 2000b, 2002; Li and Jiao, 2003]. The
solution derived here follows the linearized approach presented by Sun [1997]. It involves solving equation (1) with
w = 0 and according to the following conditions:
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d3 ð x ¼ 0; t Þ ¼
m
X
Ai cosðwi t ai Þ
ð17Þ
i¼1
@d3 x ¼ L2 ; t
¼0
@x
ð18Þ
d3 ð x; t ¼ 0Þ ¼ 0:
ð19Þ
MONTALTO ET AL.: MODEL OF MARSH WATER TABLE FLUCTUATIONS
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Figure 7. Observed and predicted water table elevations at (a) 6, (b) 12, (c) 18, (d) 24, and (e) 36 m
from the creek bank along transect 2. The y axis is in meters above the zero point on the NGVD-29
datum. The 48 m data are not shown because of missing observed data at this location due to pressure
transducer malfunction. The average marsh surface elevation along transect 2 is 1.02 m. The parameter
set values are given in Table 3.
[34] The boundary condition expressed by equation (19)
cannot be met but, as discussed later, the error associated
with this approximation is small.
[35] As mentioned previously, the water table fluctuations
in the levee well are converted into a cosine series for use as
the boundary condition at x = 0 (equation (17)). Equation (1)
is solved by writing the boundary condition in equation (17)
as the real part of an exponential function and by then
assuming a similar form for d3(x, t) that meets the boundary
conditions. The resulting solution is the ‘‘tidal solution’’
depicted graphically in Figure 6 (bottom):
d3 ð x; t Þ ¼
m
X
j¼1
x
Aj e
pffiffiffiffiffi
w Sy
j
2T
!
rffiffiffiffiffiffiffiffiffi
wj Sy
cos x
þ wj t aj :
2T
ð20Þ
[36] The value of d3(x, t) per equation (20) decreases
exponentially with increasing x, limiting the distance inland
that tidally induced fluctuations in the water table can be
transmitted. Assuming tidal characteristics, transmissivity,
and specific yield values typical for this kind of site, we can
expect d3 (x, t) ! 0 for x > 20 m. Therefore the boundary
condition in equation (18) is satisfied for all three Piermont
Marsh transects. The value of d3 (x, t = 0) 6¼ 0 as it should
per equation (19). However, this value also decreases
exponentially with x. As will be discussed below, the error
it causes is negligible, because its magnitude for all x > 0
will always be smaller than the amplitude of the tidal
fluctuations considered.
3.4. Model Simulation
[37] A computer program was developed to run the
model. Using the creek bank levee well observations as
the boundary at x = 0, water table fluctuations at 6, 12, 18,
24, 36, and 48 m along transects 2 – 4 were simulated. The
10-min time step of the simulation was equal to the interval
over which water table observations and precipitation measurements were made. The total daily PET values were
converted into 10-min incremental values by distributing
them between sunrise and sunset using a sinusoidal function. On each 10-min time step of the simulation, d1, d2, and
d3 were computed individually for each distance, x, from the
creek bank levee well being simulated. The predicted water
table position, d(x, t), was the superposition of d1, d2, and
d3, plus the average elevation of the levee boundary
condition, davg, for that interval as per equation (6).
[38] On any time step, tk, during which the elevation of
the boundary condition at x = 0, i.e., d3(x = 0, tk) + davg in
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Figure 8. Observed and predicted water table elevations at (a) 6, (b) 12, (c) 18, (d) 24, (e) 36, and
(f) 48 m from the creek bank along transect 3. The y axis is in meters above the zero point on the
NGVD-29 datum. The average marsh surface elevation along transect 3 is 1.03 m. The parameter set
values are given in Table 3.
process 3, was found to be greater than the average marsh
surface elevation along the transect, the marsh was assumed
inundated, and d(x, tk) was set equal to the water level at
x = 0. Then when the elevation of the boundary condition at
x = 0 fell below the average transect marsh surface elevation, time was reset to zero, all initial and boundary
conditions reset for the three processes, and a new davg
and set of harmonic constants in equation (17) were
computed for the x = 0 process 3 boundary condition for
the time interval ending at the next surface inundation
event. Extensive amounts of precipitation also triggered
the resetting of the initial conditions if more than three
of the water table predictions made along the transect
exceeded the elevation of the marsh surface.
[39] If along the transect, d(x, t) per equation (6), was less
than the quantity d3(x = 0, t) + davg (the elevation of the
levee water table), and d(x, t) was also less than the elevation
of the average marsh surface elevation, then d(x, t) was set
equal to the elevation of the boundary condition at x = 0.
This stipulation was an ad hoc attempt to model the effect of
fast drainage through preferential flow pathways that extend
from the creek bank into the marsh substrate.
[40] At the beginning of each time interval between the
inundation events, the computer program computed the
average value of the levee water table, davg, for that interval.
The program also used Fourier analysis to derive all the
harmonic constants, Ai, wi, and ai, in equation (17) to
represent the levee well data on that interval.
[41] We will treat in detail here how we calculated d2
since the computational steps were cumbersome. Equations
(14) and (16) were used to compute d2 as follows: Starting
on day 1 for each 10 min time step d2u(x, 0 < t < 1) was
computed directly from equation (14). At the end of the day,
equation (16) was used to reconcile any deviations from the
prescribed height at x = 0, as described earlier, with ho1 set
equal to the cumulative sum of all 10-min w(t) values
occurring on day 1. The simulation then continued
by augmenting d2c1(x, t = 1) stepwise, by the amount,
d2u(1 < t < 2), such that d2(x, 1 < t < 2) = d2c1(x, t t1 =
1) + d2u(1 < t < 2), with the history of d2u extending back to
the beginning of day 2. At t = 2 days, d2(x, t = 2) is again
calculated directly, this time by solving equation (16) twice,
considering that two days have passed since the effects of
ho1 were initiated, and one day has passed since the onset of
ho2 and so on.
[42] Several different sets of simulations were performed.
In the first simulation set, (the initial ‘‘model runs,’’) input
parameters were selected based on values measured at the
site or derived from the literature. The second set of
simulations constituted a sensitivity analysis on the model,
for which selected input parameters were varied above and
below the initial ‘‘model run’’ values, as described in the
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Figure 9. Observed and predicted water table elevations at (a) 6, (b) 12, (c) 18, (d) 24, (e) 36, and
(f) 48 m from the creek bank along transect 4. The y axis is in meters above the zero point on the NGVD-29
datum. The average marsh surface elevation along transect 4 is 1.01 m. The parameter set values are given
in Table 3.
following. Finally, the model was run with the input
parameter sets that the sensitivity analysis suggested would
lead to the lowest error, and the results compared.
3.5. Estimation of Model Input Parameters
[43] To run the model, input parameters and boundary
conditions needed to be specified. Because of the precision
of the surveying techniques employed, the marsh width, and
average marsh surface elevation were easily measured. The
Fourier analysis used to approximate the levee water table
with a cosine series also produced negligible errors. The
fitting parameters used to approximate the process 1 initial
condition water table profile were also easily and accurately
estimated based on field observations (Figure 5). The
boundary condition at the midpoints between the creeks
for transects 2, 3, and 4 were at 269 m, 188 m, and 559 m,
respectively.
[44] Estimation of the specific yield and substrate transmissivity was more difficult. The specific yield is a particularly difficult property to measure, and can demonstrate
considerable spatial variability in the landscape [Ritzema,
1994]. The transmissivity is the product of the depth of the
aquifer and the saturated hydraulic conductivity, both of
which are not known with a high degree of accuracy. For
these reasons, we opted to estimate these parameters for the
initial ‘‘model run’’. In the subsequent sensitivity analysis
on the model, we varied them independently and systematically to test our initial assumptions and also to determine
which pairs of specific yield and transmissivity values
yielded reasonable predictions along the three different
transects. The procedure employed in selecting the initial
‘‘model run’’ parameters and sensitivity ranges for these
values is described below.
[45] The initial ‘‘model run’’ estimate for the specific
yield was based on literature values. Specific yield values
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Table 1. Variability of Standard Error, Mean Deviation, Mean Cumulative Error, and Maximum Deviation With Different Pairs of
Transmissivity and Specific Yield at Each Distance on Each Transecta
Transect 2b
Distance,
m
6
6
6
6
12
12
12
12
18
18
18
18
24
24
24
24
36
36
36
36
48
48
48
48
ID
m.r.
1
2
3
m.r.
1
2
3
m.r.
1
2
3
m.r.
1
2
3
m.r.
1
2
3
not available
Transect 3c
SE,
cm
MD,
cm
MCE,
cm
MAX,
cm
6.5
2.9
2.4
2.4
5.6
3.2
3.2
3.8
4.5
2.2
2.3
3.6
4.2
2.2
2.2
3.6
2.6
2.4
1.6
2.4
5.6
2.5
2.0
1.9
4.5
2.6
2.4
2.9
3.6
1.8
1.9
2.9
3.7
1.9
1.9
3.0
2.4
2.0
1.4
2.0
4.9
0.8
0.4
0.6
3.7
0.1
0.5
1.9
3.5
0.4
0.3
2.0
3.7
0.8
0.1
1.6
2.3
1.1
0.8
0.6
12.1
8.2
8.4
8.8
13.2
13.1
12.5
12.5
12.2
12.2
12.2
12.2
11.0
11.0
11.0
11.0
11.9
11.9
11.9
11.9
Transect 4d
ID
SE,
cm
MD,
cm
MCE,
cm
MAX,
cm
ID
SE,
cm
MD,
cm
MCE,
cm
MAX,
cm
m.r.
4
5
6
m.r.
4
5
6
m.r.
4
5
6
m.r.
4
5
6
m.r.
4
5
6
m.r.
4
5
6
4.6
5.2
4.2
4.9
3.8
3.8
3.9
3.8
2.4
2.4
2.5
2.7
2.2
2.5
3.0
2.9
2.3
2.1
2.4
2.2
2.2
2.4
2.2
2.6
3.7
4.3
2.9
4.0
2.6
2.7
2.8
3.1
1.6
1.7
1.8
2.2
1.6
2.0
2.4
2.4
2.0
1.8
2.0
1.8
1.8
2.0
1.8
2.2
0.55
1.1
0.8
2.9
1.4
0.4
1.8
0.5
1.2
0.1
1.5
0.7
1.0
0
2.3
1.08
1.8
0.5
1.9
1.6
1.8
0.3
1.0
1.67
19.7
19.7
19.7
19.7
21.1
21.1
21.1
21.1
19.2
19.2
19.2
19.2
16.7
16.7
16.7
16.7
17.0
17.0
17.0
17.0
17.2
17.2
17.2
17.2
m.r.
7
8
9
m.r.
7
8
9
m.r.
7
8
9
m.r.
7
8
9
m.r.
7
8
9
m.r.
7
8
9
4.2
4.4
3.0
3.3
3.0
3.0
1.9
2.0
2.6
2.4
1.7
1.4
2.9
2.8
2.6
2.4
2.3
2.1
2.2
2.2
2.7
2.4
2.6
2.7
3.3
3.5
2.4
2.7
2.4
2.3
1.5
1.7
2.0
1.8
1.4
1.2
2.5
2.4
2.2
2.1
2.0
1.8
2.0
2.0
2.2
2.1
2.2
2.3
2.8
3.1
0.5
1.3
1.4
1.4
0.3
0.8
1.7
1.5
1.1
0.8
2.1
2.0
1.8
1.7
1.9
1.7
1.9
1.9
2.2
2.0
2.2
2.2
14.2
14.7
13.0
14.1
7.3
7.1
5.6
6.3
5.7
5.2
5.2
5.2
7.0
7.0
7.0
7.0
7.2
7.2
7.2
7.2
7.6
7.5
7.6
7.7
a
SE, standard error; MD, mean deviation; MCE, mean cumulative error; MAX, maximum deviation; T, transmissivity; SY, specific yield; m.r., initial
‘‘model run’’ with T = 1.7 m2/d and SY = 0.6 for all transects.
b
Transect 2 for point 1, T = 9.0 m2/d and SY = 0.3; point 2, T = 30 m2/d and SY = 0.7; point 3, T = 70 m2/d and SY = 0.9.
c
Transect 3 for point 4, T = 0.7 m2/d and SY = 0.42; point 5, T = 0.007 m2/d and SY = 0.6; point 6, T = 20 m2/d and SY = 0.94.
d
Transect 4 for point 7, T = 0.5 m2/d and SY = 0.2; point 8, T = 4.0 m2/d and SY = 0.6; point 9, T = 8.07 m2/d and SY = 0.98.
reported in the literature for peat range from 0.09 to 0.84,
with higher values more typical of undecomposed or living
peat, and lower values more representative of decomposed
strata [Price and Schlotzhauer, 1999]. Boelter [1965] also
measured the specific yield associated with varying degrees
of decomposition for peat samples taken in Minnesota bogs
and reported values of 0.52– 0.79 for the top 25 cm of
undecomposed sphagnum mosses. Because the water table
fluctuations in Piermont Marsh occurred in the largely
undecomposed, peaty root zone, we estimated a specific
yield of 0.6 for use in the initial ‘‘model run’’ predictions
along the three transects. For the sensitivity analysis, the
specific yield was varied at 0.02 unit intervals from 0.02 to
0.98. Although this range likely includes values that were
too high and too low to describe the peat at Piermont Marsh,
they represent the entire range of specific yield values
reported for peat in the literature.
[46] Estimates of the aquifer depth and the saturated
hydraulic conductivity are needed to calculate transmissivity.
Assuming the top of the organic silty clay layer as an
impermeable boundary to flow, an aquifer depth of 3 m was
assumed. Because the saturated hydraulic conductivity
values measured in the creek bank substrate were lower
than those measured more internally [Montalto et al., 2006],
it was assumed that the creek bank rates control horizontal
flux. Ks values measured in the creek banks of transects 2
and 3, were 0.46 103 and 0.87 103 cm/s, respectively (no measurements were made along transect 4). The
average of these two values, 6.6 104 cm/s, multiplied by
the 3 m effective substrate depth, results in an estimated
transmissivity of 1.7 m2/d, the value used in the initial
‘‘model run’’ for each transect.
[47] In the sensitivity analysis the transmissivity values
were varied to reflect a range of Ks values for marsh peat
found in the literature (see summary by Montalto et al.
[2006]) and also a range of effective depths extending from
1 m to the 12 m deep core extracted by Wong and Peteet
[1999]. T was varied on 1 m2/d intervals from a low of 5 103 m2/d, (based on an effective depth of 1 m and a
saturated hydraulic conductivity of 6.3 106 cm/s) to
25 m2/d and on 5 m2/d intervals from 25 m2/d to a high of
265 m2/d, (based on an aquifer depth of 12 m and a
conductivity of 2.6 102 cm/s). The higher-resolution
sampling for lower transmissivities is because of greater
variability of the results in that range, as will be shown later.
Table 2. Percent of All Timesteps During Which the Preferential
Flow Approximation Was Employed
Percent of All Timesteps
Distance, m
Transect 2
Transect 3
Transect 4
6
12
18
24
36
48
6.5
2.5
1.1
0.2
0.2
0.2
7.6
4.1
2.1
0.4
0.2
0.2
4.1
2.1
1.2
0.5
0.5
0.5
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Figure 10. Results of the standard error sensitivity analysis for transect 2.
Varying the specific yield and transmissivity incrementally
within these ranges, 3626 simulation runs were made for
each well location along each transect.
4. Results
[48] To assess the closeness of fit of the predictions to the
observations, and to assess whether the model has a
tendency to overpredict or underpredict the water table
elevation, three error calculations were computed: the
standard error, the mean deviation, and the mean cumulative
error. The formulas used to compute these are provided in
equations (21) – (23):
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
ðoð x; t Þ d ð x; t ÞÞ2
SE ¼
n2
MD ¼
P
joð x; tÞ d ð x; t Þj
n
P
ðoð x; t Þ d ð x; tÞÞ
MCE ¼
n
ð21Þ
ð22Þ
ð23Þ
where o(x, t) are the observations, d(x, t) are the predictions,
n equals the number of observations in the simulation, and
SE, MD, and MCE refer to the standard error, the mean
deviation, and the mean cumulative error, respectively. The
rationale for computing these three errors is as follows: The
standard error and the mean deviation are both used to
assess the model’s general goodness of fit. The mean
cumulative error allows tracking of the direction in which
the model is biased. Also computed to aid in model
validation, is the time step corresponding to the maximum
absolute difference (MAX) between the predicted and
observed water table elevations for each well in each
simulation. This is used to identify when the model
predictions deviate most from the observations.
4.1. Model Runs
[49] The predicted water table heights for the ‘‘model
runs’’ of transects 2, 3, and 4 are shown in Figures 7, 8,
and 9. Table 1 lists the all the computed errors. The standard
errors in the model run predictions were under 3 cm (the
accuracy of the observations) for the 36 m well along
transect 2, all wells located 18 m from the creek bank
along transect 3, and all wells 12 m from the creek bank
along transect 4. The remaining wells displayed slightly
higher errors.
[50] As expected, the computed mean deviations are
slightly smaller than the standard errors computed, for a
given simulation. The mean cumulative errors computed
ranged from 4.9 cm to 0.55 cm. In all except the
predictions made at the 6 m well location along transect 3,
the mean cumulative errors were negative, indicating that
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Figure 11. Results of the standard error sensitivity analysis for transect 3.
with the initial ‘‘model run’’ input parameters, the model
overestimates the position of the water table, along all three
transects. The simulations that most overpredict (and most
underpredict) the water table are predictions made at 6 m
locations, indicating greater overall bias in model predictions at those locations.
[51] The maximum deviation between observed and predicted water tables varied in the simulations from approximately 6 to 21 cm, and corresponded usually with marsh
surface inundation. For example, all of the transect 3
maximum deviations occurred on 9 June between 5:30
and 6:30 PM. This is because the model assumes that the
whole marsh inundates instantaneously, while in reality the
inundation advances with the rising tide. In addition, large
deviations of the 6 m, 12 m, 18 m, and 24 m predictions
along transect 2 were caused by model underpredicting the
drawdown between inundation events.
[52] Table 2 lists the number of time steps over which the
model invoked fast drainage through preferential flow paths
for each transect. At most, the preferential flow approximation was used on 7.6% of the predictions made on the 6 m
predictions along transect 3. It was used for less than 1 – 2%
of all the predictions made at the other well locations along
all of the transects.
4.2. Sensitivity Analysis
[53] The purpose of the sensitivity analysis was to determine if different input parameter sets could improve the
model predictions. Figures 10 –15 are contour diagrams of
the standard errors and mean cumulative errors, for sensitivity analysis simulations. In Figures 10– 15, the transmissivity axis is on a log scale so as to facilitate the visibility of
results throughout the range of values tested. No standard
error contours are shown for the 48 m well along transect 2
because no observations were available for comparison due
to a pressure transducer malfunction at that location. We do
not show contour diagrams of the mean deviations, because
of their similarity to the standard error contours.
[54] The standard error contour diagrams (Figures 10–12)
indicate that the ‘‘model run’’ input parameters do not
necessarily lie within the parameter regions that correspond
to minimum possible standard errors in the predictions at
the various well locations. For example, Figure 10 suggests
that an order of magnitude higher transmissivity than
estimated for the initial ‘‘model run’’ would generate better
predictions across the transect if the specific yield was held
at 0.6. Regions of <3 cm standard error (the accuracy of
the observations) can be found at almost all well locations
along all three transects. The only exceptions are the 6 m
and 12 m predictions along transect 3, where the minimum
possible standard errors are 4.1 cm and 3.7 cm, respectively.
Because the contours were generated at 1 cm intervals, the
minimum error contour shown on the charts, at the 6 m and
12 m locations, are 5c m and 4 cm, respectively.
[55] Along all three transects we observe that the range of
input parameter sets corresponding to reasonable (i.e.,
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Figure 12. Results of the standard error sensitivity analysis for transect 4.
<3 cm) standard errors increases with distance from the
creek. We also observe that in general, the greatest overall
errors are in the region of low specific yield and high
transmissivity.
[56] Figures 13– 15 are mean cumulative error contours
for all predictions made along the three transects. A 0 cm
mean cumulative error contour appears on all the plots
shown. As expected, high transmissivity values will tend
to underestimates of the position of the water table, because
of increased lateral flow. The greatest overestimation of the
position of the water table occurs at low to medium
transmissivity values with medium to high specific yield.
This is due to the fact that the lateral movement of water
that entered the marsh substrate during inundation is
inhibited by the transmissivity, while the higher specific
yield values translate evapotranspirative losses into only
minimal water table drawdown.
[57] From inspection of the data used to construct the
contours in Figures 10 – 15, pairs of transmissivity and
specific yield values that correspond to lower overall error
across the transects were selected, for additional model runs.
Three different pairs of Sy and T values were selected for
each transect. These values are listed in Table 3.
[58] To visualize these model results, Figures 7 – 9 display
the predicted water tables at each well location obtained by
running the model with the T and Sy values listed in Table 3.
Also shown in Figure 7– 9 are the actual observations at
each well location, as well as the initial ‘‘model run’’
predictions. Table 1 lists the standard error, mean deviation,
mean cumulative error, and maximum deviation associated
with each simulation along each transect.
[59] The predictions made with pairs of transmissivity
and specific yield values selected from inspection of the
sensitivity analysis fit the observations fairly well, and in
general are better than the initial ‘‘model runs’’. For
example, at 2.3 cm, the average standard error across
transect 2 using parameter set 2 is 50% of the standard
error calculated using the ‘‘model run’’ parameter set
(4.7 cm). The water table drawdown that occurs during
the period of neap tides is well simulated, as is the effect of
spring tide inundations on the water tables. It is interesting
to note that in most cases, all three parameter pairs closely
replicate these phenomena, even though the T and Sy values
vary widely.
[60] As expected from the results of the sensitivity
analysis, standard errors, and mean deviations of fewer than
3 cm (the error in the observations) were obtained for all
simulations along all transects, with only the 12 m well
along transect 2, and the 6 m and 12 m wells along transect 3
as exceptions. All of the mean cumulative errors obtained
from the model runs made with the parameters listed in
Table 3 were also within ±3 cm of the observations. Minor
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Figure 13. Results of the mean cumulative error sensitivity analysis for transect 2.
differences in the maximum deviation from the model run
were computed.
5. Discussion
[61] This study indicates that the model, as presented, can
be used to predict overall water table fluctuations in
Piermont Marsh to within a reasonable degree of accuracy
(i.e., to within ±3 cm of the observations). As an aid to
identifying the optimum parameter spaces for each of the
three transects, we have traced the parameter spaces
corresponding to the overlap in the 3 cm error contours
for the standard errors (Figure 16a), the mean cumulative
error (Figure 16b), and the mean deviation (Figure 16c)
values computed. To construct Figures 16a and 16b, in
cases where the 3 cm contour was not available, the next
lowest contour line was substituted. Notes in Figure 16
address specifics about this process. In Figure 16d, we have
overlapped the standard error, mean deviation, and mean
cumulative error optimum parameter regions, and indicated
the position of the initial ‘‘model run’’ and Table 3 parameter sets, with little circles. The optimum mean deviation
parameter space includes that of the standard error. It is
therefore a less conservative measure of goodness of fit, and
is for this reason not discussed further.
[62] Figure 16 suggests that the initial specific yield and
transmissivity values selected were reasonable for transects
3 and 4. An order of magnitude higher transmissivity would
have yielded better predictions for the water table fluctuations along transect 2. Because of the physical proximity of
transect 2 and 3, and their similar vegetative, topographic,
morphological, and tidal characteristics, it is difficult to
understand why this was the case. One possibility is that
there is a difference in subsurface soil strata present in the
two sites. Only additional field research can clarify this
point. Another possibility is the different meteorological
conditions witnessed during each of the two observation/
simulation periods, as discussed below.
[63] In Figure 17, the areas of the regions included within
the 3 cm standard error contour (if it was available) for all
simulations made along all transects are shown. In general,
the parameter space corresponding to fewer than 3 cm
standard error increases with distance from the creek bank.
Similarly, and with only a few exceptions, the range of
transmissivity values for which appropriate specific yield
values will result in minimal standard error in the predictions, also increases with distance from the creek. These
observations demonstrate decreasing importance of the
transmissivity in determining the water table profile
decreases with distance of creek. This is likely because of
the greater overall complexity of water table dynamics in
the creek bank zone over the marsh interior.
[ 64 ] The shape of the optimum parameters spaces
depicted in Figure 16 implies a wide range of transmissivity
and specific yield pairs that will give the same predictions.
The transect 3 space, for example, suggests that a marsh
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Figure 14. Results of the mean cumulative error sensitivity analysis for transect 3.
characterized by a uniform substrate with a specific yield as
low as 0.4 and a transmissivity below 0.01 m2/d will display
similar patterns of water table fluctuation as another marsh
characterized by highly porous soils and a transmissivity
several orders of magnitude higher. Figures 10– 12 suggest
that this is especially true further away from the creek bank.
[65] There also appears to be a seasonal influence on the
sensitivity of the model to input parameter selection. This
effect is discussed by comparing the standard error contours
at the 36 m well locations along the three transects. The
36 m location is the first observation well outside of the
creek bank zone, as it was defined previously. The data in
Figure 17 indicate that at the 36 m well locations, the
parameter region included within the 3 cm standard error
contour is largest for transect 4, smallest for transect 3, and
an intermediate value for transect 2. The transect 4 observations were made during the winter; the transect 3 observations were made during a very hot summer; and the
transect 3 observations were made during average, springlike conditions. During the winter, PET rates are lowest, and
marsh water table dynamics are simplest, and hence reasonable predictions are obtained for transect 4 observations
over the widest range of values. Conversely, during the
summer, PET rates are at their highest annual values. As a
result, there is more movement of water through the tidal
marsh substrate, and accurate modeling of water table
dynamics in the marsh interior along transect 3 requires
more precise specification of transmissivity and specific
yield values. During the spring, there are intermediate PET
rates, and the model has intermediate sensitivity to parameter selection.
[66] The sensitivity analysis above is a good example of
the phenomena of equifinality [Beven and Binley, 1992;
Beven, 1993; Freer and Beven, 1996; Brazier et al., 2000],
but its conclusion that similar water table predictions are
obtained over a wide range of transmissivity and specific
yield values seems to contradict the recent findings of other
researchers. Using a numerical model, Ursino et al. [2004]
and Silvestri and Marani [2004] found that the substrate
hydraulic conductivity is an important determinant of soil
saturation levels in a hypothetical tidal marsh. Specifically,
these researchers suggest that at a relatively low saturated
hydraulic conductivity, a persistent unsaturated zone is
present below the soil surface, even after inundation of
the tidal surface. Although the transmissivity of the hypothetical site and of Piermont Marsh are of the same order of
magnitude, the hypothetical marsh is much narrower than
Piermont. The total width of the hypothetical marsh was
10 m, while at Piermont transects almost 40– 120 times
wider. Figures 10– 15 demonstrate a lower sensitivity of the
present model to substrate parameters beyond the creek
bank zone. However, because it is narrow, the hypothetical
marsh would appear to behave entirely as a ‘‘creek bank.’’
This is significant since wider marshes have greater pore
water ‘‘reserves’’ to supplement water that has left the
marsh as lateral flow the creek bank walls.
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Figure 15. Results of the mean cumulative error sensitivity analysis for transect 4.
[ 67 ] Other differences between the two modeling
approaches that appear to be relevant in explaining the
contradictory results are (1) the tidal boundary condition
considered: regular, sinusoidal, and monochromatic leading
to regular surface inundation in the case of the hypothetical
simulation, yet irregular at Piermont and (2) a shallower
overall water table at Piermont owing to the presence of a
seepage face. No seepage face was simulated in the hypothetical studies, as observed by Wilson and Gardner [2005].
It would be interesting to see if Ursino, Silvestri and
coworkers would obtain similar results to those presented
here if their numerical model was used to simulate Piermont
Marsh conditions.
[68] This said, Figures 7 – 9 and 16, as well as Table 1,
also reveal some limitations to the present model’s predictive abilities. One of the obvious weaknesses is that some
input parameter pairs lead to better predictions closer to the
creek, and some further back. This result is a relic of the
assumption of uniform soil properties and a uniform marsh
surface elevation across the transect in the model. The
model’s inability to reflect soil and topographic heterogeneity is likely one explanation for the poor predictions
at the 6 m and 12 m locations along transect 3, between
27 May and 6 June. Similarly poor predictions were found
at the 6 m, 12 m, and 18 m locations, between 28 April
and 1 May along transect 2. During both of these instances,
the boundary condition water table was not as high as the
elevation assumed for the average marsh surface. Yet the
observations suggest that an inundation occurred. It is
possible that the marsh surface was actually slightly lower
at these locations and local inundations did, in fact, occur in
the creek bank region, raising the water table there and there
alone. Another possible explanation is that some unknown
flow process or spatially variable soil properties facilitated
conveyance of water from noninundating high tides into the
marsh at these locations. In either case, the model presented
is not capable of simulating these phenomena.
[69] Also not well modeled is the series of sharp water
table drawdown observed in the immediate postinundation
water table profile for all wells along transect 3, between
10 and 17 May. Interestingly, the same phenomena are well
Table 3. Transmissivity and Specific Yield Pairs Used in
Graphical Presentation
Transect
Parameter Set
Specific Yield
Transmissivity, m2/d
2
2
2
3
3
3
4
4
4
1
2
3
4
5
6
7
8
9
0.30
0.70
0.90
0.42
0.60
0.94
0.20
0.60
0.98
9.0
30
70
0.7
0.007
20
0.5
4.0
8.0
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Figure 16. Overlap in error contours along each transect. (a) Standard error, (b) cumulative error,
(c) mean deviation, and (d) combined error contours. In most cases the 3 cm contours at each well
location were used. Notes give details on the contour overlaps shown. In the combined error contour
diagrams the standard error parameter spaces have been colored green, and the mean deviation parameter
spaces have been colored blue. The numbers in Figure 16d refer to the parameter sets in Table 3. MR is
the parameter set for the ‘‘model run.’’
modeled along transect 2 (see, for example, the 6 m
predictions between 14 and 19 May). Multiple attempts to
improve the transect 3 predictions, using a wide range of
specific yield and transmissivity pairs were unsuccessful.
[70] Other imprecise predictions relate to irregular fluctuations in the well water table observations. For example,
the model is not capable of predicting small amplitude
oscillations of the creek bank water table found in some
wells (for example, the 6 m predictions along transect 4
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Figure 17. Area (relative units) of the region bounded by
the 3 cm standard error contours along the three transects. If
there was no 3 cm contour, the area shown is zero.
between 22 and 27 December), or the infrequent daily
drawdown of water table detected in the 24 m water table
on transect 2 between 24 and 27 April. The latter appears to
have occurred during daytime low tide events.
[71] To test whether the model’s ability to predict poorly
simulated localized phenomena such as water table fluctuations due to noninundating high tides or small-amplitude,
high-frequency well water table fluctuations could be
improved, model predictions were made for transect 3 using
a specific yield of 0.6 and an unrealistically high transmissivity (T = 100 m2/d). While this input parameter set did, in
fact, allow the predicted water table to better simulate these
fluctuations, the greater transmissivity led to greater overall
horizontal drainage, which resulted in predicted water tables
across the transect that were approximately 15 – 20 cm
lower, on average, than the observations. Thus higher
transmissivity values lead to better creek bank predictions
and lower transmissivity values lead to better predictions
inland. Although we did not model it, at a reasonable
transmissivity value, we would expect that a low specific
yield in the creek bank would be more appropriate for creek
bank predictions, while a higher specific yield would be
more appropriate for interior predictions. These effects
cannot be well simulated with the model presented, again
because of its requirement of uniform soil properties.
[72] We also tested whether errors in model structure
might be significantly skewing the results obtained. Specifically, we used the model to predict the boundary conditions
at the creek banks and the marsh midpoint. At the interior of
the marsh, an over-height error is introduced as a result
of the neglecting nonlinear effects in the periodic solution of
Flow System 3 [Parlange et al., 1984; Li and Jiao, 2003].
However, for superposition to work, linearization of the
equations is a requirement. The error associated with
linearization was easily calculated by the procedure outlined
by Parlange et al. [1984] and later by Li and Jiao [2003].
With an amplitude of 30 cm estimated for the levee well
boundary condition, and an aquifer depth of 2.7 m, the error
introduced by linearization is, at most, approximately 1 cm
at x = L/2, and nearly insignificant given the observation
error.
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[73] The second potential error associated with model
structure errors is at the creek bank, and specifically the
boundary condition embodied by equation (19) (i.e.,
d3(x, 0) = 0). As pointed out previously, d3(x, 0) as
expressed by equation (20) does not equal zero. To quantify
this error, we simulated d3 at x = 0 for transect 3, and
plotted the entire set of predictions with the observations in
Figure 18, where this error would be the greatest. As shown
in Figure 18 and its two insets, the model adequately
replicates the boundary condition, suggesting that the error
associated with this model structure inaccuracy is negligibly
small, and even so, diminishes rapidly with time.
[74] A final potential source of error in model structure is
the ad hoc approximation used to simulate the fast drainage
during noninundating high tides by preferential flow
through macropores in the creek bank walls. To assess the
error associated with this assumption, we recalculate the
transect 3 predictions at x = 6 m without the preferential
flow approximation. We picked this location along this
transect because as shown in Table 2, this was the simula-
Figure 18. Quantification of the error induced by using
equation (20) to simulate the boundary condition of
equation (19).
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tion that invoked the preferential flow approximation to the
greatest extent. This exercise produced negligible differences in the errors computed, likely because of its infrequent implementation in the simulation (see Table 2).
6. Conclusions
[75] The mathematical model developed is an approximation of the major mechanisms of water flow observed at
Piermont Marsh. As an irregularly flooded wetland in the
temperate zone, its hydrology may differ from tidal wetlands in other places, with different hydroperiods. Nonetheless, for this site, the results of the model sensitivity
analysis are informative in both assessing the limits of this
model’s validity, and in explaining a number of aspects of
the observed tidal marsh hydrology observations presented
by Montalto et al. [2006].
[76] Within a wide range of specific yield and transmissivity values, the model developed can generally predict the
temporal and spatial fluctuations of the water table over
much of Piermont Marsh to within the error threshold of the
initial observations. The range of parameter values
corresponding to reasonable standard errors appears to
increase with distance from the creek bank. This is
explained by the fact that subsurface flow dynamics are
more complex closer to the creek. Seasonally, it appears that
a more restricted set of transmissivity and specific yield
parameter pairs is required for acceptable predictions in the
summer, than in the winter, likely as a result of the increased
importance of evapotranspirative fluxes during the warmer
weather.
[77] Although the model, as presented, can be used to
compare, in general terms, the implications of broad differences in transmissivity, specific yield, marsh width, and
marsh surface elevation on the water table fluctuations in
tidal wetlands like Piermont Marsh, its use in more detailed
ecohydrologic research will require additional refinement,
especially in cases where small differences in the position of
the water table may have important impacts on, for example, oxygen availability and wetland biogeochemistry. An
example where this might be the case involves vegetation
zonation, which is itself likely a function of multiple factors
including soil saturation, but also hydroperiod, and salinity
[Silvestri et al., 2005]. One way of accomplish this goal
might involve comparing our results with numerical modeling efforts that can simulate the spatial heterogeneity of
topographic and edaphic properties at Piermont Marsh.
[78] Model validation revealed a nondeterministic quality
of the linearized, one-dimensional Boussinesq equation as
solved. The equifinality result needs to be considered in any
analysis attempting to use this model to make generalizations about hydrologic repercussions of a given set of
marsh substrate characteristics. In a more general sense, the
finding of equifinality here likely also carries implications
for other applications with a (perched) groundwater table
close to the surface and periodic rain events.
Notation
d(x, t)
d1(x, t)
elevation of the water table.
solution 1: the gradual decay in water table
elevation during periods of marsh surface
exposure.
d2(x, t)
d2u(t)
d2c(x, t)
d3(x, t)
davg
D
hoj
Ks
L
P(t)
PET(t)
T
t
w(t)
x
Ai, wi, ai
x1, b, m
Sy
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solution 2: the net effect of precipitation and
evapotranspiration on the water table elevation.
step 1 of solution 2: the cumulative sum of all
net meteorological fluxes through time, t,
assumed to act uniformly on the transect
water table.
step 2 of solution 2: the position of the water
table as a result of all net.
meteorological fluxes through the marsh
surface and all associated movement of
water through the creek bank occurring during
a given day.
solution 3: tidally induced oscillations of the
marsh water table.
the average elevation of the levee boundary
condition for the interval between inundation
events.
constant aquifer thickness.
the total daily precipitation minus the total
daily evapotranspiration on day j.
saturated hydraulic conductivity (assumed
uniform, constant, isotropic).
distance between the two parallel tide creeks.
precipitation.
potential evapotranspiration.
aquifer transmissivity.
time.
net accretion into the control volume,
P(t)-PET(t).
direction measured perpendicular to the creek
bank.
amplitude, angular velocity, and phase angle
of the periodic boundary condition at the
creek bank.
fitting parameters used to describe the initial
condition water table profile
specific yield.
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