Improved method for calibration of exchange flows for a physical transport box model of Tampa Bay, FL USA

Abstract

We report the results for both sequential and simultaneous calibration of exchange flows between segments of a 10-box, 1-dimensional, well mixed, bifurcated tidal mixing model for Tampa Bay. Calibrations were conducted for three model options having different mathematical expressions for evaporative loss. In approaching this project we asked three questions: does simultaneous calibration or sequential calibration yield better box model performance; which evaporation option best predicts observed salinities; and how well does model performance compare to more complex hydrodynamic models. Sequential calibration followed the classical salt balance and steady state approach. The nonlinear parameter estimator (PEST) was used for simultaneous calibration. The sequential approach proved useful in evaluating the three evaporation options. However, simultaneous calibration proved superior in predicting observed salinities but was ineffective in discerning differences between evaporation options. The simultaneously calibrated model produced residence times that fell within the range of more complex hydrodynamic models of Tampa Bay.

1. Introduction

A physical water transport model is at the heart of most estuarine water quality models is. The development of such models incorporates various levels of sophistication dependent on the specific utility in application. Three main categories, tidal prism, box, and full hydrodynamic encompass+ the range of models. They differ in complexity, data needs, calibration challenges, and computational run times (Kremer et al. 2010). The simplest method is the tidal prism (Officer 1976). A volume of water equal to the prism, the difference between low and high tide volume, is added to an estuary during flood tide from waters external to the estuary. Complete mixing of the estuary is assumed before the next ebb tide that removes a volume of mixed water equal to the prism to complete the tidal exchange. Early versions assumed negligible fresh water inputs (Abdelrhman 2007), required only data on tidal ranges and bathymetric measurements, and returned no portion of the ebb tide water on the flood tide. These conditions essentially limited use of the prism approach to small estuaries with little or no riverine input and strong alongshore currents outside the mouth. In later versions, the prism approach was modified for small estuaries with significant freshwater inputs and/or instances of return flow (Luketina 1998; Sheldon and Alber 2006).

Hydrodynamic models based on the laws of physics are at the other end of the complexity spectrum. These models are used to understand the very fine scale movement of water within an estuary. Time steps less than a minute and scales of less than 100 m (Galperin et al. 1991; Kremer et al. 2010) are often used to resolve important aspects of the physical transport. These models generally require data for tides, wind speed, precipitation, river and stream flow, and air pressure as well as bathymetric data (Weisberg and Zheng 2006; Meyers et al.2007). Data collected at or near the model time step best support model development. Hydrodynamic models generally require powerful modern computers or computer clusters. A ROMS hydrodynamic model of Providence River and Narragansett Bay, RI USA took 9 days of computer time on a PSSC Labs Beowulf cluster for a 77 day model simulation (Kremer et al. 2010). The long run times for hydrodynamic models are exacerbated when they are coupled to ecological models. Even when stored hydrodynamic data are used rather than simultaneously running the hydrodynamics with the ecological model the computational burden can be enormous (Kremer et al. 2010).

The box model approach lies betwixt these two bookends of complexity. Box models divide the water body of interest into a limited number of segments (boxes, also termed tanks or compartments). Estuary box models range from as little as three or four boxes (Lehrter 2008), up to as many as 59 boxes (Walker 1999). Box model assume complete mixing of box contents following each time step. Box models sacrifice the detail of physics based models for simplicity in coupling with ecological process based models that are in large part based on daily time steps. The box model approach has a long history of development stretching back to the segmented tidal prism model of Ketchum (1951). Several modifications of the approach dealing with methods of segmentation have subsequently been developed (Dyer and Taylor 1973; Wood 1979; Miller and McPherson 1991). Officer (1980) was the first to show that if salinity was assumed to be at steady state and freshwater inputs entered through the head of the estuary, exchange coefficients could be derived from a single layer system of N linear salt balance equations with N exchange coefficients. Officer also introduced stratified box models that vertically divide the estuary at the halocline. Using a modification of the Officer (1980) approach, Hagy et al. (2000) described a linear box model for the Patuxent River estuary. Of particular interest was the inclusion of multiple riverine inputs and the inclusion of precipitation and evaporation in the freshwater balance. These approaches with some modification have been extended to non-linear bifurcated estuaries (Camacho-Ibar 2003; Babson et al. 2006; Hagy and Murrell 2007; Lehrter 2008). Exchange flows between boxes are computed by solving equations of salt and water balance. Using only salinity, freshwater input, and constant segment volume, equations for daily exchange flows were solved two at a time beginning at the head of the estuary and ending at a terminal border (Officer 1980; Hagy et al. 2000; Lehrter 2008).

One challenge raised by these approaches is that sequentially solving the salt balance equations could lead to error propagation from up estuary to the down estuary boxes. Smith and Hollibaugh (1997) found salinity differences <0.3 resulted in abnormally high exchange volumes (both positive and negative) in Tomales Bay, California that cycles from below oceanic salinity in the winter to above oceanic salinity in the summer. The anomalies occurred in the transitions between seasons when ocean and bay salinities neared equivalency. Salinity and water level variability can also lead to instability of ill-conditioned models (Sakov and Parslow 2004). Soetaert and Herman (1995) suggested that smoothing salinity data before exchange coefficients were determined reduced the effects of variable salinities.

Inverse modeling approaches using optimization techniques, based on the minimization of sum of squared residuals, have been employed (Gilcoto et al. 2001; Sakov and Parslow 2004; Soetaert and Herman 1995) as an alternative approach for computing exchange flows from salt balance equations. Still others have adapted results from fine scale hydrodynamic models to establish exchange coefficients for less complex box models (Kremer et al 2010; Larsen et al. 2013). These approaches have been used for over 30 years by numerous researchers (see Kremer et al. 2010 for short review); however, rarely has the conversion process been effectively assessed or reported. In a comparison of a Regional Ocean Model System (ROMS) hydrodynamic model and a box model for Narragansett Bay, Kremer et al. (2010) detail the conversion process and provide an in-depth comparison of model results. The box model that divided Narragansett Bay into 15 boxes compared favorably to the ROMS model simulations of salinity and was, cost effective, convenient and reduced computational burden. The complex hydrodynamic model took nine days on a powerful computing cluster for a simulation run of 77 model days whereas the box model took less than five seconds on a desk top computer. However since each box is assumed to be well mixed after each time step the Box model is not able to resolve important aspect of the physics of mixing baroclinic or residual circulation (Kremer et al. 2010). Unfortunately this approach requires the development of the complex hydrodynamic model first.

In this study we compare and contrast two separate approaches for computing exchange flows for a 1-dimensional box model constructed for the Tampa Bay estuary. The first approach sequentially solves the steady state salt and water balance equations. Exchange flows are computed algebraically (Officer 1980). In the second exchange flows are calibrated simultaneously using the parameter estimation tool PEST (Doherty 2001). The PEST approach uses a particularly robust variant of the Gauss-Marquardt-Levenberg method to estimate exchange flows by minimizing the weighted sum of squared errors (Ф) between predicted and observed salinities through a series of iterative calibrations. Both methods were examined against three box models having different mathematical expressions for evaporation. Our overall objectives were to determine: (1) whether simultaneous calibration yields better model performance than by sequentially solving equations; and (2) which of the three evaporation options best predicted observed salinities; and (3) how model performance compares to more sophisticated hydrodynamic modeling efforts.

2. Methods

2.1. Study area description

Tampa Bay is on the Gulf coast of central Florida, USA (Fig. 1). It is a large (1,031 km²) shallow estuary (Zervas 1993) with a mean depth of 3.7 m and a total water volume of 3.2 km³. The Bay is generally divided into four major sections; Old Tampa Bay (OTB), Hillsborough Bay (HB), Middle Bay (MB), and Lower Bay (LB). Four major rivers (Alafia, Hillsborough, Little Manatee and Manatee) and several small streams (including Brooker, Bullfrog, Delaney Rocky Creek and Sweetwater) contribute to the freshwater flow into the Bay. The Alafia and Hillsborough rivers flow directly into HB. Ungauged watersheds, point sources, rain and groundwater are also sources of freshwater. Groundwater directly enters coastal waters through seepage areas located near the intertidal zone or in some cases further off shore (Bokuniewicz 1992; Taniguchi et al. 2002). Tampa Bay also has a long coast line and as a result groundwater is a significant input to the Bay (Brooks et.al. 1993; Kroeger et al. 2007). Rain is a significant freshwater input and evaporation is a significant freshwater loss because of the Bay’s large surface area.

Figure 1.

Tampa Bay, Florida illustrating gauged (dark shade) and ungauged (light shade) drainage basins. Gauged basins are AL - Alafia above Lithia, AN- Alafia North Prong, BC- Brooker Creek, BU- Bullfrog Creek, DE- Delaney Creek, HM- Hillsborough River above Morris Bridge, HZ-Hillsborough River above Zephyrhills, LM- Little Manatee River, RC-Rocky Creek, SW- Sweetwater Creek. Ungauged basins are numbered to indicate into which Bay Section (Fig. 2) they drain. (1 column)

Freshwater inflow comes from a small watershed of 6,483 km² which results in Tampa Bay having one of the smallest freshwater input to tidal prism ratios (0.03) of the Gulf of Mexico Estuaries (Solis and Powell 1999). Physical mixing of the Bay is primarily tidal and wind driven. An analysis of water level variance near St. Petersburg (Weisberg and Zheng 2006; http://tidesandcurrents.noaa.gov/harcon.html?id=8726520) showed that about 24% of the variance was associated with semi-diurnal tides, 42% with diurnal tides and 31% with long-term non-tidal forces (wind and steric effects). Higher tidal harmonic and seiche effects contributed the remaining 3%.

2.2. 1-Dimensional Box Model

A 1-dimensional, 10-box model (Fig. 2) previously used by Wang et al. (1999) was selected to characterize the tidally averaged mixing of the Bay. Boxes will be referred to as segments from this point forward. Segments were based on grouping of monitoring sites having similar salinities as previously reported by Wang et al. (1999). The period of record for salinity data spanned January 1985 to December 1994. Data from 1985–1991 was used for model calibration and data from 1992–1994 was used for validation. The ten segments were grouped into four Bay sections: OTB (I-III), HB (IV-VI), MB (VII-VIII), and LB (IX-X). Segments were assumed to be vertically and horizontally well mixed and of constant volume. Data collected monthly by the Environmental Protection Commission of Hillsborough County (EPCHC) from 1985 to 1994 showed little difference between surface and bottom waters for OTB, HB, and most parts of LB and MB (Wang et al. 1999).

Figure 2.

Model segmentation of Tampa Bay, salinity monitoring sites and exchange flow number designations. Old Tampa Bay (I-III), Hillsborough Bay IV-V), Middle Bay (VI-VIII), and Lower Bay (IX-X). Monitoring sites are marked by black dots. Exchange flows are marked by arrows and circles. Column)

Simulistics ® software (Simile) was used to create a C++ model code to compute daily changes in salinity from the box model equations 2 and 3. Simile is an object- based dynamic system modeling and simulation software that auto-generates C++ model code. A conceptual representation of the Simile model based on daily time steps is shown in Fig. 3. The model was run using Euler integration.

Figure 3.

Simile model diagram showing data inputs (ovals), compartments (squared boxes), parameters (rounded box) and influences (dotted lines). The central dark arrow calculates the daily change in salinity from Equations 2 and 3 for each segment and adds these changes to the salinity compartment. Influence lines connect parameters and data to calculations and indicate the sequence of calculations. Predicted daily salinities are converted to average monthly salinities (AMS) that are used in calibration. (2 columns)

The salinity in segment i at the end (S_Ei) of each daily cycle was calculated from the salinity at the start (S_Si) plus the change in salinity (ΔS_Si) during the time step (Equation 1). At the beginning of each cycle the previous S_Ei becomes the new S_Si. At T₁ the new time step (S_So) was set to the Dec. 1984 AMS for each of the segments. The ΔS_Si was calculated on a daily basis using Equation 2 for all Bay Segments, except for Segment VII when Equation 3 was used. The parameter R_k was introduced to identify three different options (R₁, R₂, and R₃ as defined below) for handling evaporation in Equations 2 and 3. Parameters for all equations are as described in Table 1.

Table 1.

Description of parameters used in the model.

Term	Definition	Units
Equations 1,2, 3, 6 and 7
SEi	salinity in Segment i at the end of the current time step
Δt	time step (daily)	day
Vi	volume of Segment i	m3
SSi	salinity in Segment i at the beginning of the current time step
SSi-1	salinity in Segment i-1 at the beginning of the current time step
SSi+1	salinity in Segment i+1 at the beginning of the current time step
Qei	exchange flow between Segment i and i+1	m3d−1
Qei-1	exchange flow between Segment i and i-1	m3d−1
Qei-4	exchange flow between Segment I and i-4	m3d−1
Qri	Σ Direct gauged, ungauged, precipitation, groundwater and point source inputs to Segment i	m3d−1
Qrai	Σ up bay inputs into Segment i	m3d−1
Qrah	Σ up bay inputs for Segments 4, 5, and 6.	m3d−1
Qrao	Σ up bay inputs for Segments 1, 2, and 3.	m3d−1
Qvi	evaporative losses from Segment i	m3d−1
Qvai	Σ evaporative losses from Segments up bay from Segment i	m3d−1
Equation 4
Flow	nonpoint source flow	(m/month)
L1	fraction of landscape that was urban
L2	fraction of landscape that was agriculture
L3	fraction of landscape that was wetlands
L4	fraction of landscape that was forests
Rn0	rainfall current month	(m/month)
Rn1	rainfall previous month	(m/month)
Rn2	rainfall 2 months previous	(m/month)
Equation 5
Pj	estimated total monthly precipitation for the jth drainage basin	(m/month)
Kj	number of National Weather Stations (NWS) within 50 km of the geographic center of the jth drainage basin	none
PK	total monthly precipitation recorded at the kth NWS station	(m/month)
DK	distance between the geographic center of the jth drainage basin and the kth NWS station	m

$$S_{Ei}=S_{Si}+\Delta S_{Si}$$ Eq. 1

$$\Delta S_{Si}=\frac{\Delta t}{V_i}\left[\left(Q_{rai}\right.\left(S_{Si-1}-S_{Si}\right)\right.+Q_{ei-1}\left(S_{Si-1}\right.-\left.S_{Si}\right)+Q_{ei}\left(S_{Si+1}\right.-\left.S_{Si}\right)-Q_{ri}\left(S_{Si}\right.)+R_k]$$ Eq. 2

Bay Segment VII represents the branch point of the Bay and ΔS_Si was calculated using Equation 3:

$$\Delta S_{Si}=\frac{\Delta t}{V_i}\left[Q_{rah}\right.\left(S_{Si-1}\right.-\left.S_{Si}\right)+Q_{ei-1}\left(S_{Si-1}\right.-\left.S_{Si}\right)+Q_{rao}\left(S_{Si-4}\right.-\left.S_{Si}\right)+Q_{ei-4}\left(S_{Si-4}\right.-\left.S_{Si}\right){+Q}_{ei}\left(S_{Si+1}\right.-\left.S_{Si}\right)-Q_{ri}\left.S_{Si}+R_k\right]$$ Eq. 3

Where R₁=0 and Q_ri = Σ Direct gauged, ungauged, rain and groundwater inputs to Segment i – evaporative losses from Segment i and R₂ = Q_viS_Si+1 and Q_ri = Σ Direct gauged, ungauged, rain and groundwater inputs to Segment i and R₃ = Q_viS_Si+1+Q_vai(S_Si+1−S_Si) and Q_ri = Σ Direct gauged, ungauged, rain and groundwater inputs to Segment i Separate daily exchange flows (Q_eis) for each of the ten Bay Segments (Fig. 2) were used for dry and wet seasons and were provided to the model as natural logs.

2.3. Salinity and freshwater inputs

Monthly salinity data for each of the monitoring sites in Fig.2 were obtained from Environmental Protection Commission of Hillsborough County (http://www.epchc.org/index.aspx?NID=219. Monthly salinity data associated with monitoring sites for specific segments (Fig. 2) were averaged resulting in ten data sets, each containing 120 average monthly salinities (AMS).

Freshwater inputs to each Bay Segment were calculated as the sum of flows from gauged rivers and creeks (R) and ungauged flows (UF), precipitation (P), ground water (GW) and point sources (PS) (F = R + UF + P + GW + PS).

Freshwater inflows (m³ per month) were obtained from the USGS National Water Information System (http://maps.waterdata.usgs.gov/mapper/index.html) for gauged rivers and creeks at points above tidal influence. Gauge sites were Alafia at Lithia (02301500), Brooker Creek near Tarpon Springs (02307359), Bullfrog Creek near Wimauma (02300700), Delaney Creek near Tampa (02301750), Hillsborough River at Morris Bridge near Thonotosassa (02303330), Little Manatee River near Wimauma (02300500), Rocky Creek near Sulphur Springs (02307000), and Sweetwater Creek near Tampa (02306647). Complete data sets were obtained for all but Rocky Creek and Sweetwater Creek. Data were not available from January to September of 1985 for those two creeks. Data for these months were estimated by averaging monthly flows for a particular month using data available for 1986 −1991. These gauged river basins comprised 46% of the Tampa Bay watershed. The remaining 54% were considered ungauged basins (Fig. 1).

Monthly freshwater flows from ungauged basins were estimated from the basins landscape characteristics (http://www.swfwmd.state.fl.us/data/gis/layer_library/category/physical_dense) and estimated rainfall using Equation 4 (Zarbock et al., 1994; see Table 2 for parameter definitions):

$$\log \left(Flow\right)=c_1{L}_1+c_2{L}_2+c_3{L}_3+c_4{L}_4+b_0{Rn}_0+b_1{Rn}_1+b_2{Rn}_2$$ Eq. 4

Table 2.

Summary of average monthly salinities, freshwater inputs to and evaporation from Tampa Bay segments from October 1985 through December 1991. Values are 10³ m³d⁻¹.

Segment		O-Ia	O-II	O-III	H-IV	H-V	H-VI	M-VII	M-VIII	L-IX	L-X
Salinity
Season
Dry	Ave	24.4	24.8	26.2	26.0	26.2	27.2	27.6	29.6	31.8	33.5
	SD	4.0	3.8	3.8	4.2	3.9	3.4	3.5	3.8	3.9	3.9
Wet	Ave	22.8	23.4	25.8	23.6	23.9	25.8	26.6	28.4	30.4	32.9
	SD	3.7	3.5	3.1	4.6	3.8	3.1	3.3	3.7	4.3	4.7
	Range	11–31	11–31	12–31	15–30	16–30	8–31	16–32	26–32	28–34	32–35
Precipitation (m3d−1)
Dry	Ave	96	203	177	106	137	210	257	285	296	254
	SD	79	168	146	87	111	171	218	241	243	206
Wet	Ave	193	417	367	212	274	435	556	620	643	558
	SD	116	253	226	125	161	271	403	441	406	336
Major Rivers and Creeks (m3d−1)
Dry	Ave	64	0	0	444	476	714	0	0	0	0
	SD	52			384	337	520
Wet	Ave	181	0	0	741	1065	1725	0	0	0	0
	SD	156			727	1030	1644
Ungauged Basins (m3d−1)
Dry	Ave	62	53	15	516	321	20	35	238	0.3	19
	SD	54	42	9	416	225	16	30	450	0.3	14
Wet	Ave	277	209	60	1065	511	125	164	380	6	287
	SD	317	315	65	764	492	175	297	976	9	355
Point Sources (m3d−1)
Dry	Ave	59	21	0.6	251	36	0.3	7	22	2	31
	SD	3	3	0.2	16	5	0.2	2	5	2	2
Wet	Ave	63	24	0.8	275	35	0.4	6	29	2	35
	SD	5	3	1	22	7	0.3	2	6	0.8	3
Groundwater (m3d−1)
Dry	Ave	49	37	71	71	50	7	9	8	17	13
	SD	2	1	2	2	2	0.2	0.3	0.3	0.6	0.4
Wet	Ave	40	32	63	96	68	17	20	18	21	16
	SD	1	1	1	1	1	0.2	0.3	0.3	0.3	0.2
Evaporation (m3d−1)
Dry	Ave	162	354	307	180	224	340	417	460	477	407
	SD	58	126	109	64	80	121	148	163	169	144
Wet	Ave	184	401	348	205	254	386	472	521	540	461
	SD	21	45	39	23	28	43	53	58	60	52

^aSegments: O = Old Tampa Bay; H = Hillsborough Bay; M = Middle Tampa Bay; L = Lower Tampa Bay. Segment numbers refer to segments shown in Figure 1.

The seven coefficients were estimated using a full multiple linear regression model applied to the gauged basins (SAS software) with no intercept (following Zarbock et al., 1994). The model was fitted to an 84 month time series of flow totals from the above eight freshwater inputs plus two additional gauged sub-basins at Hillsborough River near Zephyrhills (02303000) and North Prong Alafia River (02301000). Flows were adjusted by subtracting any point sources and dividing the resulting monthly volume (m³/month) by the surface area of a particular basin to give m/month. Landscape fractions were determined from 1990 GIS summary maps as described by (Zarbock et al., 1994).

Monthly precipitation rates (m/month) were estimated using Equation 5 that is based on the inverse distance-weighted average of the k_i weather stations within a 50 km radius from the i^th basin center.

$$P_i=\frac{\sum _{k=1}^{k_i}{P}_k\left(\frac{1}{D_k^2}\right)}{\sum _{k=1}^{ki}\left(\frac{1}{D_k^2}\right)}$$ Eq. 5

Monthly precipitation totals were obtained from the National Climate Data Center of the National Weather Service (http://lwf.ncdc.noaa.gov/oa/climate/stationlocator.html) from 1985 to 1994. Precipitation totals were obtained from 22 long-term stations (Appendix A.) within or near the Tampa Bay watershed. The data set was divided into four subsets: (1) wet season rain (Jul, Aug, Sept, Oct) plus landscape percentage for basins with <19% urban; (2) wet season rain plus landscape fraction for basins with >19% urban; (3) dry season rain (Jan, Feb, Mar, Apr, May, Jun, Nov, Dec) plus landscape fraction for basins with <19% urban; and (4) dry season rain plus landscape fraction for basins with > 19% urban. Four sets of coefficients were estimated from gauged basins and used to calculate ungauged flows from ungauged basins using the landscape characteristics and estimated precipitation rates for each of the ungauged basins (Fig. 1). Monthly precipitation rates directly to the ten Bay Segments were also calculated with equation 5 using the geographic centers of each of the Segments to calculate D_k. Ground water flow data calculated from Darcy’s equation were obtained from Zarbock et al. (1994). Flow data were reported for OTB, HB, MB and LB. Flows were parsed to the ten Bay Segments based on percent of total shore-line. Groundwater associated with river and streams and ungauged watersheds were not included in ground water calculations (Zarbock et al. 1994). The resulting monthly groundwater flows were kept constant over the seven-year calibration and three-year validation model runs.

Monthly point source flows were obtained through the Tampa Bay Estuary Program’s point source database. A complete listing of the 36 sources can be found in Zarbock et al. (1994).

2.4. Evaporation

Pan evaporation rates measured at the Lake Alfred Experimental Station, Lake Alfred, Florida, east of the Hillsborough drainage basin (http://my.sfwmd.gov/dbhydroplsql/show_dbkey_info.main_menu ) were used to calculate monthly evaporation rates, based on surface area, for each of the ten Bay Segments. A factor of 0.77 was used to adjust pan evaporation to evaporation from open water (Linacre 1994).

2. 5. Sequential Exchange Flow Calibration

Daily exchange flows (Q_eis) were calculated by a traditional steady state approach using a spread sheet format where exchange flows were sequentially calculated from the head to the mouth of the estuary (Miller and McPherson 1991; Hagy et al. 2000; Lehrter 2008) using equations 6 and 7. Equations 6 and 7 were derived from Equations 2 and 3 by setting ΔS_Si equal to 0. Daily exchange flows for each of the 84 months were calculated separately using seasonally averaged AMS for wet and dry seasons and daily freshwater inputs and evaporative losses. Daily freshwater inputs and evaporative losses were derived by dividing monthly inputs and losses by days per month. Unfortunately using the 84 month AMS data set for each segment led to some unrealistic large (greater than the volume of the segment) exchange flows that led to unstable model runs. Unrealistic exchange flows occurred when the AMS of neighbor segments differed by less than 0.3 Unrealistic exchange flows were eliminated by replacing wet season AMS with the average of the 28 AMS of the wet season (Jul, Aug, Sept and Oct) and by replacing dry season AMS with the average of the 56 AMS of dry season (Jan, Feb, Mar, Apr, May, Jun, Nov, and Dec) for each of the ten segments. These calculations resulted in 28 wet season and 56 dry season exchange flows per Bay Segment. The wet and dry exchange flows were then averaged to yield one final wet and one final dry exchange flow for each Bay Segment. At the Bay heads where only freshwater enters the Bay (i =1 and 4) Q_rai, Q_vai, Q_ei−1 and S_Si−1 were set to zero.

$$Q_{ei}=\frac{\left(S_{si}\right.-\left.S_{si-1}\right)\left(Q_{rai}\right.\left.{+Q}_{ei-1}\right)+Q_{ri}{S}_{si}-R_k}{\left(S_{si+1}\right.-\left.{Ss}_i\right)}$$ Eq 6

$$Q_{ei}=\frac{\left(S_{si}\right.-\left.S_{si-1}\right)\left(Q_{rah}\right.+\left.Q_{ei-1}\right)+\left(S_{si}\right.-\left.S_{si-4}\right)\left(Q_{rao}\right.+\left.Q_{ei-4}\right)+Q_{ri}{S}_{si}-R_k}{\left(S_{si+1}\right.-\left.S_{si}\right)}$$ Eq 7

2.6. Simultaneous exchange flow calibration

The simultaneous method coupled the Simile based non steady state model (Fig. 3) and the parameter estimation software PEST. PEST is a robust variant of the Gauss-Marquardt-Levenberg method of parameter estimation (Doherty and Johnston 2003). Optimization was based on minimizing the weighted sum of squared residuals (Ф):

$$\Phi =\sum _i^n{\omega }_i{\left(O_i-P_i\right)}^2$$ Eq.8

where n = the number of observations, O_i = the ith of n observed AMS, P_i = the ith of n predicted AMS and ω_i is inversely proportional to the standard deviation of the O_i data set. This approach is best described as a steepest assent hill climbing technique that initiates from initial estimates of the parameter space. The method is iterative and the relationship between model parameters and model-generated observations are linearized by formulating a Taylor expansion around the current best parameter set at the initiation of each iteration. Derivatives of each parameter are then calculated by the forward difference method. See Doherty (2001) for an in-depth review of the mathematics behind PEST. After each iteration the cost function φ is assessed to determine if the derivatives should be calculated by central difference to further optimize the calibration process. The calibration process is terminated and the calibrated parameters are reported once φ had reached a predetermined percent decrease from the previous iteration.

PEST directly interacts with a Tidal Mixing Model through the model input and output files. Simile software links to PEST through a menu driven process that allows the user to identify input parameters, initial parameter estimates and a calibration data set, and allows the user to provide upper and lower limits for Q_eis. For the purposes of PEST, all Q_ei parameters were natural log transformed (lnQ_eis). Without log transformation Ф increased to levels not supported by PEST and the calibration runs were automatically terminated early with an error notification. The initializing Q_ei parameter set was calculated by dividing the volume of the upstream Bay Segment of a segment pair (see Fig. 2) by ten. These values were used for both dry and wet conditions The PEST output generates an initial and a final Φ at the end of the complete calibration run.

2.7. Skill metrics

In addition to Φ, two additional metrics, average error (AE) and the reliability index (RI), were used during the calibration process to assess model skill at the segment level:

$$AE=\frac{\sum _{i=1}^n\left(P_i-O_i\right)}{n}$$ Eq. 9

$$RI=e^{\sqrt{\frac{1}{n}\sum _{i=1}^n{\left(ln\frac{O_i}{P_i}\right)}^2}}$$ Eq. 10

Where n = the number of observations, O_i = the ith of n observations, P_i = the ith of n predictions.

Average error (Equation 9) is an index of aggregate model bias (Stow et. al. 2009). The reliability index (Equation 10; Leggett and Williams 1981) calculates the factor by which model predictions differ from observations. A value near one indicates a close fit to observations whereas a value of two for example would indicate an average two-fold difference from observations.

2. 8. Model validation

Validation was conducted by extending the model for an addition three year validation period, 1992–1994. The lnQ_eis estimated through the calibration process were incorporated into an extended version and freshwater inputs (gauged, ungauged, and groundwater) and evaporative losses were extended three additional years. Predicted AMS obtained from model runs (1992–1994) were compared to observed AMS using skill metrics (AE and RI). This approach has been termed predictive validation (Power 1993).

2.9. Residence times

Residence times for Tampa Bay produced by our model were compared to residence times produced by other modeling approaches to further validate the model. Residence times are useful in understanding the transport of materials (e.g., nutrients or contaminants) within an estuary and are an important physical control of ecological processes including nutrient cycling. The fraction of nutrient loads exported from an estuary has been shown to be inversely proportional to residence time (Nixon et al. 1996). The methods of Miller and McPherson (1991) were used to calculate the seasonal estuarine residence time (ERT) for all the water in Tampa Bay using evaporation option R₃ and simultaneously calibrated exchange flows. For ERT and segment-based residence times (RT) calculations the initial conservative tracer was set to a mass of 1.0 for all segments. Head waters entering the Bay and the Gulf waters were set to zero. Residence times were calculated using the e-folding approach where residence time was the length of time (model days) for the total residual tracer to reach e⁻¹ times the original tracer mass. To examine the effects of wet and dry conditions on residence time, tracer was added to each of the four individual Bay sections in separate model runs begun on Feb 1, 1987 (dry) and Sept 1, 1985 (wet).

3. Results

3.1. Salinity and Freshwater Input

Salinity increased from an overall average of 24 at the upper most segment of OTB to 33 at the last seaward segment of LB (Table 2). Average monthly salinities of OTB (Segments I-III) and HB (Segments IV-VI) were the most variable ranging from11–31 and 15–30 with highest salinities in the dry season and lowest in the wet. Salinity variability increased from MB (Segments VII-VIII) to LB (Segments IX-X) with Segment X ranging from 32–35.

Precipitation accounted for 49 % of the freshwater to the Bay overall, but varied as a fraction of freshwater across segments and between wet and dry seasons (Table 2). Precipitation contributed more than 50 % to those segments with no direct freshwater inflows from monitored rivers or creeks. Precipitation contributed 88 to 90 % of freshwater to Segment IX. As a fraction of total freshwater input, precipitation contributed only 8–14 % of the freshwater to Segments IV and V. Direct inputs of freshwater from rivers and creeks occurred in Segments I, IV, V, and VI and ranged from 26 to 51 % of the freshwater. Segment IV received the most freshwater from all other sources (i.e., groundwater, point sources, and ungauged flows) (63–64%) and Segment VIII received the least (10–12 %).

Evaporation was a major contributor to water balance. For those Segments (II, III, VII, VIII, IX, X) receiving no freshwater from monitored rivers or creeks, evaporation was greater than freshwater inputs for more than half of the months in the dry season (Table 3). During the wet season evaporation exceeded freshwater input in only 25 to 32 % of the months. For Segments IV and V in HB where major river inputs occur, evaporation never exceeded freshwater input.

Table 3.

Number of months from October 1985 through December 1991 when evaporation exceeded freshwater inputs. Months with no available gauged river flow data were excluded from this survey.

Evaporation Exceeds Fresh Water Input
Bay Segment	Season		Total
	Drya	Wetb
I	5	0	5
II	30	6	36
III	32	9	41
IV	0	0	0
V	0	0	0
VI	17	1	18
VII	34	7	41
VIII	26	7	33
IX	39	9	48
X	37	5	42
Months Included in Study
Total	50	25	75

^aJan, Feb, Mar, Apr, May, Jun, Nov, Dec

^bJul, Aug, Sep, Oct

3.2. Calibration of Exchange Coefficients

The variability in lnQ_eis (Table 4) was dependent on both calibration approach and evaporation option. Values of lnQ_eis derived from the sequential approach were, in general, lower than lnQ_eis derived from simultaneous calibrations. Exceptions were noted for Segment IV during the dry season and Segments IV, VII and X during the wet season. With respect to evaporation options, sequential calibration dry season lnQ_eis were the most variable. lnQ_eis for Segments I and III decreased from option R₁ to R₃ by as much as 2 Ln units. All three of the dry season lnQ_eis for OTB (I, II, and III) R₃ were 2–5 Ln units less than the corresponding simultaneous lnQ_eis. Wet season lnQ_eis for Segments I and II were also lower but not to the same extent. We suggest that these low values primarily from the dry season contributed to the under prediction AMS observed in OTB (Fig. 4).

Table 4.

Natural log of Q_E1s (m³d⁻¹) produced from sequential and simultaneous calibration approaches. lnQ_eis are listed for the three evaporation options.

Segment	Sequential			Simultaneous
	R1	R2	R3	R1	R2	R3
Dry
I	17.475	16.826	16.014	17.941	17.951	18.273
II	14.494	14.279	14.169	16.856	16.851	16.942
III	13.898	13.706	12.225	17.583	17.582	17.182
IV	18.839	18.838	18.838	17.644	17.669	17.610
V	17.646	17.753	17.737	17.928	17.920	17.662
VI	18.997	19.102	19.085	19.221	19.229	18.828
VII	17.186	17.181	17.079	17.573	17.557	17.637
VIII	17.114	17.273	17.105	17.909	17.965	17.857
IX	17.406	17.616	17.390	18.751	18.814	18.683
X	15.897	17.825	17.953	19.437	19.437	19.372
Wet
I	16.835	16.014	16.826	18.063	18.052	18.213
II	15.984	15.936	15.913	17.393	17.338	17.488
III	17.388	17.394	17.361	17.420	17.505	17.108
IV	19.098	19.097	19.097	17.364	17.370	17.349
V	17.759	17.866	17.850	17.458	17.463	17.310
VI	18.865	18.971	19.461	18.837	18.875	18.612
VII	18.431	18.429	18.726	17.663	17.656	17.674
VIII	18.394	18.484	18.723	18.586	18.624	18.581
IX	18.403	18.511	18.679	18.559	18.615	18.530
X	19.027	18.843	19.711	18.364	18.384	18.315

Values of lnQ_eis produced by simultaneous calibration varied at most 0.3ln units or less with only a few exceptions. Also the cost function φ for PEST calibration varied less than 2 percent across evaporation options (R₁ = 9,833; R₂ = 9,746; R₃ = 9,948).

Figure 4.

Observed AMS and predicted AMS are compared over the calibration period Jan. 1985 through Dec. 1991. Model predicted AMS were produced using sequentially calibrated lnQ_eis (left) and simultaneously calibrated lnQ_eis (Right). Observed AMS are represented by closed circles (●) and predicted salinities are represent by dotted lines for evaporation option R₁, dashed lines for R₂ and solid lines for R₃. Bay Segments in descending order are examples of Old Tampa Bay, Hillsborough Bay, Middle Bay, and Lower Bay. (2 columns)

Values of lnQ_eis produced by simultaneous calibration varied at most 0.3ln units or less with only a few exceptions. Also the cost function φ for PEST calibration varied less than 2 percent across evaporation options (R₁ = 9,833; R₂ = 9,746; R₃ = 9,948).

3.3. Predicted versus observed AMS

Simultaneous calibrated exchange coefficients proved the better product when predicted AMS derived from each of the six sets of exchange coefficients were compared to observed AMS by the skill measures, RI and AE (Table 6) and graphical representation (Fig. 4).

Table 6.

Average error (AE) and reliability index (RI) for validation model runs.

	Sequential						Simultaneous
Seg.	R1		R2		R3		R1		R2		R3
	AE	RI	AE	RI	AE	RI	AE	RI	AE	RI	AE	RI
1	−8.08	1.53	−7.23	1.43	−7.16	1.44	1.53	1.08	1.54	1.08	1.56	1.08
2	−8.35	1.54	−7.07	1.41	−7.31	1.45	1.27	1.07	1.28	1.07	1.26	1.07
3	−8.10	1.49	−6.13	1.32	−6.73	1.38	0.43	1.05	0.45	1.05	0.44	1.05
4	−6.67	1.46	−2.91	1.18	−3.19	1.21	0.23	1.07	0.25	1.07	0.25	1.08
5	−6.07	1.41	−2.27	1.15	−2.56	1.18	1.41	1.10	1.41	1.10	1.42	1.10
6	−7.26	1.46	−3.33	1.18	−3.63	1.21	0.39	1.04	0.40	1.04	0.62	1.05
7	−7.75	1.47	−3.77	1.19	−4.08	1.22	−0.02	1.04	−0.01	1.04	0.34	1.04
8	−7.92	1.44	−3.14	1.14	−3.43	1.17	0.46	1.04	0.79	1.04	0.81	1.04
9	−8.01	1.40	−2.39	1.10	−2.45	1.10	0.84	1.04	0.82	1.04	0.85	1.04
10	−8.45	1.38	−2.03	1.07	−1.68	1.06	−0.13	1.02	−0.17	1.02	−0.11	1.02

Simultaneous calibration produced reasonably small RI values ranging from 1.12.to 1.18 and AE values generally showed a slight negative bias ranging from 0.75 to −1.52 whereas sequential calibration RI values were larger ranging from1.12 to 1.87 and AE values showed a strong negative bias ranging from −1.27 to −8.31. With respect to the three evaporation options RI and AE values from simultaneous calibration were the least variable. Reliability index values were same at two significant figures. Small differences were observed between AE values that followed a common pattern for the three options. Segments I and II showed a slight positive bias and the remaining segments showed a negative bias with segment X by far the largest value. Graphically little or no difference could be seen between the three options (Fig. 4; right side).

Sequential calibration produced a poor fit of predicted to observed salinities for all three evaporation options (Table 5 and Fig.4; left side). The poorest fit occurred for OTB (Segments I, II and III) where RI ranged by a factor of 1.5 to 1.87 and AE ranged from −8.31 to −6.07, for option R1. Options R₂ and R₃ produced similar results for OTB with AE showing a slightly reduced negative bias. For the remaining segments the degree of fit decreased from option R₃ to R₁ (Table 6). Graphical representation showed a similar pattern with R₃ producing the best fit. In some cases R₁ produced a markedly different pattern of predicted AMS. This was especially evident for Segment X (Fig. 4).

Table 5.

Comparison of average error (AE) and reliability index (RI) derived from observed and predicted AMS for the three evaporation options using lnQ_eis calibrated sequentially or simultaneously.. Values are listed for each segment.

	Sequential						Simultaneous
Seg.	R1		R2		R3		R1		R2		R3
	AE	RI	AE	RI	AE	RI	AE	RI	AE	RI	AE	RI
1	−8.31	1.87	−8.10	1.87	−7.71	1.84	0.74	1.18	0.75	1.18	0.69	1.18
2	−8.17	1.81	−7.49	1.75	−6.89	1.70	0.59	1.17	0.60	1.17	0.55	1.17
3	−6.07	1.50	−5.19	1.44	−4.37	1.43	−0.34	1.16	−0.34	1.16	−0.30	1.16
4	−2.83	1.27	−1.60	1.22	−0.51	1.21	−0.24	1.19	−0.25	1.19	−0.22	1.19
5	−3.30	1.26	−2.05	1.20	−0.96	1.18	−0.12	1.16	−0.12	1.16	−0.08	1.16
6	−3.53	1.25	−2.33	1.18	−1.16	1.15	−0.71	1.13	−0.70	1.13	−0.65	1.13
7	−3.59	1.25	−2.42	1.18	−1.32	1.16	−0.07	1.12	−0.08	1.12	−0.03	1.12
8	−3.79	1.24	−2.29	1.17	−1.43	1.15	0.00	1.14	0.00	1.14	0.00	1.14
9	−3.85	1.23	−2.10	1.14	−1.39	1.12	−0.88	1.14	−0.90	1.14	−0.82	1.14
10	−3.98	1.21	−1.96	1.12	−1.27	1.12	−1.51	1.15	−1.52	1.15	−1.46	1.15

3.3. Validation of Calibration.

The validity of the two calibration options was examined by extending the time series from seven years (1985–1991) to ten years (1985–1994). Predicted AMS using simultaneously calibrated LnQ_eis produced a better fit to observed AMS than was produced from sequential calibration (Table 7 and Fig. 5). Consistent with the calibration results (Fig. 4 right side) all three evaporation options produced very similar results (Fig. 5 right side) and predicted AMS generally following the contours of the validation AMS data set. The RI metric for simultaneous calibrations fell within a 1.02–1.10 range for validation which was markedly less than the 1.12–1.19 range observed in calibration Tables 6 and 7. Except for Segments VII and X, the prediction metric AE showed a slight positive bias (Table 7). Using sequentially produced LnQ_eis resulted in AMS (Fig. 5 left side) biased to under prediction (Table 7). Evaporation option R₁ was the least predictive with the AE metric ranging from −8.45 to −6.06 and RI ranged from 1.54 to 1.38 (Table 6). Options R₂ and R₃ produced better fits to observed AMS with the greater improvements occurring for sections HB, MB and LB (Table 7 and Fig. 5). However, unlike simultaneous calibration the differences in skill levels supported a slightly better fit of predicted to observed for R₂ over R_3..

Table 7.

Residence time (in model days) for Tampa Bay (ERT). Residence time was defined as the time required for reducing a conservative tracer to e⁻¹ of the initial level. Model runs were initiated by adding 1 unit of tracer to each Bay Segment.

	Bay Segment
	I	II	III	IV	V	VI	VII	VIII	IX	X	ERT
	Conservative Tracer Residence Time (model days)
Dry Season
	156	156	143	129	128	123	118	60	12	2	110
1986	147	147	134	121	120	115	109	62	14	3	100
1987	107	107	99	86	90	91	90	64	15	3	75
1988	152	151	135	121	120	114	109	63	14	3	100
1989	155	155	142	129	127	122	58	58	14	3	109
1990	153	153	140	126	125	120	115	60	13	3	110
1991	150	130	135	116	116	115	113	64	14	3	105
AVE	145.7	142.7	132.6	118.3	118.0	114.3	101.7	61.6	13.7	2.9	101.3
SD	17.3	18.1	15.2	15.0	13.1	10.9	21.7	2.3	1.0	0.4	12.4
Wet Season
1985	128	129	118	106	105	101	97	60	42	19	85
1986	138	139	127	115	114	109	105	67	43	17	95
1987	148	148	134	120	119	114	111	72	43	16	100
1988	63	66	66	63	64	62	58	41	34	21	45
1989	152	152	137	123	122	118	114	69	41	16	104
1990	166	165	147	135	133	127	123	73	43	16	114
AVE	132.5	133.2	121.5	110.3	109.5	105.2	101.3	63.7	41	17.5	90.5
SD	36.4	35.1	28.9	25.1	24.1	22.9	22.9	12.0	3.5	2.1	24.3

Figure 5.

Observed AMS and model predicted AMS are compared for the validation period Jan.1992 through Dec. 1994. Model predicted AMS were produced using sequentially calibrated lnQ_eis (left) and simultaneously calibrated lnQ_eis (Right). Observed AMS are represented by closed circles (•) and predicted salinities are represent by dotted lines for evaporation option R₁, dashed lines for R₂ and solid lines for R₃. Bay Segments in descending order are examples of Old Tampa Bay, Hillsborough Bay, Middle Bay, and Lower Bay.

3. 4. Confidence Limits.

The PEST tool for simultaneous calibration reports out a 95% confidence limit as a product of the calibration process. Confidence limits were largely the same for all three evaporation options. The results for R₃ are shown in Fig. 6. The least confident Q_eis for the dry season were Q_e1 (OTB) and lnQ_e6 (HB) (Fig. 6). For the wet season Q_e2 and Q_e6were least confident. The most confident Q_eis were associated with MB and LB.

Figure 6.

95% Confidence limits for calibrated Q_Eis derived evaporation option R₃ for the Dry (upper panel) and Wet (lower panel) Seasons. (1.5 columns)

3.6. Residence Times.

Residence times of Bay water were determined for each Segment (RT) and for the entire Bay (ERT) by adding a unit conservative tracer to all Segments. Residence times of Tampa Bay Segments consistently decreased from the head to the mouth of the bay (Table 7). On average ERT for the Bay and RT for segments above and including Segment VIII were greater in the dry than in the wet season. However below Segment VIII RT were shorter during the dry season. The shortest ERT was observed for the 1988 wet season and for an unusually wet dry season in 1987. Consistent with the larger freshwater inputs from rivers in HB, RT for segments in OTB were longer than RT for Segments in HB.

The movement of water within the Bay was followed by adding the conservative tracer separately to specific Bay sections (Fig. 7). In a broad sense the results are consistent with the retention times listed in Table 7. The results are also consistent with the freshwater inputs to each of the Bay sections. For example tracer was lost at a faster rate from HB (Segment V) than from OTB (Segment II). This correlated well with the rapid transfer of tracer from HB TO MB (Segment VII) which reached at a peak near 35% within a few days. Transfer of tracer from OTB to MB was markedly slower and reached a maximum level of 20% after a few weeks. The transfer of tracer initially added to MB to HB ramped up faster and to a great degree than transfer to OTB. This pattern of tracer transport was consistent with the much larger exchange coefficient (Table 5) between MB and HB (lnQ_e6) than between MB and OTB (lnQ_e3) during both the dry and wet seasons.

Figure 7.

Distribution of tracer over time in model days during the dry (left side) and wet (right side) seasons of a conservative tracer added to the segments in each of the four major sections of the Bay. Separate model runs were initiated on Feb 1, 1987 (dry) and Sept 1, 1985 (wet) by setting the initial tracer AMS to 1.0 in the appropriate segments and setting the AMS to zero in all other segments. Tracer AMS was then reported for Segment II (Old Tampa Bay, ), Segment V (Hillsborough Bay, ), Segment VII (Middle Bay, ), and Segment X (Lower Bay, ). (2 columns)

4. Discussion

Estuaries have been classified as positive if freshwater inflow exceeds evaporation, inverse if evaporation exceeds freshwater inflow (Pritchard 1952) and neutral if neither freshwater inflow nor evaporation dominates. Tampa Bay is representative of a neutral estuary where a portion of the Bay trends towards an inverse estuary and another portion trends to the positive. For OTB, MB, and LB evaporation dominated freshwater inflow during the dry season whereas freshwater inflow dominated during the wet season. In contrast HB was dominated by freshwater during both seasons. The HB receives most of Tampa Bay’s riverine freshwater inflow (Schmidt and Luther 2002). The neutral nature of Tampa Bay presented a challenge to developing a box model describing the movement of water within Tampa Bay.

We were led to this research by our inability, using classical box model techniques, to obtain exchange coefficients for a box model originally developed by (Wang et al. 1999) for Tampa Bay. The classical methods developed by Officer (1980) were based on salinity, water level, freshwater input and water transport (Sakov and Parslow 2004 and references there in). Evaporation was generally not included in model development (Miller and McPherson 1991; Prego and Fraga 1992; Soetart and Herman 1995; Sheldon and Alber 2002; Sakov and Parslow 2004). When evaporation (Q_vi) was considered it was simply subtracted from the total freshwater (Q_fi) input (Hagy et al. 2000; Hagy and Murrell 2007; Lehrter, 2008) as described for evaporation option R₁. However upon reflection option R1 proved not to be mathematically consistent with a steady state approach based on both salt and water balance. This became apparent when Eq. 2 was modified by inserting (Q_fi−Q_vi) for Q_ri in evaporation option R_1. Substituting into Eq. 2 and collecting terms yields

$$\Delta S_{Si}=\frac{\Delta t}{V_i}\left[\left(Q_{rai}\right.\left(S_{Si-1}-S_{Si}\right)\right.+Q_{ei-1}\left(S_{Si-1}\right.-\left.S_{Si}\right)+Q_{ei}\left(S_{Si+1}\right.-\left.S_{Si}\right)-Q_{fi}\left(S_{Si}\right.)+\left(Q_{vi}\right)\left(S_{Si}\right.)]$$ Eq.11

Equation 11 infers that evaporative water losses Q_vi from a given segment are replaced by saline water Q_vi(S_Si) from the same segment which is physically impossible. Option R₁ is an example where an unrecognized false assumption, the subtracting evaporative loss from the freshwater input, can led to an inappropriate mathematical expression. It is important to point out here, however, that when the ration of Q_vi to Q_fi becomes very small Equation 11 approaches evaporation option R₁. However, option R₃ would be the better choice for future box models.

To address this problem we developed two additional options that including evaporation in a manner akin to negative or inverse estuarine models. When evaporation exceeds freshwater inputs internal circulation is inverse to the classical positive estuary (Wolanski 1986; Nuns Vaz et al. 1990). In simple terms evaporation draws water from the ocean boundary to the estuary. In our second option R₂ evaporation in each segment draws and equal amount of water from the adjacent down bay segment (i.e. evaporation from segment V would draw water from segment VI). Each segment was acting as a negative estuary. This option was considered an approximation because it did not support a complete water balance. In the third option R₃ we added a second term to fully complete water balance. This term accounts for the cascade effect that results from drawing water to replace evaporative losses from up bay segments (i.e. water drawn from segment II to replace evaporative losses from segment I must be replace by water from segment 3 and so on).

The sequential approach is successful when the system of linear equations is well-conditioned. This approach can fail however for a number of reasons. Two problems dominated here. The first occurs when AMS in adjacent segments differ by less than 0.3 (Smith and Hollibaugh 2004). Small differences in the denominators of Eq. 6 and 7 can lead to exchange flows larger than the volume of the segment (Sakov 2004). The second occurs when the down bay segment of a segment pair is less saline than the up bay segment. The denominators become negative resulting in negative exchange coefficients which can lead to negative tracer concentrations during model runs (Sakov 2004). All three evaporation options became ill-conditioned when AMS, daily freshwater inputs and evaporative losses for a specific month were used to calculate each of the 84 exchange coefficients. Both potential problems were encountered and led to large negative and unrealistically large positive seasonally averaged exchange coefficients. Both problems were eliminated by using of seasonally averaged AMS. However the effect of evaporation still persisted. This is exemplified by the large negative bias in predicted over observed AMS observed in OTB where fresh water inputs were low relative to evaporation and the much smaller negative bias observed for HB where freshwater inputs were high with respect to evaporation. The smaller negative bias observed in MB and LB can be attributed to the freshwater inputs to HB although evaporation was often greater than the direct input of freshwater to these segments. This apparent anomaly results from the sequential calculation of exchange flows from the upper reach to the mouth of the bay. The much greater freshwater input to HB over shadows that from OTB.

Despite the poor fits of predicted to observed AMS the sequential approach did provide insight in discriminating between the three evaporation options. Of the three options R₁ was the least effective in predicting observed AMS. By nearly all (validation being the exception) metrics examined option R₃ produced the best results. Of the three options R₃ described a complete mass balance which is key to the sequential approach.

Simultaneous calibration based on the iterative calibration tool PEST was examined as an alternative to the sequential approach. PEST adjusts model wet and dry exchange flows (Fig. 3) until the fit between model prediction AMS and observed AMS are optimized to predetermined criteria based on the sum of weighted squared errors (φ). It was apparent from a comparison of skill metrics (AE and RI), the cost function Фs and graphical comparisons of predicted and observed AMS that simultaneous calibration was the better method. However it is somewhat disconcerting that we were unable to readily differentiate between the three evaporation options. Only for option R₃ do the small reductions observed in negative AE bias provide limited support. The minimal differences observed for all exchange flows strongly suggests that the different evaporation options little affected the simultaneous calibration process.

Testing a model’s reasonable prediction of observed data that was withheld from the calibration data set falls under the commonly held rubric of validation. While the use of this term is common in the scientific literature, models are inherently only approximations of the truth. For this reason, the validation process may be better expressed as confirmation or corroboration (Reckhow and Chapra 1983). Predictive validation (Power 1993) was conducted by extending the model time frame three years. Validation confirmed simultaneous calibration as the preferred calibration tool. AE values indicated a change from a general small negative bias to a small positive bias. The marked reduction in RI resulted from the smaller monthly fluctuations in the observed AMS for validation. However, once again little difference was observed between evaporation options. Validation of the sequential approach again illustrated the negative impact of presupposing that freshwater inputs equals freshwater minus evaporation (R₁).

It is important to note here that two of the options, R₁ and R₂, are based on assumptions in addition to constant volume. The first assumes that evaporation can be subtracted from freshwater inputs to maintain water balance and the second assumes that evaporation from upper reach segments do not impact segments near the mouth of the Bay. The third option only assumed constant volume and was the best mathematical representation of the box model.

The reliability of the model as a predictive tool can also be assessed by comparing results from other modeling approaches. Residence time is a calculation common to many tidal mixing and circulation models created for Tampa Bay. Using mean data from the mid-1980s and the freshwater fraction method, Solis and Powell (1999) estimated an ERT of 150 days. The first to use a numerical model to estimate ERTs, Burwell et al. (2000) compared Eulerian and Lagrangian approaches using a version of the Princeton Ocean Model adjusted for Tampa Bay (Vincent et al., 1997). A Eulerian approach that uses a passive tracer subroutine produced the shorter 75 day ERT whereas a Lagrangian approached thatuses a particle tracking subroutine produced a 159 day ERT. These values were suggested as the upper and lower limits for Tampa Bay ERTs. Weisberg and Zheng (2006) using an FVCOM based model reported a lower residence time of 100 days using 2001 (8/24/2001 to 11/30/2001) inflow data from the Egmont Channel at the mouth of the Bay. Meyers and Luther (2008) using an ECM3D based model and following particle movement within the Bay reported ERTs of 156 days (dry case) and 36 days (wet case). The dry case was consistent with Burwell’s upper limit. In this former case, freshwater discharges averaged over two 90 day periods starting 5/31/2001 (dry) and 9/09/2003 (wet) were used for model execution whereas Burwell’s model was initiated on 10/1/1997 at the beginning of the dry season. It should be noted that none of these studies included evaporation in model development. In a more recent publication, Zhu et al. (2015) reexamined the Weisberg and Zheng model (2006) using a finer resolution grid and the same river discharge data set. Using the same Egmont Channel approach a 69 day ERT was calculated. They attributed this lower value to a better measure of the channel dimensions. We determined ETRs by simply inserting a conservative tracer at a specific date in the middle of the dry and wet seasons. This effectively reduced dry season ERT because sequential months tended to become wetter, whereas the wet season ERT would be increased because sequential months would be drier. Assuming that the residence times of Meyers and Luther (2008) represent upper and lower limits our ERTs fall near the middle to lower portion of the range. The ETR of 45 days for our wettest season (1988) was slightly higher than their wet case residence time. Our simpler box model produced results that were not inconsistent with the more complex physical models and provided residence times over a continuous long term time frame of seven years.

We can gain an understanding of the influence of freshwater on the daily movement of tracer through the estuary by drilling down through the ERT approach and assessing the level of tracer remaining in individual Bay segments. Of the two branches of the estuary (HB and OTB) HB (segments IV, V, and VI) had the greater daily influx of freshwater and the greater efflux of tracer. OTB (segments I, II, and III) had a much lower influx of freshwater and therefore a lower rate of efflux of tracer. Using a high resolution hydrodynamic model Zhu et al. (2015) observed a similar pattern of tracer efflux from OTB and HB. When rivers were included RT varied from <40 days in LB to 91 days in upper HB and >91 days for OTB. Zhu also observed that the retention time at the end of 91 model days was greater than 91 for almost the entire bay when freshwater inputs were removed from the model. In combination these results emphasizing the importance freshwater in flushing the Bay even though the freshwater to tidal prism ratio for Tampa Bay is one of the lowest of the Gulf Estuaries (Solis and Powell 1999).

Freshwater however was not the whole story. Some of the tracer leaving OTB moves through MB to HB through tidal exchange. In a similar fashion tracer from HB moves through MB to OTB. The path of least resistance for the movement of tracer would be from Segment VI to VII and then back into Segment III. Tracer transferred from HB to OTB will linger longer in OTB than tracer transferred from OTB to HB because of the lower exchange from between segments III and VII (lnQ_e3) than between VI and VII (lnQ_e6). By extrapolation a portion of a nutrient added to HB from the Hillsborough and Alafia rivers will impact OTB.

Collectively our results would also suggest that for models based on a daily time step appropriate for most ecological models a simple box model will suffice. Ecological rates and coefficients are generally measured or inferred on a daily basis. Monitoring data meant as input to the model or used in calibration is often collected monthly (i.e. salinity) or in some cases daily but not often at less than a minute. Time scales less than a minute best support the time scales of physics based hydrodynamic models. Short time scales of necessity lead to large and expensive computer clusters and extended computation times. Often to circumscribe computational limitations, fine-scale hydrologic models based on physical measurements have been used to define the exchange coefficients for simpler box models (Kremer et al. 2010). Kremer et al. (2010) documented their approach in converting a ROMS model of Narragansett Bay to a box model. To mix the Narragansett Bay for 77 model days using the box model required less than five seconds of computer processing on a personal computer. The ROM model took roughly nine days using 20 nodes of a PSSC Labs Beowulf cluster. Computed salinities of the two models were within 1–5 % and were far less than would be expected for ecological processes. Although the final outcome was a simple and computationally efficient mixing model its development required significant time, expense, computing power. In our approach we have combined physical measurements of water depth, a monthly salinity data base from a long term monitoring effort and the calibration tool PEST to produce an efficient mixing model in less time, at lower cost and greatly reduced computing power. The model runs in less than five seconds for seven model years and calibration runs are completed in less than one minute. Computed AMS fell within 12–19 % range of observed values following calibration, and validation resulted in a 2–10 % range. This simpler approach that only required a desktop computer achieved similar results to approaches initiated with fine-scale hydrodynamic models. However, one should always remember that what we gain in reduced computation time we lose in the finer scales of salinity distribution.

5. Conclusions.

We have demonstrated a new, more efficient, and accurate approach to calibrate exchange flows for a 1-dimensional box model using Tampa Bay as a surrogate for large shallow bays where evaporation can play a large role in water balance. It is not uncommon for evaporation to exceed precipitation in Gulf Coast estuaries (Solis and Powell 1999). The model simulated daily salinity changes for ten model years in less than four seconds on a desk top computer. Physical hydrodynamic models in contrast take approximately 11 to 35 days computation time on a powerful computing cluster to cover the calibration period used in this effort (Galperin et al. 1991; Warner et al. 2005; Kremer et al. 2010). The short execution time proved very useful for PEST estimations of lnQ_eis. Less than 100 simulations were generally needed per calibration. The model independent Gauss-Marquardt-Levenberg method of parameter estimation software PEST was essential in simplifying this effort.

Appendix A.

Table A.

Long-Term National Weather Service (NWS) Precipitation Stations.

NWS Site Number	Site Name
228	Arcadia
478	Bartow
520	Bay Lake
940	Bradenton
945	Bradenton 5ESE
1046	Brooksville
1163	Bushnell
1632	Clearwater
1641	Clermont
3153	Ft. Green
3986	Hillsborough
4707	Lake Alfred
5973	Mountain Lake
6065	Myakka
6880	Parrish
7205	Plant City
7851	St. Leo
7886	St. Petersburg
8788	Tampa International Airport
8824	Tarpon Springs
9176	Venice
9401	Wauchula