Time series modeling and prediction of salinity in the Caloosahatchee River Estuary



[1] Complex three-dimensional (3-D) numerical models are often used for simulation and forecasting of salinity for estuarine water resources management. However, the effort which goes into these models is significant, partly due to the difficulty associated with model development and the prolonged computation time required for model runs. Modern water resources management sometimes requires a quick turnaround time for long-term simulations or short-term forecasts of estuarine salinity conditions. This paper presents an innovative approach for the development of an alternative salinity model based on time series analyses of salinity data. The structure of the model consists of an autoregressive term representing the system persistence and an exogenous term accounting for physical drivers including freshwater inflow, rainfall, and tidal water surface elevation that cause salinity to vary. An analogy to 1-D physical models reveals that major physical processes of salt transport are implicitly embedded in the time series model. Model calibration and validation using up to 20 years of measured data collected in the Caloosahatchee River Estuary, Florida indicate that the time series model offers comparable or superior performance compared with its 3-D counterpart. This model has been used as a tool for water resources management projects relating to ecosystem restoration and water control in south Florida. A special case of model application is included to demonstrate how the model has been used for salinity forecasting to support weekly operation of water control infrastructure and water resources management decision making. Similar modeling tools can be developed using this approach for other estuaries.

1. Introduction

[2] Estuaries, the interface between terrestrial and coastal waters, are among the most productive natural habitats in the world [Costanza et al., 1993]. Salinity in estuaries reflects the shifting balance between freshwater inflow from the land and the continual exchange of sea water with the coastal ocean. The interplay of these drivers creates a transition zone with lower salinities near the source water of upstream watershed and higher salinities toward the ocean mouth of the downstream marine environment. In this land-ocean continuum, salinity becomes fundamental to the health of estuarine ecosystems. The existing literature has documented salinity as an important factor affecting estuarine water quality [Damme et al., 2005], the composition and distribution of vegetative communities [Doering et al., 2002], and the life history of most estuarine animal species [Peebles and Greenwood, 2009]. “In many ways, salinity is a master ecological variable that controls important aspects of community structure and food web organization in coastal systems” [Meyers and Ewel, 1990].

[3] As estuaries concentrate waters from very large land surfaces into relatively small water bodies, salinity conditions within estuaries are strongly dependent upon the quantity and timing of freshwater inflows and thereby the management of water resources in the upstream watersheds [Alber, 2002]. Human activities affect estuarine salinity directly by either changing the amount or timing of freshwater inflow or altering the waterbody hydrodynamics and the rate of mixing with the coastal ocean. Watershed alteration and land development have resulted in more short duration, high volume peak flows, and less life-sustaining base flows to estuaries [Chamberlain and Doering, 1998; Alber, 2002; Wan et al., 2006]. These changes in the pattern of freshwater inflows have resulted in rapid, drastic, high-magnitude fluctuation in estuarine salinity, and as a result, areas within the optimum salinity ranges for various species may no longer coincide with structural features of the estuary that favor the growth and survival of the biological species. Concerns about this ecological functioning call for a holistic water resources management approach to emphasize the integration of hydrologic flow patterns with estuarine ecosystem services [Wolanski et al., 2004; Granek et al., 2010]. Therefore, water resource managers must be able to predict changes in estuarine salinity that can occur as a result of activities occurring in the watershed or proposed for protection or restoration of coastal ecosystems.

[4] To predict the response of coastal and estuarine ecosystems to anthropogenic and natural changes, computer models for simulation and forecasting of salinity are an important tool in the assessment and management of water and ecological resources. Due to the complexity of estuarine circulation and mass transport processes, salinity is typically simulated through the development of physically based numerical models to account for a variety of physical processes and a wide range of spatial and temporal scales. Two of the most widely used hydrodynamic models are the 3-D Curvilinear-grid Hydrodynamic (CH3D) circulation model originally developed by Sheng [1987, 1990] and the Environmental Fluid Dynamics Code (EFDC) developed by Hamrick [1992]. Both models require detailed treatment of irregular coastal geometry, complicated bottom bathymetry, tides, and wind effects. For estuaries in south Florida, the CH3D model has been applied in the Indian River Lagoon and St. Lucie Estuary [Sheng et al., 2002], Loxahatchee River and Estuary [South Florida Water Management District (SFWMD), 2006], Tampa Bay [Sheng et al., 1996], and Caloosahatchee Estuary and Estero Bay [Qiu, 2006; Qiu et al., 2007]. The EFDC model has been applied in the St. Lucie Estuary [Ji et al., 2007], Florida Bay [Hamrick and Moustafa, 2003], and Caloosahatchee Estuary [Xia et al., 2010].

[5] While these numerical models are powerful tools in simulating coastal and estuarine processes such as circulation pattern and salinity field in a fine scales of space and time, the effort which goes into model development and application is significant in terms of the resources required for setting up, compiling, configuring, and executing such models. Furthermore, these models often require significant computational resources, particularly when long-term simulations (e.g., over 40 years of simulation period) are performed [Qiu et al., 2007]. Although recent advances in computer technology have made it possible to use these sophisticated numerical techniques to get accurate solutions, in today's water resources management world, technically sound adaptive management decisions often demand a quick turnaround time for long-term simulations or near-term forecasting of estuarine salinity conditions. Modeling tools addressing this need can take two approaches. The first approach is to simplify the scales of time and space in mathematical representation of the physical system. Savenije [2012] provided an overview of 1-D physical models that have been successfully applied to many alluvial estuaries [e.g., Savenije, 1986, 1988, 1989; Risley et al., 1993]. Such 1-D models, while maintaining the physical processes of estuary hydrodynamics, mixing and salt intrusion, work only in estuaries that are well mixed with a regular geometry. The second approach is empirical, strongly relying on available field data. Examples of these data-driven methods include artificial neural networks [e.g., Maier and Dandy, 1996] and statistical models [e.g., Sun and Koch, 2002]. The disadvantage of data-driven methods is that they fail to contain underlying physics of the system.

[6] The objective of this paper is to document an alternative salinity model developed with a novel approach integrating time series modeling and salinity forcing function analysis with data collected from the Caloosahatchee River Estuary (CRE), Florida. Section 2 of the paper provides the background of the CRE system and its water management issues. Section 3 describes data used in this study along with an analysis of salinity forcing functions that are used for model development. Section 4 details the formulation, calibration, and validation of the model. An analogy to the 1-D model of Savenije [1986] is made to provide physical explanations and theoretical backing of the salinity model developed in this paper. The last section demonstrates the application of the model for salinity forecasting to support weekly operation of water control infrastructure and water resources management decision making.

2. The Caloosahatchee River Estuary-Watershed System

[7] The Caloosahatchee River Estuary, located on the southwest coast of Florida, is a subtropical riverine estuary that has been the subject of water quantity and water quality concerns since the 1950s. The estuary is connected to the larger Charlotte Harbor system in the downstream and to Lake Okeechobee through 67 km canal (C-43 Canal) in the upstream (Figure 1). The CRE drains a land area of about 3450 km2 including the C-43 basin (70%) and the tidal basin (30%). Irrigated agricultural lands are among the major land use types in the watershed, and they create significant demand for water during the dry season.

Figure 1.

Location of the Caloosahatchee River Estuary and associated water management structures.

[8] The system has undergone a number of alterations to facilitate navigation, flood control, and regulatory releases from Lake Okeechobee. A navigation channel was dredged in the estuary, and in the 1960s a causeway was built across the mouth of San Carlos Bay. Historic oyster bars upstream of Shell Point have been mined and removed for road construction. Three locks and dams were constructed to control flow and water levels in Lake Okeechobee and the watershed. The Moore Haven Lock (S-77), located on the southwest shore of Lake Okeechobee, regulates Lake Okeechobee waters. The Ortona Lock (S-78) aids in control of water levels on adjacent lands upstream and separates the C-43 Basin into two distinct hydrologic units, the East and West Basins. The W.P. Franklin Lock and Dam (S-79) is the most downstream structure and marks the beginning of the estuary. The S-79 structure maintains specified water levels upstream, regulates freshwater discharge into the estuary, and acts as an impediment to saltwater intrusion to the upstream portion of the river.

[9] The C-43 Canal (between S-79 and S-77) conveys both basin runoff and releases from Lake Okeechobee to the estuary at S-79. The pattern and magnitude of flows is highly variable, influenced by the subtropical climate of south Florida, regulatory discharges from Lake Okeechobee, and withdrawals for irrigation and water supply for agricultural and urban uses. The Lake Okeechobee releases are made according to a comprehensive regulation schedule used by the U.S. Army Corps of Engineers (USACE) to manage the lake water levels [SFWMD et al., 2010]. Regulatory releases into the CRE are made when the lake levels are relatively high, and release rates generally increase as water levels rise. These high levels of releases, which are often harmful to the estuarine ecosystem, occur typically in the wet season. During the dry season, however, low level releases of lake water may be made to benefit the estuarine ecosystem. These environmental releases are often in conflict with permitted agricultural and urban water supply needs.

[10] The CRE provide a wide range of habitats for ecologically important estuarine biota. Low salinity reaches in the upper estuary support submerged aquatic vegetation (SAV) that provide protection from marine predators and serve as an important nursery area for larval and juvenile fishes and invertebrates including commercially and/or recreationally important species. The prevailing SAV within the upper estuary is tape grass (Vallisneria americana), which occurs in well-defined beds in shallow water [Doering et al., 1999; SFWMD, 2000]. Lower reaches of the estuary are characterized by a shallow bay, where sparse to moderately dense beds of shoal grass (Halodule wrightii) and substantial oyster reef exist [Doering et al., 2002; Volety et al., 2009]. Tape grass, shoal grass, and oysters have been used as indicators of estuarine conditions for water resources management and ecosystem restoration [Chamberlain and Doering, 1998; Doering et al., 2002; Volety et al., 2009].

[11] In recent years, the estuary has become the focus of several comprehensive ecosystem restoration and water resources management programs. The South Florida Water Management District (SFWMD) established a Minimum Flow and Level (MFL) for the CRE based on salinity tolerance of tape grass in the upper estuary [SFWMD, 2000]. To help meet the MFL criteria, regional storage reservoirs have been proposed as a restoration strategy under the Comprehensive Everglades Restoration Plan [USACE and SFWMD, 2010] and the Northern Everglades Estuarine Protection Program [SFWMD, 2009]. The Lake Okeechobee Adaptive Protocols (LOAP) was developed to guide low level environmental releases from the lake to the CRE [SFWMD et al., 2010]. An essential step to support these programs is the simulation and forecasting of estuarine salinities under varying flow conditions and restoration alternatives. The SFWMD has employed both a 3-D hydrodynamic model (CH3D) and a time series model as management tools to conduct such analyses. The Caloosahatchee Estuary CH3D Model was described in detail by Qiu [2003, 2006] and Qiu et al. [2007]. This paper documents the development and application of the time series model.

3. Data Sources and Salinity Forcing Function Analysis

3.1. Data Sources

[12] The SFWMD initiated a continuous salinity monitoring program in the CRE in 1992. Temperature and specific conductivity data are collected at 15 min intervals at two depths of seven locations along the estuary: S-79, BR31, Val I-75 (I-75), Ft. Myers, Cape Coral, Shell Point, and the Sanibel Causeway (Figure 2). Water surface elevation data with the National Geodetic Vertical Datum of 1929 (NGVD 29) are collected at I-75 and Shell Point. Salinities of the surface layer collected at four stations including I-75, Ft. Myers, Cape Coral, Shell Point were selected for modeling analyses. These four locations were selected because of their association with critical ecological habitats and ongoing ecosystem restoration and water resources management programs of the estuary (Table 1). The 15 min data were averaged to obtain daily means of salinity and water surface elevation to be consistent with the daily time step used in hydrologic modeling and ecological evaluation [SFWMD, 2009; USACE and SFWMD, 2010].

Figure 2.

Salinity monitoring network in the Caloosahatchee River Estuary.

Table 1. Selected Salinity Monitoring Stations and Associated Ecological Habitats in the Caloosahatchee River Estuary
LocationDistance From Headwater Structure S-79 (km)Associated Ecological HabitatPreferred Salinity Range (psu)Water Resources Management Mandates or Plans
Val-I7511.5Low salinity zone0–5Lake Okeechobee operation adaptive protocols [SFWMD et al., 2010]
Ft. Myers20.6Tape grass0–10Caloosahatchee River Estuary minimum flows and levels [SFWMD, 2000]
Cape Coral32.8Oysters10–25Caloosahatchee River watershed protection plan [SFWMD, 2009]; C-43 reservoir integrated project implementation report [USACE and SFWMD, 2010]
Shell Point41.9Oysters and Seagrasses15–30Caloosahatchee River watershed protection plan [SFWMD, 2009]; C-43 reservoir integrated project implementation report [USACE and SFWMD, 2010]

[13] Freshwater discharges are measured at structures S-79 and S-77. Flows of five tributaries in the tidal basin have been monitored since 2008, and these data were used to calibrate the Tidal Caloosahatchee Basin Hydrologic Model to estimate the freshwater inflow from the tidal basin into the estuary [Konyha and Wan, 2011]. Daily rainfall and evaporation data were obtained from the SFWMD DBHYDRO database.

3.2. Estuarine Salinity and Hydrology

[14] The salinity structure within an estuary is typically controlled by a combination of factors including tidal dynamics, freshwater input, and wind forcing, varying with spatial and temporal scales of interest [Orlando et al., 1993; Ward and Montague, 1996]. When the tidal signal in salinity is smoothed on a daily basis, freshwater inflows are probably the most important factor affecting salinities. Figure 3 depicts daily changes in rainfall, total freshwater inflows, and salinities in the estuary during the period from 2007 to 2012. Salinity in the estuary is clearly driven by hydrology on multiannual, seasonal, and daily time scales. The distinct dry and wet seasons of subtropical climate, which is typical in south Florida, are keyed into precipitation and runoff in the watershed, and thereby salinity in the estuary. Rainfall in the area averages about 1340 mm yr−1 (from 1965 to 2005), ranging from 860 mm yr−1 (in 2000) to 1700 mm yr−1 (in 1995). About 70% of annual precipitation falls during the wet season (from June to October) and 30% during the dry season (from November to May) (Figure 3a). This rainfall pattern causes a large seasonal signal in freshwater discharge to the CRE. On average, runoff from the C-43 basin east of S-79 is more than three times higher in the wet season than in the dry season. The system often receives low or no flows from S-79 during the dry season, and this allows salt to intrude up to the head of the estuary at S-79. Salinity sometimes reaches above 20 psu in the upper estuary (e.g., during the 2007 drought, Figure 3b), causing mortality of tape grass and brackish water organisms that normally inhabit this area [Chamberlain and Doering, 1998; SFWMD, 2000; Doering et al., 2002]. In contrast, low levels of salinity prevail during the wet season. Extremely high flows, which are typically associated with Lake Okeechobee releases, can drive the entire system nearly fresh (Figure 3c). Salinity at such low levels stresses marine biota in the lower estuary [Chamberlain and Doering, 1998; Doering et al., 2002; Volety et al., 2009].

Figure 3.

(top) Time series of rainfall and total inflows, (middle) salinities measured at I-75, Ft. Myers, and (bottom) Cape Coral and Shell Point from January 2007 to January 2012.

[15] A simple regression analysis demonstrated an apparent inverse relationship between estuarine salinity and freshwater inflow with strong variance of salinity at a given inflow (Figure 4). A few notes deserve our attention with respect to this analysis. First, in spite of the apparent inverse relationship between salinity and inflow as found in other estuaries [Orlando et al., 1993; Ward and Montague, 1996], such a simple statistical relationship is unable to account for the temporal variation of salinity in the estuary. This is partly because salinity distribution in the estuary has a memory of its past. The response of salinity to a given inflow depends strongly on its antecedent conditions, which can be further complicated by the lag between freshwater signal and the corresponding response of salinity somewhere in the estuary [Ward and Montague, 1996].

Figure 4.

Regression between salinity and total inflow at (a) I-75, (b) Ft. Myers, (c) Cape Coral, and (d) Shell Point (data period from January 2006 to February 2012). Lines are best fit regression lines.

[16] Second, since salinity generally increases in a downstream direction as distance from S-79 and proximity to the Gulf of Mexico increase, the influence of freshwater inflow on salinity changes with respect to locations in the estuary (see different regression equations at different locations in Figure 4) [Savenije, 2012]. The upper estuary can become fresh rapidly after a certain threshold of inflow (about 100 cubic meters per second (cms)) is exceeded, reflecting the displacement of salt by freshwater. Beyond this threshold, the segment remains fresh with further increases in inflow (Figures 4a and 4b). The magnitude of the threshold inflow increases with increasing proximity to the Gulf since it takes much more fresh water to push the salt out of the system in the lower estuary. The geomorphology of the estuary is also a factor [Savenije, 2012]. In contrast to the narrow and dredged reach upstream of I-75, the downstream reach becomes much wider. As freshwater diffuses into a larger surface area and volume, its impact becomes gradually reduced.

[17] Third, salinity variance at a given inflow is especially pronounced in the low inflow regime. A close examination of the temporal variation of salinity with inflow and rainfall during the dry season reveals that salinity may continue to increase while inflows are extremely low (Figure 5), reflecting the influence of “tidal mixing” and net deficit of rainfall (high evaporation) on saltwater intrusion [Savenije, 1988; Savenije and Pages, 1992]. Saltwater intrusion is especially pronounced during abnormally dry periods. Under this condition, the effect of rainfall becomes significant in comparison to periods of high river inflow when the influence of the freshwater plume on salinity is predominant and the direct effect of rainfall becomes negligible.

Figure 5.

Salinity changes with rainfall and freshwater inflows during the dry season of 2011. Note salinity increase during periods of no flows at S-79.

3.3. Estuarine Salinity and Water Surface Elevation

[18] Fluctuation of salinity affected by changes in water surface elevation, which is driven by tide, current, longshore pressure gradient, and wind forcing, can best be seen during a tidal cycle. When data are averaged on a daily basis, the influence of water level changes can be masked by that of other factors such as freshwater inflow. However, we examined the temporal variation in daily mean water surface elevation measured at I-75 and salinities in the estuary during the dry season when the S-79 structure was closed. Figure 6, as an example of this analysis, indicates that during 18 October to 15 December 2006, daily salinity clearly fluctuated with water surface elevation along with the general dry season uptrend noted earlier. This fluctuation is indicative of the influence of “stacking up” of coastal water and density current on salt transport in the estuary [Ward and Montague, 1996]. Also note that the salinity fluctuation gradually dissipated in the upper estuary, especially at I-75.

Figure 6.

Changes of salinity with fluctuation of water surface elevation measured at I-75 from 18 October 2006 to 18 December 2006 when no discharge was made at S-79.

4. Model Development

4.1. Model Formulation

[19] The salinity model was developed based on a combination of empirical analyses of salinity forcing functions described above and time series analysis. Autoregressive modeling has been extensively used in the field of hydrology for prediction of surface water processes such as precipitation and stream flows [e.g., Lettenmaier and Wood, 1993; Salas, 1993]. Its application to estuarine salinity modeling has been limited with exception of Sun and Koch [2001] who applied autoregressive integrated moving average models to simulate salinity in the Apalachicola Bay, Florida. This type of analysis assumes that the present state of a system is a function of the present and past values of its inputs [Box and Jenkins, 1976]. Thus, the model is able to account for lags in the system, making it suitable for simulating the response of salinity to hydrologic signals. The salinity model developed in this study resembles an autoregressive exogenous model, which relates the current value of the salinity time series with (1) its past values and (2) the current and past values of the driving (exogenous) series (that cause salinity change). The general form of this model can be stated algebraically as:

display math(1)

where St, St−1, St−2 is the salinity time series, Et, Et−1, Et−2 is the exogenous time series, which are inflows, rainfall, and tidal water surface elevation in this study, and t is the error term.

[20] In our application, we did not follow the traditional statistical procedure of Box and Jenkins [1976]. Instead, our previous analysis of the interactions between salinity, freshwater inflow, rainfall, and water surface elevation is factored into the formulation of the salinity model. Equation (1) is simplified into a three-regime first-order threshold model with exogenous inputs in a form of:

display math(2)

where α1, β1, α2, β2, α3, and β3 are model coefficients. St is the salinity on day t, and St−1 is the salinity of the previous day t − 1. δSt is the nonlinear function accounting for salinity changes induced by freshwater inflow (δSQ), rainfall (δSR), and tidal water level (δSH). The flow regime is defined by total inflow on day t, Qt, according to the upper and lower thresholds, QUT and QLT, which vary with the location in the estuary.

[21] Salinity changes induced by flow (δSQ), local rainfall (δSR), and water level (δSH) are additive, and their relative contribution depends on the flow regime. In the high flow regime, (Qt ≥ QUT), an estuary segment can be almost fresh and δSt is negligible. In the low flow regime, (Qt ≤ QLT), the influence of freshwater inflow becomes negligible. Salinity change is mostly due to localized rainfall and tidal mixing, which can be written as:

display math(3)

where So is the salinity of ocean water (35 psu), a is the scaling factor; math formula is the moving average daily rainfall prior to day t; and math formula is the salinity step impulse as a function of the moving average rainfall. The number of days included in the moving average represents the lag time of salinity response to a rainfall event. During an extended period of dry conditions (net deficit of rainfall), salinity can approach So or larger than So, leading to hypersaline conditions [Savenije, 1988; Savenije and Pages, 1992].

[22] In the intermediate flow regime, freshwater inflow is the dominant factor attributing to change in salinity. The daily change in salinity is simulated with:

display math(4)

where math formula is the moving average flow prior to day t. The number of days for the moving average reflects the local hydrologic characteristics and the corresponding salinity response in the estuary. In the CRE, 3–15 days of moving average is used, and the lag time increases as it moves from upstream to downstream. math formula is the salinity step impulse as a function of the magnitude of math formula and salinity rising or falling regime. math formula is the reference state salinity at given flow Q on day t. For the CRE, the reference state salinity at a particular location was obtained with regression analyses of flow and salinity data or steady state simulations of a range of constant inflows using the Caloosahatchee CH3D Model if field data were not available. An exponential decay function is obtained for the CRE to estimate math formula with a given flow Q on day t:

display math(5)

where A and B are constants. In essence, math formula represents a dynamic process of salinity shift from the previous day's salinity toward the reference state salinity under today's flow Qt.

[23] Water surface elevation was included as a variable to account for the influence of tide, long-shore current, and wind forcing on the transport of salt in the estuarine system. Salinity change due to the difference in tidal elevation is expressed as:

display math(6)

where Ht − Ht−1 is the difference of water surface elevation between day t and day t − 1; H is the maximum tidal range; and KS is the scaling factor relating to maximum salinity difference during a tidal cycle.

4.2. Analogy to 1-D Salt Transport Model

[24] While the reader can refer to Sheng [1987, 1990] and Hamrick [1992] for detailed 3-D salt transport equations in a curvilinear or Cartesian coordinate system, we take the 1-D model of Savenije [1986] to demonstrate that major salt transport processes are implicitly embedded in each of the terms described above. The Savenije 1-D salt intrusion model on a daily time step can be written following

display math(7)

where s is the salinity, Q is the freshwater inflow, A is the cross-sectional area, D is the longitudinal dispersion coefficient, R is the net rainfall (rainfall – evaporation), and H is the water depth. All variables are functions of time t and distance x. The x axis has it origin at the mouth of the estuary. The upstream direction is taken as positive.

[25] Equation (7) can be compared with equation (2). The first term in equation (7) is the rate of change of salinity and it has to do with the autoregressive term in equation (2). The second term represents the advective transport of salt by freshwater inflow. The third term represents net upstream salt transport through density and tidal driven dispersion. The right-hand member is the source term due to net rainfall (rainfall – evaporation). All these terms are related with the physical drivers in equation (2). For a given estuary geometry, the relative size of each term depends on flow regimes [Savenije and Pages, 1989]. When the discharge is low (Qt ≤ QLT) and evaporation exceeds rainfall, the source term is positive, contributing to an increase in salinity [Savenije, 1988]. Note that effect of net rainfall in equation (7) depends on the salinity (s) and the depth (H), similar to that in equations (3) and (6).

[26] During high flows, the second term (advection) and the third term (dispersion) in equation (7) are large, and the source term can be neglected. Effective dispersion and advective transport counteract with each other. Savenije [2012] demonstrated that for an exponential variation of the cross-sectional area, variation of D in the longitudinal direction can be written:

display math(8)

where Do is the dispersion at the estuary mouth, Ao is the cross-sectional area at the estuary mouth, K is the dimensionless Van den Burgh's coefficient, Q is the freshwater inflow, and a is the convergence length. Apparently, the dispersion is highest near the estuary mouth, decreasing in upstream direction. It also decreases with increasing the magnitude of Q. When the discharge is very high advection overwhelms dispersion at upstream locations and the salinity is zero in spite of the salinity gradient between upstream and downstream reaches of the estuary. This is the same with the high flow regime in equation (2) with (Qt ≥ QUT).

[27] In the intermediate flow regime (QLT < Qt < QUT), advection and dispersion compete for dominance [Savenije, 1988], depending on the magnitude of Q and location in the estuary. At a given location, advection can dominate over dispersion if math formula in equation (4) < 0, leading to a decrease in salinity. Similarly, dispersion may dominate over advection if math formula in equation (4) > 0, leading to an increase in salinity. Thus, math formula in equation (4) can be considered as a parameter adjusting the rate of increase or decrease. In the time scale analysis of system response toward the steady state to variation of Q by Savenije [2012], the time scale is defined as

display math(9)

where S is the steady state salinity (equivalent to math formula), s is the unsteady state salinity (equivalent to St−1), Ts is the time scale for the system response. An analogy between equations (9) and (4) reveals that the physical meaning of math formula in equation (4) has to do with the system response time, which depends on the magnitude of Q and regimes of increase or decrease in salinity or flow. In essence, equation (4) represents a special case solution of equation (7).

[28] Under the steady state with ∂s/∂t = 0, one can rewrite equation (7) with the sources term neglected and integrate with respect to x under the boundary condition that S = 0 (the freshwater salinity) and ∂s/∂x = 0 when x → ∞. This yields:

display math(10)

[29] Integrating equation (10) with respect to x such that x=X yields essentially equation (5) in which constant A is related with X and constant B is related with the cross-sectional area and dispersion.

4.3. Model Calibration and Validation

[30] Four models were calibrated and validated with measured data collected at I-75, Ft. Myers, Cape Coral, and Shell Point. Measured salinity data collected since 2006 were used for model calibration, which involves the adjustment of model parameter values so that the deviations between the model results and the measured data are minimized and are within some acceptable ranges of accuracy. Available data collected prior to 2006 were used for model validation, which is the subsequent testing of the calibrated model. This second independent data set was used to further examine the model's ability to realistically represent the real world. This long-term data set encompasses a sufficient range of hydrologic events and conditions including the average, wet, and dry years to activate all salinity response processes during the calibration and validation periods.

[31] Five statistical parameters are used to evaluate the performance of the model: the coefficient of determination (R2), the Root–Mean-Square Error (RMSE), the Relative Root-Mean-Square Error (RRE), the Nash-Sutcliffe efficiency (NSE) coefficients, and the percent bias (PBIAS). These statistics are widely used in the calibration and validation of hydrologic and hydrodynamic models [e.g., Legates and McCabe, 1999; Moriasi et al., 2007; Ji, 2008]. The coefficient of determination is an indication of the percent of variation of the observed salinities being explained by simulated salinity. The RMSE is a measure of the deviation of the simulated salinities from the measured salinities. The dimensionless form of RMSE is the RRE which is defined as the ratio of RMSE to the range of observed salinities. The NSE indicates how well the plot of observed versus simulated data fits the 1:1 line. NSE ranges between −∞ and 1.0, with NSE = 1 being the optimal value. Percent bias (PBIAS) measures the average tendency of the simulated data to be larger or smaller than their observed counterparts. The optimal value of PBIAS is 0.0, with low-magnitude values indicating accurate model simulation. Positive values indicate model underestimation bias, and negative values indicate model overestimation bias. During model calibration, an objective function to minimize the value of ((1 − R2) + (1 − NSE) + RRE + Abs (PBIAS)) was used (Abs (PBIAS) is the absolute value of PBIAS).

[32] The mathematical equations of these parameters are given below:

display math(11)
display math(12)
display math(13)
display math(14)
display math(15)

where n is the number of data point (days) during the period of evaluation, Oi is the observed daily salinity, math formula is the mean of the observed daily salinity,Omax is the maximum value of observed salinity, Omin is the minimum value of observed salinity, Pi is the simulated daily salinity, and math formula is the mean of the simulated daily salinity.

[33] These statistical parameters of model calibration and validation are presented in Table 2. Graphic comparisons between observed and simulated salinities are also presented in Figures 7 and 8 for visual presentation of the goodness of fit. Overall, the performance of the model is considered excellent according to the calibration and validation criteria proposed by Moriasi et al. [2007] and Ji [2008]. For example, the R2 values ranged from 0.89 at Shell Point to 0.95 at Ft. Myers and I-75 during the calibration period. The RMSE values ranged from 1.38 at I-75 to 2.41 at Shell point, with corresponding REE values ranging from 5.47 to 7.14%. During the validation period, comparable model performance is achieved in spite of slightly higher REE and PBIAS values. Because there were no measured data collected at I-75 prior to 2006, measured data from 1992 to 2005 of the adjacent station—BR31 were used for model validation. In addition to the expected over estimation associated with using an upstream station, the validation results were still considered acceptable. Slight modification of selected model parameters yielded an excellent model for BR31 should application at this location be desired.

Table 2. Summary Statistics of Model Performance for Model Calibration and Model Validation
Locationn math formula (psu)Omin (psu)Omax (psu)R2RMSE (psu)RRE (%)NSEPBIAS (%)
  1. a

    No validation data available at I-75. Measured data collected at the adjacent station—BR31 were used for the similarity between the two data sets. Over estimation is expected with the use of BR31 data.

Ft. Myers
Cape Coral
Shell Point
Figure 7.

Plots of measured salinity versus simulated salinity during the model calibration period. Lines are the best fit 1:1 regression lines.

Figure 8.

Comparison of simulated salinity using the time series models (solid line) with measured salinity (dots) and modeled salinity using the Caloosahatchee CH3D Model (dashed line).

[34] Further validation of the model was conducted by comparing simulated salinity by the time series model with modeled data using the Caloosahatchee CH3D Model. As shown in Figure 8, the simulated salinities using the time series model and the Caloosahatchee CH3D Model agreed well with each other. The statistics of model calibration and validation presented in Table 2 are comparable or superior, at selected locations, to these of the Caloosahatchee CH3D Model for its pervious calibration [Qiu, 2006; Qiu et al., 2007] and the updated calibration (Table 3). The CH3D model currently runs on 16 parallel servers with a total of 176 CPU's spread (1.9–2.9 GHz and 8–24 GB RAM). A 10 year simulation at a time step of 90 s normally takes about 10–12 h, depending on the specific server on which the model is executed. In comparison, the time series model takes less than 1 min to run on a personal computer, offering a much more efficient means for long-term simulations.

Table 3. Summary Statistics of Model Performance of the Caloosahatchee CH3D Model (Calibration Period: January 2006 to March 2011)
LocationR2RMSE (psu)RRE (%)NSEPBIAS (%)
Ft. Myers0.922.318.250.917.50
Cape Coral0.942.306.920.93−2.63
Shell Point0.872.697.210.87−0.48

5. Model Application

[35] Due to its simplicity, the time series model has been applied by the SFWMD to aid in water resource management and evaluation in several ways. In addition to long-term simulations (from 1965 to 2005) for development and evaluation of restoration alternatives [e.g., USACE and SFWMD, 2010; Buzzelli et al., 2013], coupling the salinity model with the Lake Okeechobee Operations Screening (LOOPS) Model [Neidrauer et al., 2006] was implemented to design, test, and evaluate multiple Lake Okeechobee alternative operating protocols [SFWMD et al., 2010]. The salinity model was also restructured in the position analysis mode to examine the probability distribution of salinities in the estuary in a context of regional climate outlooks given a specific state of the system as the initial condition [Wan, 2012]. In this paper, we present a special case of model application to demonstrate how the model is used for short-term or near-term salinity forecasting and hindcasting to support implementation of the LOAP [SFWMD et al., 2010].

[36] The LOAP was developed to manage water resources associated with Lake Okeechobee. Protection of the CRE ecosystem is one of the objectives that are integrated into the protocols for lake operation. Every week, an interdisciplinary team of scientists and engineers from the SFWMD and USACE, and other federal and state agencies meet to review the status of the regional hydrologic system. Adaptive operation actions are discussed for implementation with the aim to balance multiple objectives of water management within the existing constraints of the physical system. Special consideration is given to conditions that may be stressful to particular ecosystems where operational adjustment can be made to minimize the impacts. Forecasting salinity at I-75 is one important component of this effort. The 30 day moving average salinity at I-75 is used as a performance measure of the LOAP which calls for releases of freshwater from S-79, and from Lake Okeechobee at S-77 if flow at S-79 is insufficient, to protect the low salinity zone habitat in the upper estuary. If the 30 day moving average salinity is forecast to exceed 5 psu within the next 2 weeks, then freshwater is required to maintain the low salinity zone.

[37] An explicit salinity forecast model for current or future salinity at I-75 as a function of past and future values of salinity, inflow, rainfall, and tidal water level was formulated to aid in the adaptive management process. Similar forecasting tools were also developed by Risley et al. [1993] and Sun and Koch [2001]. Figure 9 is an example of weekly salinity forecast with 2 weeks lead time for the period of 30 November to 13 December 2010. The format of the actual forecast diagram used in SFWMD's weekly operation meetings was kept with flows at S-79 and from the tidal basin expressed in cubic feet per second (cfs). In the forecast mode, the model performs the one-step-ahead forecast using the previous time step true salinity (measured) to predict salinity at the current time step. One observes that a close match between the forecast and observed salinity data series when one-step-head forecast is conducted. When forecasts with longer lead time such as 14 days are performed, the previous time step simulated salinities along with forecasted rainfall and flow are used. The dry season forecast typically involves two scenarios: with and without releases at S-79 (Figures 9a and 9b). In 2010, the dry season started early in October. Daily salinity at I-75 increased above 5 psu soon after discharge at S-79 ceased in mid-October. Implementation of LOAP initiated a series of pulsed releases averaging 12.74 cms or 450 cfs at S-79. These releases were shown to be effective in keeping salinity at a level that is beneficial to low salinity zone biota inhabiting in the upper estuary (Figure 9). The forecast indicated that the 30 day moving average salinity would increase above 5 psu without releases of freshwater at S-79.

Figure 9.

Salinity forecast at I-75 of 2 week lead time for the period from 30 November 2010 to 14 December 2010: (a) no discharge at S-79 and (b) 7 day pulse release at S-79 averaging 450 cfs (12.7 cms). One-step-ahead forecast was conducted using previous time step true salinity.

[38] For a retrospective analysis of the benefit of the LOAP releases, the model was set to run in the hindcast mode assuming that the releases at S-79 under the LOAP were not made while all the rest of input data were kept unchanged. Figure 10 depicts how salinities at I-75 and Ft. Myers would have increased if the seven pulse releases as shown in Figure 9 had not been made during the period from 29 October to 16 December 2010. The hindcast salinities were compared with the actual salinities observed in the field. This modeling exercise clearly shows that without these releases, salinity would have continued its uptrend beginning late October, eventually reaching about 15 psu at I-75 and 20 psu at Ft. Myers in mid-December. Salinities this high are detrimental to tape grass in the upper estuary [Chamberlain and Doering, 1998; Doering et al., 1999.

Figure 10.

Retrospective analysis of the benefit of managed pulse releases at S-79, comparing actual salinity and hindcast salinity at (top) I-75 and (bottom) Ft. Myers assuming the seven pulse releases had not been made during 29 October 2010 to 16 December 2010 (shaded area).

6. Conclusions

[39] As a management tool for rapid assessment and forecasting, the time series salinity model developed in this study has a number of clear advantages over complex, process-based hydrodynamic models, not the least of which is its simplicity and computational speed. The model has relatively few data requirements and can be easily calibrated. The model predicts and is calibrated against, daily salinity data, a time scale sufficient for many applications. Model runtimes are almost instantaneous compared with its 3-D counterpart, which requires computations in time steps of seconds or minutes. The model provides a relatively inexpensive, robust means of evaluating or forecasting the impact of freshwater inflow management on estuarine salinity. Although this approach is documented herein only for the CRE, similar modeling tools can be developed using the approach for other estuaries where freshwater inflow serves as a key factor influencing salinity [Wan, 2012].


[40] We would like to thank Calvin Neidrauer for constructive feedback on the model application for salinity forecasting. Reviews by Peter Doering, Susan Gray, Calvin Neidrauer, Paul Trimble, and three anonymous reviewers are also greatly appreciated.