4. Case studies

Two case studies, the application of a hydrodynamic model in a lake and a water quality model in a river, are presented in this section to show how the respective models are applied for real world surface water systems.

#### 4.1. Case study 1: Southern Lake Michigan

#### 4.1.1. Background

Water quality in the nearshore region of southern Lake Michigan had a problem with contamination by fecal bacteria from various sources. For Great Lakes beaches, fecal pollution attracted a great deal of attention from both beach managers and the public due to its potential risk to human health, triggering many beach closures and advisories every year with a consequent significant loss to local economies along the shoreline. Predictive modeling has been suggested as an effective approach to enhancing measurements of water quality both temporally and spatially, thus reducing the damage caused by improving the beach management. This case study was therefore conducted to develop a nearshore transport model for fecal pollution in Lake Michigan. The resulting model can be used to inform beach goers promptly in order to protect them from any potential exposure to waterborne pathogens and to help develop a better understanding of the key processes and factors influencing the fate and transport of fecal pollution in the nearshore reaches of the lake. As the indicators of fecal bacteria, Escherichia coli (EC) and enterococci (ENT) were used to evaluate the water quality at a recreational beach. In such environments, modeling nearshore, wind-driven circulation and the transport of EC and ENT are particularly challenging due to the interactions with complex lake-wide circulation. Originally, a 2D depth-averaged model was developed to simulate the entire lake with finer meshes close to the shoreline to emphasize the nearshore region [21]. The modeling results show that the current in nearshore region flew predominately along the shoreline direction and the cross-shoreline flow was fairly weak. Therefore, within an acceptable error tolerance range, the entire lake can be simplified as a narrow channel along the shoreline to significantly save computation efforts for nearshore process investigations. With this simplification, current flows only in clockwise or anticlockwise direction along the shoreline, and the exchanging process of mass and momentum in crossshoreline direction only occur within the channel [26]. Later investigation indicates that the current in vertical direction sometimes cannot be neglected in such lake environment, and thus, a stratified 3D model may be necessary to explain the hydrodynamic and transport processes along the water depth. In this case study, a new 3D model for the domain model was developed based on the aforementioned channelized 2D model. Some 3D modeling results are reported in the following sections.

#### 4.1.2. Investigation domain and model mesh

This case study focused on the nearshore regions along approximately 100 km of the shoreline of southern Lake Michigan. Based on the channelization simplification [26], the computational domain included a 5-km wide channel throughout the entire Lake Michigan shoreline (Figure 1). To fairly delineate the complex shoreline, a finite element model with a 3D nonuniform mesh was used. The mesh was gradually refined from a resolution of approximately 1–2 km at the locations far from the research domain to 100 m in the research area. Four streams are the primary tributaries discharging into southern Lake Michigan: Trail Creek (TC) at the Michigan City Harbor (USGS 04095380), Kintzele Ditch (KD) nearby Michigan City, Burns Ditch at Portage, IN (BD, USGS 04095090), and Indiana Harbor Canal (IHC) at East Chicago. As KD and TC both discharge combined sewer flows (CSOs), these are the most significant sources of EC and ENT so Mt. Baldy beach, which is located between the two, was chosen as the nearshore beach for this investigation (Figure 1).

## 4.1.3. Data collection

where C is the concentration of the contaminant or temperature, A is the cross-sectional area, As is the storage cross-sectional area, Di is the turbulent diffusion coefficient in the i direction, G is a general term representing sources or sinks, and k is the first order decay or reproductive rate coefficient [23]. These equations represent the first order case; other reaction rates (such as zero order or second order) are described using different expressions for the source term. The

Contaminants may be released into a water system via one of two modes: constant release or pulsed release. For both, the initial condition is C xð Þ¼ ; t ¼ 0 C0, where C<sup>0</sup> is the initial concentration that can be assigned as a constant such as the background concentration. The boundary

n represents the coordinates in the direction normal to the boundary. Other observed concen-

For a water quality model, numerous parameters and data are needed as input. Initial values for these parameters are obtained from the literature, measured directly, or determined via model calibration. The modeling results for various values of the parameter are then compared against the observed data, and the value that achieves the best match is selected for further

Water quality models are fairly complex due to the multiple interrelationships among the many processes controlling the fate and transport of contaminants. Many of these processes can only be described using empirical formulations, which need adequate data for verification and calibration. Data quantity and quality are thus keys for developing and applying a water quality model. In the remainder of this chapter, finite element model RMA10 is used as the

Two case studies, the application of a hydrodynamic model in a lake and a water quality model in a river, are presented in this section to show how the respective models are applied

Water quality in the nearshore region of southern Lake Michigan had a problem with contamination by fecal bacteria from various sources. For Great Lakes beaches, fecal pollution attracted a great deal of attention from both beach managers and the public due to its potential

hydrodynamic model and RMA11 as the water quality model in the discussion [23].

<sup>∂</sup><sup>n</sup> <sup>¼</sup> 0, where

condition for the free surface, bottom and side surfaces is the no flux condition, <sup>∂</sup><sup>C</sup>

terminology used for the other variables is same as that used in Section 2.

trations are assigned to the inflow and outflow boundaries.

3.3. Parameters and data for the water quality model

3.2. Boundary and initial conditions

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modeling runs.

4. Case studies

4.1.1. Background

for real world surface water systems.

4.1. Case study 1: Southern Lake Michigan

Major factors considered in the model included the bathymetry, the shape of the lake shoreline, wind stress, hydrological flows from the tributaries, and water temperature. Bathymetric data

4.1.4. Hydrodynamic and water quality model

(e.g., a(1)–a(3)) nodes from the lake surface to the bottom.

4.1.4.2. Initial and boundary conditions

4.1.4.3. Fate and transport of EC and ENT

k Ið Þ ; <sup>T</sup>; vs |fflfflfflfflffl{zfflfflfflfflffl} ð Þ1

¼ f <sup>P</sup> vs

Although a 2D depth-averaged model based on the finite element model RMA10 [21, 26] can well describe the wind-driven circulation at the particular location in Lake Michigan during the specific period, the transport phenomenon along the water depth is fairly significant for other contaminants at other locations [27]. Therefore, a stratified 3D hydrodynamic and water quality model was developed to address the vertical current and transport. The model was employed to simulate current velocity, water depth, water surface elevation, water temperature, and EC. The model principles and setup, including the governing equations, boundary conditions, and initial conditions, were those described earlier in Sections 2 and 3. The governing equations were thus Eqs. (11)–(16) for the hydrodynamic model and Eq. (21) for the water quality model. A constant

the nearshore region. The water body was evenly divided into three layers, and thus, there are totally either five nodes (for the corner points in a finite element) or three nodes (for the midpoints in a finite element) at each location throughout the water depth. Figure 1f shows the nodes constituting a typical finite element in the 3D mesh. A quadratic finite element consists of corner nodes (solid dots such as A, B, C, and D) and middle nodes (empty dots such as a, b, c, and d). In this two-layer mesh, four and two nodes are placed under each corner and middle node at the water surface, respectively. Therefore, there are totally five (e.g., A(1)–A(5)) or three

The initial conditions used to model EC were the lake at rest and a background value of 3 CFU/ 100 mL, based on general observations. The observed water temperatures at the different stations were interpolated into the mesh for the thermal model when t = 0. The boundary conditions for the hydrodynamic model included the no leakage condition across the surface and the bottom, zero pressure and wind stress at the free surface, drag at the bottom surface, and the flow loading from the tributaries. For the water quality model, EC and ENT data were loaded for both TC and KD, and the loading rates were monitored at the stream mouths.

In addition to determining the hydrodynamic transport processes in the water body, estimating the activation rate of EC and ENT is another key issue in the model. The common factors affecting the fate and transport of EC and ENT include sunlight, nutrient content, salinity, suspended solids, sedimentation, water temperature, pH, and predation. Based on the important inactivation mechanisms reported in the literature, a time-dependent inactivation rate was used, which considered solar insolation, sedimentation, and water temperature, as shown below:

where k Ið Þ ; T; vs is the overall inactivation rate, kI is the inactivation rate for light, I tð Þ is the measured solar insolation, θ is the temperature correction factor (usually 1.07), f <sup>P</sup> is the

vs <sup>H</sup> <sup>θ</sup>ð Þ <sup>T</sup>�<sup>20</sup> |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} ð Þ2

<sup>þ</sup>kII tð Þθð Þ <sup>T</sup>�<sup>20</sup> |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} ð Þ3

(22)

<sup>H</sup> <sup>þ</sup> kII tð Þ h iθð Þ <sup>T</sup>�<sup>20</sup> <sup>¼</sup> <sup>f</sup> <sup>P</sup>

/s was used for both the eddy viscosity and the turbulent diffusion coefficient in

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4.1.4.1. Model description

value of 2.0 m2

Figure 1. (a) Whole lake model mesh (2D), (b) channelized mesh (2D), (c) channelized mesh (3D), (d) Google Earth satellite image with the channelized mesh, (e) enlarged research domain with mesh, and (f) scheme of node distribution in quadratic a 3D element mesh.

with a resolution of 3 arc-seconds were obtained from the NOAA's National Geophysical Data Center. As inputs of the model, hourly meteorological data, including wind speed and direction, air temperature, dew point temperature, air pressure, and sunlight insolation were obtained from six stations run by NOAA's National Climatic Data Center and two buoys belonging to NOAA's National Data Buoy Center (NDBC). Data on current velocity and WSE were also collected from several stations for model calibration.

#### 4.1.4. Hydrodynamic and water quality model

#### 4.1.4.1. Model description

Although a 2D depth-averaged model based on the finite element model RMA10 [21, 26] can well describe the wind-driven circulation at the particular location in Lake Michigan during the specific period, the transport phenomenon along the water depth is fairly significant for other contaminants at other locations [27]. Therefore, a stratified 3D hydrodynamic and water quality model was developed to address the vertical current and transport. The model was employed to simulate current velocity, water depth, water surface elevation, water temperature, and EC. The model principles and setup, including the governing equations, boundary conditions, and initial conditions, were those described earlier in Sections 2 and 3. The governing equations were thus Eqs. (11)–(16) for the hydrodynamic model and Eq. (21) for the water quality model. A constant value of 2.0 m2 /s was used for both the eddy viscosity and the turbulent diffusion coefficient in the nearshore region. The water body was evenly divided into three layers, and thus, there are totally either five nodes (for the corner points in a finite element) or three nodes (for the midpoints in a finite element) at each location throughout the water depth. Figure 1f shows the nodes constituting a typical finite element in the 3D mesh. A quadratic finite element consists of corner nodes (solid dots such as A, B, C, and D) and middle nodes (empty dots such as a, b, c, and d). In this two-layer mesh, four and two nodes are placed under each corner and middle node at the water surface, respectively. Therefore, there are totally five (e.g., A(1)–A(5)) or three (e.g., a(1)–a(3)) nodes from the lake surface to the bottom.

#### 4.1.4.2. Initial and boundary conditions

The initial conditions used to model EC were the lake at rest and a background value of 3 CFU/ 100 mL, based on general observations. The observed water temperatures at the different stations were interpolated into the mesh for the thermal model when t = 0. The boundary conditions for the hydrodynamic model included the no leakage condition across the surface and the bottom, zero pressure and wind stress at the free surface, drag at the bottom surface, and the flow loading from the tributaries. For the water quality model, EC and ENT data were loaded for both TC and KD, and the loading rates were monitored at the stream mouths.

### 4.1.4.3. Fate and transport of EC and ENT

with a resolution of 3 arc-seconds were obtained from the NOAA's National Geophysical Data Center. As inputs of the model, hourly meteorological data, including wind speed and direction, air temperature, dew point temperature, air pressure, and sunlight insolation were obtained from six stations run by NOAA's National Climatic Data Center and two buoys belonging to NOAA's National Data Buoy Center (NDBC). Data on current velocity and WSE

Figure 1. (a) Whole lake model mesh (2D), (b) channelized mesh (2D), (c) channelized mesh (3D), (d) Google Earth satellite image with the channelized mesh, (e) enlarged research domain with mesh, and (f) scheme of node distribution

were also collected from several stations for model calibration.

in quadratic a 3D element mesh.

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In addition to determining the hydrodynamic transport processes in the water body, estimating the activation rate of EC and ENT is another key issue in the model. The common factors affecting the fate and transport of EC and ENT include sunlight, nutrient content, salinity, suspended solids, sedimentation, water temperature, pH, and predation. Based on the important inactivation mechanisms reported in the literature, a time-dependent inactivation rate was used, which considered solar insolation, sedimentation, and water temperature, as shown below:

$$\underbrace{k(I, T, v\_s)}\_{(1)} = \left[f\_P \frac{v\_s}{H} + kI(t)\right] \theta^{(T-20)} = \underbrace{f\_P \frac{v\_s}{H} \theta^{(T-20)}}\_{(2)} + \underbrace{k\_I I(t) \theta^{(T-20)}}\_{(3)}\tag{22}$$

where k Ið Þ ; T; vs is the overall inactivation rate, kI is the inactivation rate for light, I tð Þ is the measured solar insolation, θ is the temperature correction factor (usually 1.07), f <sup>P</sup> is the fraction of pathogens attached to the suspended sediment, vs is the settling velocity, and H is the water column depth.

#### 4.1.5. Model calibration

The hydrodynamic model was calibrated by adjusting the Manning's roughness coefficient, n after examining the sensitivity of model currents to the horizontal viscosity. The modeling results were compared with Acoustic Doppler Current Profiler (ADCP) data collected at Burns Ditch (July 2–August 14). Figure 2 shows these comparisons. A constant bed roughness value of 0.1 leading to a minimum root mean square error (RMSE, Eq. (23)) was finally used in the model. The modeling results for the current velocity and WSE were generally consistent with known circulation patterns in southern Lake Michigan.

$$RMSE = \sqrt{\frac{1}{N} \sum\_{i=1}^{N} \left[ \log\_{10}(\mathbf{X}\_{sim}) - \log\_{10}(\mathbf{X}\_{obs}) \right]^2} \tag{23}$$

different depth at the sampling locations were compared with the observed data (Figures 3 and 4). Time-dependent inactivation rate based on temperature, sedimentation, and observed solar insolation (Eq. (22)) was used to simulate EC inactivation. The same parameter values as those in the 2D models were used for the 3D model, which include a kI value of

different depth showed a similar pattern of temporal variation, which can reasonably describe the data. Generally, both the speed and the magnitude of velocity components tend to decrease

Figure 3. Comparison of E. coli between data and 3D modeling results at (a) Mt. Baldy and (b) Central Avenue.

Figure 4. Comparison of temperature between data and 3D modeling results at (a) Mt. Baldy and (b) Central Avenue.

, a fraction f <sup>P</sup> of 0.1, and a vs value of 5 m per day. The modeling results at

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0.0026 W<sup>1</sup>

m2 d<sup>1</sup>

where Xsim, Xobs are the simulated results and the observed data, (referring to the concentrations in this case study), respectively.

#### 4.1.6. Results and discussion

In addition to the previous published 2D modeling investigation of the domain, we examined the 3D modeling results. The modeling concentrations of EC and water temperatures at

Figure 2. Comparison of (a) velocity component and (b) speed between data and 3D modeling results (lake surface).

different depth at the sampling locations were compared with the observed data (Figures 3 and 4). Time-dependent inactivation rate based on temperature, sedimentation, and observed solar insolation (Eq. (22)) was used to simulate EC inactivation. The same parameter values as those in the 2D models were used for the 3D model, which include a kI value of 0.0026 W<sup>1</sup> m2 d<sup>1</sup> , a fraction f <sup>P</sup> of 0.1, and a vs value of 5 m per day. The modeling results at different depth showed a similar pattern of temporal variation, which can reasonably describe the data. Generally, both the speed and the magnitude of velocity components tend to decrease

fraction of pathogens attached to the suspended sediment, vs is the settling velocity, and H is

The hydrodynamic model was calibrated by adjusting the Manning's roughness coefficient, n after examining the sensitivity of model currents to the horizontal viscosity. The modeling results were compared with Acoustic Doppler Current Profiler (ADCP) data collected at Burns Ditch (July 2–August 14). Figure 2 shows these comparisons. A constant bed roughness value of 0.1 leading to a minimum root mean square error (RMSE, Eq. (23)) was finally used in the model. The modeling results for the current velocity and WSE were generally consistent with

where Xsim, Xobs are the simulated results and the observed data, (referring to the concentra-

In addition to the previous published 2D modeling investigation of the domain, we examined the 3D modeling results. The modeling concentrations of EC and water temperatures at

Figure 2. Comparison of (a) velocity component and (b) speed between data and 3D modeling results (lake surface).

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

log10ð Þ� Xsim log10ð Þ Xobs � �<sup>2</sup>

vuut (23)

the water column depth.

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4.1.5. Model calibration

known circulation patterns in southern Lake Michigan.

RMSE ¼

tions in this case study), respectively.

4.1.6. Results and discussion

1 N X N

i¼1

Figure 3. Comparison of E. coli between data and 3D modeling results at (a) Mt. Baldy and (b) Central Avenue.

Figure 4. Comparison of temperature between data and 3D modeling results at (a) Mt. Baldy and (b) Central Avenue.

with depth (U, V, W and Speed in Figures 5–7). The water temperatures at water surface are higher than those below and the lake bottom has the minimum value. Figures 8 and 9 show the contour of E. coli reflecting its transport and fate and water temperature distribution at different time. The model was able to reasonably describe the observed data.

Figure 5. Distribution of calculated parameters along water depth at Julian day 210 at Mt. Baldy Beach.

4.2. Case study 2: Upper San Joaquin River

The San Joaquin River (SJR) is the second longest river in California, and its watershed within the Central Valley is one of the California's most productive agricultural areas. The river

Figure 8. Spatial distribution of E. coli at 12:00 am on Julian day: (a) 180, (b) 190, (c) 200, and (d) 210.

Figure 7. Distribution of calculated parameters along water depth at Julian day 210 at Central Avenue Beach.

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4.2.1. Background

Figure 6. Distribution of calculated parameters along water depth at Julian day 210 at Trail Creek.

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Figure 7. Distribution of calculated parameters along water depth at Julian day 210 at Central Avenue Beach.

Figure 8. Spatial distribution of E. coli at 12:00 am on Julian day: (a) 180, (b) 190, (c) 200, and (d) 210.

#### 4.2. Case study 2: Upper San Joaquin River

#### 4.2.1. Background

with depth (U, V, W and Speed in Figures 5–7). The water temperatures at water surface are higher than those below and the lake bottom has the minimum value. Figures 8 and 9 show the contour of E. coli reflecting its transport and fate and water temperature distribution at different

time. The model was able to reasonably describe the observed data.

102 Applications in Water Systems Management and Modeling

Figure 5. Distribution of calculated parameters along water depth at Julian day 210 at Mt. Baldy Beach.

Figure 6. Distribution of calculated parameters along water depth at Julian day 210 at Trail Creek.

The San Joaquin River (SJR) is the second longest river in California, and its watershed within the Central Valley is one of the California's most productive agricultural areas. The river

watershed. In this case study, alternative 3 was used for the water quality study. The water pathway is: Reach 4A—Eastside Bypass—Mariposa Bypass—Reach 4B2—Reach 5 [28].

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The same finite element mesh employed in the hydrodynamic model with resolutions ranging

Bathymetric data were collected during 2010 and 2011 by the US Bureau of Reclamation (USBR) using GPS and ADCP at a spatial interval of 6 m. The flow rate and WSE data (from January 1 to September 30 of 2011) used to verify and calibrate the model and as boundary conditions were obtained from the US Geological Survey (USGS), USBR, and the California Department of Water Resources (CADWR). The geographic boundary of the SJR was delineated using coordinates from Google Earth based on the WGS84 global reference system. The data collected from different coordinate systems were all converted and georeferenced using the same coordinate system and reference datum, namely the North American Vertical Datum NAVD 88 and California State Plane, Zone 3, North American horizontal Datum NAD 83. The

A vertically integrated hydrodynamic model has been employed using the finite element scheme RMA10 to simulate flow velocity, water depth, and WSE. The governing equations were Eqs. (5)–(8). Compared with the first case study, the Coriolis force and wind stress are insignificant and can thus be neglected for this small-scale river reach. For the initial conditions, the river was assumed to be at rest at the beginning and it took a considerable time (10 days) to reach the actual initial conditions. The boundary conditions included the upstream flow rates, the known downstream WSEs, no leakage across the surface and the bottom, a drag stress at the river bottom, no wind stress and zero pressure at the water

The hydrodynamic model has been calibrated using the 2011 discharge and WSE data set. The Manning roughness coefficient of the river reach was manually adjusted to calibrate the model using the RMSE (Eq. (23)). The optimum value of 0.035 of the coefficient led to the minimum

The flow velocity and WSE obtained from the validated and calibrated model were then used to assess the habitat suitability of the SJR reach for the spring Chinook salmon. The detailed model calibration and engineering plan comparisons have been reported by Liu and Ramirez [28]. Due to the lack of observed water quality data, the developed water quality model was used to simulate a virtual scenario as follows: four chemical species, including NH3, NO3, Organic N, and Organic P, which are common in the SJR watershed, entered the SJR at the upstream entrance of the river reach (SDP). These contaminants transport to the downstream with water flow and also undergo their own deactivation. Figure 10 shows the concentrations change with time during a 110-day period. The SDP curve represents the upstream boundary condition. EBM and FFB are the middle station and downstream station, respectively. The

from 0.5 to 50 m was used to delineate the complex boundaries of the SJR.

4.2.3. Data collection

4.2.4. Model setup

surface.

RMSE.

4.2.5. Results and discussion

water quality data are not available at this time.

Figure 9. Spatial distribution of water temperature at 12:00 am on Julian day: (a) 180, (b) 190, (c) 200, and (d) 210.

originates mainly from snowmelt and runoff in the high Sierra Nevada, eventually converging with the Sacramento River in the Sacramento-San Joaquin Delta in Northern California. The SJR has experienced considerable low flow over the years, at times ceasing to flow completely. Chinook salmon was historically abundant in the SJR, but their populations have significantly decreased due to the insufficient flow. In order to restore and sustain salmon and other fish populations, the San Joaquin River Restoration Program (SJRRP) was established in 2006 to maintain continuous flow along the entire length of the river and improve its hydrodynamic conditions from Friant Dam to its confluence with the Merced River. Suitable hydrodynamic conditions, including flow velocity and water depth, are crucial for the safe passage of migrating salmon. Due to practical limitations, the flow has been rerouted along several alternative pathways specifically designed and created as part of the river restoration effort, modifying the traditional SJR channels. These alternatives were designed and compared to support the passage of fish by providing adequate hydrodynamic conditions throughout the river reach. The first objective of the research conducted for this case study was to model the stream conditions, including current velocity, depth, and WSE, for three alternatives proposed for the SJRRP given the same hydrologic/hydraulic boundary conditions. This part of hydrodynamic research has been done and published [28]. The second objective was to further investigate how the water quality at the upstream of the SJR affects the downstream. A 2D water depth-integrated water quality model was developed corresponding to the previous hydrodynamic model to simulate and predict the fate and transport of the contaminants in the upper SJR reach. The water quality model will provide a tool for the SJR restoration and management in the future.

#### 4.2.2. Investigation domain and model mesh

The study area covered approximately 90 river kilometers from the SJR monitoring station near Dos Palos (SDP) to the SJR monitoring station at the Fremont Ford Bridge (FFB) near California Highway 140, which is located within the Middle San Joaquin-Lower Chowchilla watershed. In this case study, alternative 3 was used for the water quality study. The water pathway is: Reach 4A—Eastside Bypass—Mariposa Bypass—Reach 4B2—Reach 5 [28].

The same finite element mesh employed in the hydrodynamic model with resolutions ranging from 0.5 to 50 m was used to delineate the complex boundaries of the SJR.
