U.S. GEOLOGICAL SURVEY

Water-Resources Investigations Report 99-4003

Probability Analysis of the Relation of 

Salinity to Freshwater Discharge in the 

St. Sebastian River, Florida

Prepared in cooperation with the

St. Johns River Water Management District

By Shaun M. Wicklein and W. Scott Gain

Tallahassee, Florida

1999
U.S. DEPARTMENT OF THE INTERIOR

BRUCE BABBITT, Secretary

U.S. GEOLOGICAL SURVEY

Charles G. Groat, Director

For additional information write to: Copies of this report can be purchased

from:

U.S. Geological Survey

Branch of Information Services

Box 25286

Denver, CO 80225-0286

The use of firm, trade, and brand names in this report is for identification purposes 

only and does not constitute endorsement by the U.S. Geological Survey.

District Chief

U.S. Geological Survey, WRD

Suite 3015

227 North Bronough Street

Tallahassee, FL 32301
Contents iii

CONTENTS

Abstract.................................................................................................................................................................................. 1

Introduction ........................................................................................................................................................................... 2

Purpose and Scope....................................................................................................................................................... 2

Description of Study Area ........................................................................................................................................... 4

Approach ............................................................................................................................................................................... 6

Factors Affecting Salinity Conditions in the St. Sebastian River.......................................................................................... 8

Wind ............................................................................................................................................................................ 8

Tidal Flux..................................................................................................................................................................... 8

Freshwater Discharge .................................................................................................................................................. 11

System Flushing .......................................................................................................................................................... 12

Specific-Conductance Characteristics ................................................................................................................................... 13

The Probabilistic Relation of Salinity to Freshwater Discharge ........................................................................................... 18

Modeling Specific Conductance as a Function of Discharge...................................................................................... 18

Residual Uncertainty and Exceedance Probabilities ................................................................................................... 27

The Response of Salinity Conditions to Changing Flow Regimes ............................................................................. 31

Summary and Conclusions .................................................................................................................................................... 35

Selected References............................................................................................................................................................... 37

FIGURES

1. Map showing location of study area........................................................................................................................ 3

2. Map showing data-collection sites, St. Sebastian River basin ................................................................................ 5

3. Diagram showing wind vector summary for St. Sebastian River basin, May 1, 1992, to March 18, 1996 ............ 9

4-6. Hydrographs showing:

4. Daily mean stage at South Prong St. Sebastian River at Roseland, Fla .......................................................... 9

5. Daily mean tidal flux at St. Sebastian River basin .......................................................................................... 10

6. Daily mean freshwater discharges at South Prong St. Sebastian River at Roseland, Canal 54 near 

North Prong St. Sebastian River, and St. Sebastian River at U.S. 1 at Sebastian............................................ 11

7. Graph showing discharge duration curves for South Prong St. Sebastian River at Roseland, Canal 54 near 

North Prong St. Sebastian River, and St. Sebastian River at U.S. 1 at Sebastian ................................................... 13

8. Scatter plots showing daily range of specific-conductance values for St. Sebastian River at U.S. 1 

at Sebastian.............................................................................................................................................................. 14

9. Graph showing specific-conductance duration curves for data-collection sites ..................................................... 15

10. Graphs showing specific-conductance duration curves for selected freshwater discharge ranges at 

conductance-monitoring sites.................................................................................................................................. 16

11-13. Scatter plots showing:

11. Freshwater discharge as a function of the ratio of tidal flux to freshwater discharge at St. Sebastian 

River at U.S. 1 at Sebastian ............................................................................................................................. 19

12. Specific conductance as a function of freshwater discharge at South Prong St. Sebastian River near 

Roseland and St. Sebastian River at U.S. 1 at Sebastian................................................................................. 21

13. Relation of model residuals to freshwater discharge ....................................................................... ................ 22

14. Hydrographs showing measured and simulated daily specific-conductance values ............................................... 24

15. Graph showing fitted statistical model without lagged smoothing for specific conductance as a function 

of freshwater discharge at sites in the St. Sebastian River ...................................................................................... 26

16. Graphs showing specific-conductance duration curves for modeled and observed values..................................... 29

17. Scatter plots showing absolute value of statistical model residuals as a function of discharge and residuals 

model regression curve............................................................................................................................................ 31

18. Graphs showing specific-conductance duration curves comparing period of record to detention and retention 

for conductance data-collection sites....................................................................................................................... 32
iv Contents

TABLES

1. Data-collection sites, St. Sebastian River basin, Fla .............................................................................................. 7

2. Conductance probe measurement locations and mean stage for St. Sebastian River basin 

conductance-measurement sites, May 1, 1992, to March 18, 1996........................................................................ 7

3. Tidal flux summary for St. Sebastian River basin, May 1, 1992, to March 18, 1996 ............................................ 11

4. Daily mean freshwater discharge summary for St. Sebastian River basin ............................................................. 12

5. Daily mean specific-conductance summary for St. Sebastian River basin, May 1, 1992, to March 18, 1996....... 15

6. Seasonal and period of record lower-layer to upper-layer differences for instantaneous and daily mean 

values of specific conductance, May 1, 1992, to March 18, 1996 ......................................................................... 16

7. Statistical parameters for the model of specific conductance as a function of discharge for St. Sebastian 

River basin, May 1, 1992, to March 18, 1996 ........................................................................................................ 20

8. Statistical parameters for the model of residuals as a function of discharge for St. Sebastian River basin,

May 1, 1992, to March 18, 1996 ............................................................................................................................ 31

9. Predicted days of occurrence at a given threshold of specific conductance for the period of record, 

detention, and retention conditions, May 1, 1992, to March 18, 1996................................................................... 34

CONVERSION FACTORS, VERTICAL DATUM, ABBREVIATIONS 

AND ACRONYMS 

Temperature in degrees Fahrenheit (°F) may be converted to degrees Celsius (°C) as follows:

°C = (°F-32)/1.8.

Sea level: In this report, “sea level” refers to the National Geodetic Vertical Datum of 1929 (NGVD

of 1929)--a geodetic datum derived from a general adjustment of the first-order level nets of both the

United States and Canada, formerly called Sea Level Datum of 1929.

Acronyms and additional abbreviations used in report:

SJRWMD St. Johns River Water Management District

USGS U.S. Geological Survey

μS/cm microsiemens per centimeter at 25 degrees Celsius

Multiply By To obtain

Length

foot (ft) 0.3048 meter

inch (in) 25.4 millimeter

mile (mi) 1.609 kilometer 

Area

square foot (ft2

) 0.09290 square meter 

square mile (mi2) 2.590 square kilometer

Flow Rate

cubic foot per second (ft3

/s) 0.02832 cubic meter per second
Abstract 1

Probability Analysis of the Relation of Salinity to 

Freshwater Discharge in the St. Sebastian River, Florida

By Shaun M. Wicklein and W. Scott Gain

Abstract

The St. Sebastian River lies in the southern 

part of the Indian River basin on the east coast of 

Florida. Increases in freshwater discharge due to 

urbanization and changes in land use have reduced 

salinity in the St. Sebastian River and, consequently, salinity in the Indian River, affecting the 

commercial fishing industry. Wind, water temperature, tidal flux, freshwater discharge, and downstream salinity all affect salinity in the St. Sebastian 

River estuary, but freshwater discharge is the only 

one of these hydrologic factors which might be 

affected by water-management practices.

A probability analysis of salinity conditions 

in the St. Sebastian River estuary, taking into 

account the effects of freshwater discharge over a 

period from May 1992 to March 1996, was used to 

determine the likelihood (probability) that salinities, as represented by daily mean specific-conductance values, will fall below a given threshold. 

The effects of freshwater discharge on salinities 

were evaluated with a simple volumetric model 

fitted to time series of measured specific conductance, by using nonlinear optimization techniques. 

Specific-conductance values for two depths at 

monitored sites represent stratified flow which 

results from differences in salt concentration 

between freshwater and saltwater. Layering of 

freshwater and saltwater is assumed, and the 

model is applied independently to each layer with 

the assumption that the water within the layer is 

well mixed. The model of specific conductance as 

a function of discharge (a salinity response model) 

was combined with a model of residual variation 

to produce a total probability model. Flow distributions and model residuals were integrated to 

produce a salinity distribution and determine 

differences in salinity probabilities as a result of 

changes in water-management practices.

Two possible management alternatives were 

analyzed: stormwater detention (reducing the peak 

rate of discharge but not reducing the overall flow 

volume) and stormwater retention (reducing peak 

discharges without later release). Detention of 

freshwater discharges increased the probability of 

specific-conductance values falling below a given 

limit (20,000 microsiemens per centimeter) for all 

sites but one. The retention of freshwater input to 

the system decreased the likelihood of falling 

below a selected limit of specific conductance 

at all sites. For limits of specific conductance 

(1,000 microsiemens per centimeter or 

20,000 microsiemens per centimeter, depending 

on the site), the predicted days of occurrence 

below a limit decreased ranging from 17 to 

68 percent of the predicted days of occurrence for 

unregulated flow.

The primary finding to be drawn from the discharge-salinity analysis is that an empiricalresponse model alone does not provide adequate 

information to assess the response of the system to 

changes in flow regime. Whether a given level of 

discharge can produce a given response on a given 

day is not as important as the probability of that 

response on a given day and over a period of many 

days. A deterministic model of the St. Sebastian 

River estuary based only on discharge would predict 

that retention of discharge peaks should increase the 

average salinity conditions in the St. Sebastian 

River estuary. The probabilistic model produces a 

very different response indicating that salinity can 

decrease by a power of three as discharges increase, 

and that random factors can predominate and control salinity until discharges increase sufficiently to 

flush the entire system of saltwater.
2 Probability Analysis of the Relation of Salinity to Freshwater Discharge in the St. Sebastian River, Florida

INTRODUCTION

The St. Sebastian River lies in the southern part 

of the Indian River on the east coast of Florida (fig. 1) 

and contributes about 16 percent of the total annual 

freshwater inflow to the Indian River (Knowles, 1995). 

Changes in land use in the St. Sebastian River basin for 

agriculture and land development have increased freshwater discharges into the system (Brown and others, 

1962; Crain and others, 1975). These increases in 

freshwater inflow during storm events tend to reduce 

salinities in the Indian River, affecting the commercial 

fishing industry and resulting in loss of revenue (Indian 

River Lagoon Estuary Program, 1993).

Salinity conditions in the St. Sebastian River are 

highly dependent on patterns of mixing and circulation 

and on the volumetric contributions of saltwater and 

freshwater. The dynamic nature of streamflow and 

wind conditions causes variations in the relative contributions of saltwater and freshwater to the estuary, 

resulting in wide-ranging variations in salinity. Windspeed and direction, water temperature, tidal discharge, 

freshwater discharge, and downstream salinity generally affect salinity in an estuary. Freshwater discharge 

is the only one of these hydrologic factors which might 

be affected by water- management practices.

The St. Sebastian River estuary was predominantly a saline environment prior to development in the 

drainage basin that increased freshwater discharges. 

Large tidal influxes of highly saline water from the 

Indian River are much greater in volume than the relatively small freshwater inflows to the estuary from surface- and ground-water discharges. Density driven 

vertical stratification promotes increased saline conditions at the bottom of the St. Sebastian River by confining much of the less dense freshwater to the upper 

layers. Freshwater is preferentially transported from 

the system in the upper layers without completely mixing with, and diluting, the saltier water in lower layers.

Freshwater conditions at the bottom of the St. 

Sebastian River occur only infrequently in the undisturbed system and are associated with large stormwater 

inflows sufficient in volume and intensity to flush the 

heavier saltwater out of the river and into the Indian 

River. These freshwater conditions are of short duration primarily because the denser saltwater of the 

Indian River flows back into the lower reaches of the 

river with receding streamflow. Sustained high freshwater flows decrease salinity of the Indian River near 

the mouth of the St. Sebastian River, because the saltwater in the Indian River is being diluted. These high 

freshwater flows tend to reduce the average salinity of 

the Sebastian River also; but, dilution effects near the 

mouth of the St. Sebastian River are not observed further into the Indian River.

Changes in land use and urban development 

within the St. Sebastian River basin have increased 

both the total and peak rate of freshwater inflow into 

the river. Increased volumes and rates of inflow, in turn, 

have altered the general patterns of circulation and 

mixing within the estuary and have increased the frequency of occurrence of freshwater conditions above 

predevelopment conditions. Changes in mixing and 

circulation within the river system also affect constituent concentrations in water moving through the estuary.

Because average salinity conditions and the frequency of occurrence of freshwater conditions can 

have a defining effect on biological communities in the 

river, it is important to determine what effects changes 

in the freshwater flow regime may have on the river 

system. Uncontrolled factors, such as wind and salinity 

conditions in the Indian River and turbulent mixing in 

the St. Sebastian River, produce random variations in 

the observed response of salinity to freshwater discharge which cannot be predicted by strictly deterministic models, but which must be accounted for in any 

analysis of the effects of planned or unplanned changes 

in the river basin.

A conductance-monitoring network was established in May 1992 to provide data for calibration and 

verification of a two-dimensional deterministic hydrodynamic model of the St. Sebastian River. However, 

because of limitations of the deterministic model to 

simulate constituent diffusion due to vertical stratification of specific conductance, a statistical model was 

developed to predict specific conductance, and thus, 

salinity, from discharge. Salinity probability modeling 

is a tool used to predict salinity in an estuary that results 

from a range of hydrologic conditions. In order to 

develop a salinity probability model, the U.S. Geological Survey (USGS), in cooperation with the St. Johns 

River Water Management District (SJRWMD), conducted a study of data collected over a 4-year period to 

determine the relation of freshwater inflows on salinity 

within the St. Sebastian River estuary. 

Purpose and Scope

This report describes a probability analysis of 

salinity conditions in the St. Sebastian River estuary, 

taking into account the effects of freshwater discharge 
Abstract 3

Figure 1. Location of study area.
4 Probability Analysis of the Relation of Salinity to Freshwater Discharge in the St. Sebastian River, Florida

over a period from May 1992 to March 1996. A probability analysis is used to determine the probability that 

salinities, as represented by daily mean specificconductance values, will fall below a given threshold. 

The effects of freshwater discharge on salinities are 

evaluated with a simple volumetric model fitted to time 

series of measured specific-conductance values, by 

using nonlinear optimization techniques. The model of 

specific conductance as a function of discharge is combined with a model of residual variation to produce a 

total probability model. Flow distributions and model 

residuals are integrated to produce a salinity distribution and determine differences in salinity probabilities 

as a result of changes in watershed-management practices. There are three goals in development of the salinity-response model: (1) describe as much of the 

variation in specific conductance as possible in terms 

of freshwater discharge, (2) define a basic form of the 

model that is descriptive of all sites with a minimum 

number of degrees of freedom, and (3) eliminate bias in 

the residuals. Two possible management alternatives 

are analyzed: stormwater detention (reducing the peak 

rate of discharge but not reducing the overall flow 

volume) and stormwater retention (reducing peak discharges without later release). Tidal and freshwater 

flow conditions and probabilities are presented and 

discussed, as are the range and natural variability of 

salinity conditions in the estuary.

Description of Study Area

The St. Sebastian River lies on the boundary of 

Indian River County and Brevard County (fig. 1). The 

small town of Micco lies to the north, and the towns of 

Roseland and Sebastian lie to the south. Historically, 

swampy conditions over much of the St. Sebastian 

River basin precluded extensive development. The 

addition of a series of drainage canals led to the development of much of the southern part of the basin near 

the towns of Roseland and Sebastian. East of the St. 

Sebastian River is the Indian River, which is bordered 

to the east by a barrier island. Nearly due east of the 

mouth of the St. Sebastian River, breaching the barrier 

island, is Sebastian Inlet, one of only four connections 

between the Atlantic Ocean and the Indian River.

The shape of the land surface in the study area 

has changed little since Pleistocene time. The major 

landforms of the area are of two general types—

terraces and ridges (Brown and others, 1962; Crain and 

others, 1975). Terraces and associated scarps were 

formed by ancient seas that once covered all of peninsular Florida. Ridges represent ancient shoreline 

deposits such as offshore sandbars and relict beaches.

The barrier island forming the east boundary of 

the Indian River is the remnant of an ancient offshore 

sandbar and forms a nearly continuous barrier to saltwater inflow. The barrier island extends more than 

100 mi from Haulover Canal in the north (connecting 

the Indian River and Mosquito Lagoon) to the St. Lucie 

Inlet in the south. The barrier island ranges in altitude 

from 0 to about 25 ft and is breached in only two 

places—Sebastian and Fort Pierce Inlets. The coastal 

ridge is the first ridge west and inland of the Indian 

River, and ranges in altitude from about 5 to 50 ft. The 

natural effect of the coastal ridge is to block freshwater 

inflows from the west to the St. Sebastian River estuary. The coastal ridge extends from and forms the 

watershed boundary between the St. Johns River and 

Indian River. The natural drainage basin of the Indian 

River consists of a narrow strip of land, only a few 

miles wide in some locations, and over 100 mi long.

In recent decades, freshwater input to the Indian 

River has been increased by the channeling of water 

through manmade canals across the coastal ridge from 

the upper St. Johns River basin. Changes in the natural 

geomorphology, and consequently, in drainage characteristics of the Indian River and upper St. Johns River 

have affected the hydrology of the area, including the 

St. Sebastian River.

The subtropical climate of the Indian River basin 

is characterized by wet and dry seasons with frequent, 

intense thunderstorm activity during the wet season. 

The intensity of this thunderstorm activity creates a 

seasonal variation of up to twice the rainfall accumulation from dry to wet seasons (Knowles, 1995). Mean 

annual rainfall in the vicinity of the St. Sebastian River 

basin averages about 48 inches per year. Because of the 

seasonal variations in rainfall, freshwater inflow into 

the Indian River also varies by as much as 50 percent 

from dry to wet seasons. Seasonal variations in rainfall 

have little effect on the percentage of flow contributed 

to the Indian River by the St. Sebastian River basin.

The St. Sebastian River receives freshwater 

inflows from three major tributaries. North and South 

Prongs of the St. Sebastian River provide natural drainage from the north and south parts of the basin, respectively, and Canal 54 conveys water from areas to the 

west, much of which lie within the St. Johns River 

basin. Two smaller canals on the west end of Canal 54 

drain the Fellsmere area and an area on the west side of 

the coastal ridge (fig. 2).
Abstract 5

Figure 2. Data-collection sites, St. Sebastian River basin.
6 Probability Analysis of the Relation of Salinity to Freshwater Discharge in the St. Sebastian River, Florida

The North Prong St. Sebastian River is a small 

basin (approximately 28.5 mi2) that is relatively undeveloped through the lower reaches and has been channeled in the upland reaches to provide drainage for 

citrus groves. The South Prong basin is larger than the 

North Prong basin and is approximately 63.7 mi2

. 

Urban land use in the upper basin is increasing as a 

result of expansion of the communities of Sebastian 

and Vero Lake Estates.

Canal 54 and Fellsmere Canal are manmade drainage systems that drain the inland areas of Brevard and 

Indian River Counties (Steward and VanArman, 1987). 

This inland area is referred to as the St. Johns Marsh, 

headwaters of the St. Johns River. Canal 54 drains about 

100 to 150 mi2

 of the St. Johns Marsh, and the flow is controlled by Structure S-157. Fellsmere Canal is controlled 

by a weir structure and was dug to drain part of the St. 

Johns River Marsh for citrus cultivation.

 The quality of water in the St. Sebastian River 

basin is determined by the mixing of waters from two 

principal sources: saltwater from the Atlantic Ocean 

propagated through the Indian River and upstream into 

the St. Sebastian River, and stormwater runoff input 

from natural and developed parts of the basin. Tidal 

fluctuations can generate substantial inflow of saltwater to the river. Tides along the east coast of Florida are 

primarily semidiurnal. Tidal variations throughout 

much of the St. Sebastian River typically produce 

changes in water-surface elevations of 0.5 ft. The rising 

and falling of tides in the St. Sebastian River results in 

transient storage changes of as much as 30 percent of 

the river’s volume. Tidal fluctuations are observed 

within the estuary as far upstream as structure S-157 on 

Canal 54 and State Road 512 on the South Prong of St. 

Sebastian River (fig. 2).

APPROACH

The data-collection network consists of ten sites: 

four stage and discharge sites, located at the major 

freshwater inputs into the St. Sebastian River; one 

stage-only site, located near the confluence of the 

North and South Prongs of the St. Sebastian River; and 

five conductance-monitoring sites. The conductancemonitoring sites were located at two upstream locations, located on the South Prong and Canal 54, and 

three downstream locations, one near the confluence of 

the North and South Prongs, another downstream at the 

crossing of U.S. 1, and the third in the Indian River just 

outside the mouth of the St. Sebastian River (fig. 2).

Data-collection sites were selected to provide 

representative information on main inflows and controlling hydrologic features within the study area. 

Types of data collected at each site were selected based 

on features of the site location and need for information 

at the location. Conductance data were collected to represent the inputs and outputs of the estuary and 

describe vertical stratification within the water column. 

Stage was measured at several locations throughout the 

study area, and discharge was measured at points of 

freshwater input to the system. Meteorological data 

were collected to represent weather conditions in and 

near the estuary. Information on data-collection sites, 

along with type of data collected, is presented in 

table 1.

Stage was measured at nine locations in the study 

area, five that are within the estuary: South Prong St. 

Sebastian River near Roseland (SPcon), South Prong 

St. Sebastian River at Roseland (SPStage), Canal 54 

near North Prong St. Sebastian River (C54con), St. 

Sebastian River near Railroad Bridge (Sebcon1), and 

St. Sebastian River at U.S. 1 at Sebastian (Sebcon2) 

(fig. 2). The site SPStage was selected to represent 

stage for the entire estuary.

Discharge was determined at South Prong St. 

Sebastian River at Highway 512 near Sebastian (QSP), 

Fellsmere Canal near Micco (QFells), Canal 54 at S-157 

near Micco (QC54), and North Prong St. Sebastian 

River near Micco (QNP). Continuous measurement of 

stage and measurement of discharge (six times per 

year) were performed at three of the sites: South Prong, 

North Prong, and Fellsmere Canal. Stage and discharge 

data for the Canal 54 were measured by the SJRWMD. 

All other stage and discharge data were collected by the 

USGS using standard methods described in Rantz and 

others, 1982.

Windspeed and direction data were collected at 

two locations: Sebcon1, within the estuary, and IRcon, 

just outside of the mouth of the estuary. Mean windspeed and wind direction were recorded at 15-minute 

intervals at St. Sebastian River near Railroad Bridge 

and Indian River near Sebastian.

Specific conductance was monitored at five 

sites: C54con, SPcon, Sebcon1, Sebcon2, and IRcon. 

Conductance was measured at 15-minute intervals at 

selected depths (table 2), and the measurement was 

corrected using water temperature to obtain a measurement of specific conductance corrected to 25 °C 

(Miller and others, 1988).
Approach 7

A regression equation was developed using the 

relation of specific conductance to salinity at 25 °C 

(Neumann and others, 1966):

, (1)

where is salinity (dissolved solids), in parts per 

thousand; and

 is specific conductance at 25 °C, in millisiemens per centimeter.

Because of this relation between salinity and specific 

conductance, defined by equation 1, specific conductance is used as an indicator of salinity in this report.

Elevation of the channel bottom in the vicinity of 

the gages ranged from 3.0 to 4.5 ft below sea level 

(table 2). Conductance measurements made in the 

upper layer of the water column were made at an average of 1.5 ft below the surface. Measurements made in 

the lower layer of the water column were made at an 

average of 4.3 ft below the surface. Because of the 

locations of the gage installations, the conductance 

measurements for SPcon were made in the upper layer, 

whereas conductance measurements at C54con were 

made in the lower layer. 

Daily mean values of stage and surface area of 

the estuary were used to compute storage volumes and 

tidal flux for the system. Discharge was computed for 

the major inputs into the estuary system, and vectors of 

wind velocity and direction were computed from meteorological data collected over the study period. Daily 

mean specific conductance was used to represent salinity conditions within the system. Probabilities were 

computed for the occurrence of specific values of discharge and specific conductance from the observed 

Table 1. Data-collection sites, St. Sebastian River basin, Fla.

[mi2, square miles; data type: d, discharge; e, stage; c, conductance; t, temperature; r, rainfall; ws, windspeed; wd, wind direction; 

I, indeterminate; U, unknown; SJRWMD, St. Johns River Water Management District. Sites shown in figure 2]

Site Site name and

identification number Latitude Longitude Date established

Drainage 

area 

(mi 2)

Data type

QSP South Prong St. Sebastian River 

at State Road 512 

(02251000)

27° 46' 09" 80° 30' 22" 05-04-93 35.0 d, e

SPcon South Prong St. Sebastian River 

near Roseland 

(02251200)

27° 49' 06" 80° 30' 31" 02-09-87 61.2 e, c, t

SPStage South Prong St. Sebastian River 

at Roseland 

(02251210)

27° 49' 56" 80° 30' 01" 02-10-87 163.7 e

QFells Fellsmere Canal near Micco 

(02251767)

27° 49' 49" 80° 32' 04" 01-15-91 I d, e

QC54 Canal 54 at S-157 near Micco 

(SJRWMD)

27° 49' 49" 80° 32' 25" U I d, e, r

C54con Canal 54 near North Prong 

Sebastian River 

(275007080311200)

27° 50' 07" 80° 31' 12" 04-15-92 I e, c, t

QNP North Prong St. Sebastian River 

near Micco 

(02251500)

27° 51' 21" 80° 31' 28" 01-22-87 28.5 d, e

Sebcon1 St. Sebastian River near 

Railroad Bridge 

(275017080295600)

27° 50' 17" 80° 29' 56" 05-01-92 I e, c, t, r, 

ws, wd

Sebcon2 St. Sebastian River at U.S. 1 at 

Sebastian 

(275114080292800)

27° 51' 14" 80° 29' 28" 05-14-92 I e, c, t

IRcon Indian River near Sebastian 

(275128080291800)

27° 51' 28" 80° 29' 18" 07-01-92 I c, t, r, ws, 

wd

1

 Approximate drainage area.

S 0.0015C2 = + + 0.578C 0.019

S

C

Table 2. Conductance probe measurement locations 

and mean stage for St. Sebastian River basin 

conductance-measurement sites, May 1, 1992, to 

March 18, 1996

[Gage datum is 10.0 feet below sea level; --, not available]

Site 

Elevation in feet above gage datum

Mean stage Top 

probe

Bottom 

probe

Streambed

SPcon1 10.7 10.0 -- 6.0

C54con1 10.7 -- 6.0 5.5

Sebcon1 10.7 9.5 7.4 7.0

Sebcon2 10.8 8.7 5.9 5.7

IRcon2 10.5 8.5 6.0 5.5

1 Site has only one probe.

2 Mean stage not based on entire period.
8 Probability Analysis of the Relation of Salinity to Freshwater Discharge in the St. Sebastian River, Florida

data. A statistical model of specific conductance as a 

function of discharge was developed for the period 

from May 1992 to March 1996, and a second statistical 

model of residuals as a function of discharge was 

developed and combined with the specific-conductance model to produce estimated duration curves. By 

using this approach, probabilities of a particular specific-conductance value exceeding or failing to exceed 

a limit were computed for two alternative management 

scenarios, one for retention and the other for detention 

of freshwater discharges.

FACTORS AFFECTING 

SALINITY CONDITIONS 

IN THE ST. SEBASTIAN RIVER

Salinity conditions in the St. Sebastian River are 

highly dependent on patterns of mixing and circulation, 

and the relative volumes of saltwater and freshwater in the 

estuary. Wind, tidal flux, freshwater discharge, and system 

flushing affect specific conductance within the St. Sebastian River. Movement of water into and out of the estuary 

affects specific conductance and can be described by 

flushing characteristics of the system. The dynamic nature 

of streamflow and wind conditions cause variations in the 

relative volumes of saltwater and freshwater in the estuary 

and produce wide-ranging variations in salinity.

Wind

The shape and location of the St. Sebastian River 

estuary play a role in wind effects on tidal action within 

the system. Wind and salinity conditions in the Indian 

River and turbulent mixing in the St. Sebastian River 

produce random variations in the observed response of 

salinity to discharge which cannot be predicted but 

which must be accounted for in any analysis of salinity 

conditions in the river.

Monthly mean, mean monthly for all years of 

record, and annual mean windspeed and vector direction 

were calculated for each site and are shown graphically 

in figure 3. Mean values were calculated for normal 

components of wind velocity and direction so that the 

average direction and speed indicated is the final disposition of a wind-carried particle tracked over a monthly 

or annual time period. The wind vector summary in 

figure 3 indicates that winds across the estuary blew 

from a net northeasterly direction for the study period; 

however, this may not be indicative of a prevailing 

northeast wind. Generally, winds change throughout the 

day as offshore winds increase in the afternoon and frontal systems move through the area. Winds outside of the 

St. Sebastian River basin in the Indian River originate 

from a predominantly northwesterly direction.

Wind affects tidal transport within the confines 

of a shallow estuary (Smith, 1990). Winds prevailing 

from the north and northeast build up water within the 

St. Sebastian River estuary system. When high tides 

and strong winds combine, a large volume of water is 

forced farther upstream into the system. This combination of tide and wind is more likely to occur during the 

fall and winter months, when monthly mean winds prevail from the north and northeast and tidal storage is 

greatest. Although long-term prevailing winds can 

build up water within the estuary, tidal changes and 

freshwater discharges are still the primary factors 

affecting salinity. Short-term variable winds are more 

common in the afternoon during summer months and 

can cause turbulence within the system; this turbulence 

causes random variation in salinity during times of high 

salinities and lower freshwater flows.

Tidal Flux

Tidal influxes of highly saline water from the 

Indian River to the St. Sebastian River are much greater 

in volume than the relatively small freshwater inflows 

from surface- and ground-water sources. Tidal flux 

within the system is propagated by change in stage 

within the estuary. A graph of daily mean stage at the 

SPStage site indicates seasonal and monthly variations 

in stage that occur in close association with the Atlantic 

Ocean tides (fig. 4). Highest tides during the year generally are in October and November and lowest tides 

typically are from February through April. The stage 

change due to tides ranges from 2 to 3 ft in a typical 

year, and averages 0.5 ft over a typical tidal cycle. 

Although peak freshwater discharges from the St. 

Sebastian River commonly are observed in October 

and November, tidal flux appears to be the more important factor affecting stage within the estuary.

Storage for the St. Sebastian River was calculated 

using a model grid and a volume for each cell of the grid. 

Mean water-surface elevation (stage) in feet above sea 

level and bathymetric data were used to determine the 

depth of each grid cell. The total mean storage upstream 

from each of the four St. Sebastian River conductancemonitoring sites was calculated by summing the cell volumes upstream of the respective sites. The volume 

upstream from Sebcon2 (fig. 2) represents the total storage volume for the entire St. Sebastian River estuary.
Factors Affecting Salinity Conditions in the St. Sebastian River 9

1992

1993

1994

1995

1996

Mean

S 180°

4

2

0

miles per hour N 0°

Sebcon1

IRcon

EXPLANATION

JAN MAR APR MAY FEB JUNE JULY AUG SEPT OCT NOV DEC MEAN

Figure 3. Wind vector summary for St. Sebastian River basin, May 1, 1992, to March 18, 1996. Arrowheads 

point towards direction from which wind originates.

M J J A S O N D J F M A M J J A S O N D J F M A M J J A S O N D J F M A M J J A S O N D J

1992 1993 1994 1995 1996

9

13

10

11

12

DAILY MEAN STAGE,

IN FEET (ABOVE GAGE DATUM)

MISSING DATA

F M

Figure 4. Daily mean stage at South Prong St. Sebastian River at Roseland, Fla. (SPStage).
10 Probability Analysis of the Relation of Salinity to Freshwater Discharge in the St. Sebastian River, Florida

The computed tidal flux represents a rate (positive or negative) of water flowing through a given 

cross section of the river and is considered the average 

flow rate of water as a result of stage change that passed 

through the cross section in one day. Tidal flux is computed as the average absolute value of tidal changes in 

storage calculated as the product of surface area and 

change in stage over tide cycles adjusted for the actual 

duration of inflow. Water-surface areas upstream from 

Sebcon2, Sebcon1, and C54con were estimated based 

on 7.5-minute USGS quadrangle maps. Water-surface 

area upstream of SPStage was estimated using change 

in stage and a regression equation that relates crosssectional area to stage; this approach accounted for 

changes in surface area with changes in stage. Tidal 

flux, QT, in cubic feet per second, was calculated from 

change in stage and surface area by using the following 

equation:

, (2)

where is 15-minute stage change, in feet; and

SA is surface area upstream from the site, in 

square feet.

Flow reversals and stratification of flow due to 

density gradients are common within the St. Sebastian 

River estuary, and tidal flux (fig. 5) does not exhibit the 

seasonal variations observed in daily mean stage. Over 

a full tidal cycle, the net tidal flux approaches zero 

because the volume of water that flows into the estuary 

is approximately equal to the volume that flows out.

QT

( ) ∑ ∆H SA ( ) ⁄ 2

86 400 , = ----------------------------------------

∆H

M J J A S O N D J F M A M J J A S O N D J F M A M J J A S O N D J F M A M J J A S O N D J F M

1992 1993 1994 1995 1996

1

500

2

3

4

5

7

10

20

30

40

50

70

100

200

300

400

DAILY MEAN TIDAL FLUX UPSTREAM FROM INDICATED SITE,

M J J A S O N D J F M A M J J A S O N D J F M A M J J A S O N D J F M A M J J A S O N D J F M

1992 1993 1994 1995 1996

10

1,000

20

30

40

50

70

100

200

300

400

500

700

Sebcon2

Sebcon1

IN CUBIC FEET PER SECOND

SPcon

C54con

Figure 5. Daily mean tidal flux at St. Sebastian River basin.
Factors Affecting Salinity Conditions in the St. Sebastian River 11

Maximum, minimum, and mean tidal flux at 

selected sites (table 3) indicate the variability in tidal 

flux with location. Tidal flux at the mouth of the St. 

Sebastian River (represented by Sebcon2) ranged from 

45.3 to 878 ft3

/s throughout the study period. Tidal flux 

decreased from downstream to upstream and the South 

Prong (SPStage) had a higher flux than Canal 54 

(C54con). In November 1994, tropical storm Gordon 

produced the highest recorded stage during the study, 

but the increase in tidal flux resulting from the storm 

was relatively small. If freshwater inflow is assumed to 

be equal to zero, then the tidal flux, based on the annual 

mean value for the study, could flush the system in 

about 5.5 days. Based on tidal flux at the mouth of the 

estuary (Sebcon2), the system could be flushed in 

1.3 days assuming the maximum value of tidal flux, and 

in 24.8 days assuming the minimum value of tidal flux. 

Saltwater in the upstream reaches of the river is flushed 

downstream more rapidly; the rate of flushing slows as 

the river becomes wider and deeper near the mouth.

Freshwater Discharge

Changes in land use and increased development 

within the St. Sebastian River basin tend to increase the 

total and peak rate of flow of freshwater into the river. 

Increased volumes and rates of inflow can, in turn, alter 

the general patterns of circulation and mixing within 

the estuary. Freshwater conditions at the bottom of the 

river occur only infrequently in the undisturbed system 

and are associated with large stormwater inflows sufficient in volume and rate to push the denser saltwater 

out of the St. Sebastian River and into the Indian River.

Time series of daily mean freshwater discharge 

data were compiled at the four major freshwater inputs 

(sites QSP, QFells, QC54, and QNP) into the St. Sebastian River estuary for the study period 1992-96 except 

for site QSP, which was installed in May 1993. Daily 

mean freshwater discharge data for missing periods 

were estimated using a multiple stepwise regression 

analysis on comparable discharge data at similar 

sites. Daily mean discharge record for South Prong 

St. Sebastian River was extrapolated to the beginning 

of the study period, and daily mean discharges for part 

of the fourth quarter of the 1995 water year for Fellsmere Canal were estimated. Runoff into the system 

from intermediate drainage between measured inputs 

and the mouth of the estuary is relatively insignificant. 

Thus, freshwater discharges at downstream sites are 

considered equivalent to the summation of freshwater 

discharges from upstream sites. Discharges for the 

major freshwater inputs (fig. 6) were combined to 

obtain total freshwater discharge for the entire system.

Table 3. Tidal flux summary for St. Sebastian River basin, 

May 1, 1992, to March 18, 1996

[ft2, square feet; ft3/s, cubic feet per second]

Site 

Upstream

surface area

(ft2)

Maximum 

daily

tidal flux1

(ft3/s)

1

October 11, 1995, at all sites except SPStage (September 9, 1995).

Minimum 

daily

tidal flux2

(ft3/s)

2

October 2, 1992.

Mean tidal 

flux 1992-

1996

(ft3/s)

SPStage 35,840,000

3 Median upstream surface area.

291 15.5 51.0

C54con 1,410,000 47.5 3.36 11.6

Sebcon1 14,100,000 551 29.2 118

Sebcon2 23,800,000 878 45.3 198 DAILY MEAN FRESHWATER DISCHARGE, IN CUBIC FEET PER SECOND

M J J A S O N D J F M A M J J A S O N D J F M A M J J A S O N D J F M A M J J A S O N D J

1992 1993 1994 1995 1996

1

10,000

2

5

10

20

50

100

200

500

1,000

2,000

5,000

Sebcon2

C54con (combined freshwater discharges from QFells and QC54)

SPStage

F M

(combined freshwater discharges from QNP, QC54, QFells,

and QSP)

(freshwater discharge from QSP)

Figure 6. Daily mean freshwater discharges at South Prong St. Sebastian River at Roseland (SPStage), Canal 54 near 

North Prong St. Sebastian River (C54con), and St. Sebastian River at U.S. 1 at Sebastian (Sebcon2).
12 Probability Analysis of the Relation of Salinity to Freshwater Discharge in the St. Sebastian River, Florida

Combined daily mean freshwater discharges 

ranged from 50 to 6,400 ft3

/s (during tropical storm 

Gordon, November 1994). The annual mean freshwater discharge of the system was 295 ft3

/s (table 4). A 

freshwater flux equivalent to the annual mean discharge would flush the system in 3.8 days. At minimum discharge, the freshwater flux would flush the 

system in 22.4 days. Tropical storm Gordon produced 

the maximum freshwater flow during the study; the 

freshwater flux for this storm would flush the system in 

less than 5 hours, or 0.2 day.

System Flushing

Salinity conditions in the St. Sebastian River are 

dependent on patterns of mixing and circulation of 

freshwater and saltwater. System flushing is based on 

inflow rates; flushing rates are determined by combining freshwater and saltwater inflows. Flushing periods 

presented in this report represent a minimum time in 

which the system contents will be replaced by new 

incoming water.

The time required for one system-volume of 

water in an estuary to be replaced may be expressed in 

terms of a tidal flushing period. Computation of a 

flushing period based on inflow rates assumes that 

freshwater and tidal inflows do not mix within the system, and that the first water into the system is the first 

water to leave the system. Due to stratification of flow 

observed in the estuary, these conditions typically do 

not exist. Calculations of flushing period, therefore, 

represent a minimum amount of time in which the system volume is replaced by either tidal flux, from 

incoming saltwater, or freshwater discharge from 

upstream sources. This minimum flushing period is 

calculated as the ratio of system-volume to daily flux 

for various values of tidal flux and freshwater inflow.

Flushing rates based on the combination of tide 

and freshwater discharge can be used to evaluate flushing characteristics of the system. The minimum time 

required for complete flushing of the entire system, 

based on minimum tidal flux and freshwater discharge, 

is 8.5 days. Combining tidal flux and freshwater discharge decreases the minimum time required to purge 

the system to one-third of the time required for tidal 

flux or freshwater discharge alone. Combined maximum tidal flux and freshwater discharge decreases the 

minimum flushing rate by less than 1 hour, to about 

4 hours, compared to the flushing rate for maximum 

freshwater discharge alone. The combined mean freshwater and tidal fluxes flush the system in about 

2.5 days on average. Based on the vertical stratification 

observed in this system, freshwater discharge typically 

is confined to the upper layer of flow. Therefore, the 

flushing times for upper and lower layers probably are 

significantly different. It is estimated that, during average conditions, the flushing period for the upper freshwater layer would be considerably less than 2.5 days, 

whereas the flushing time for the lower layer (due to 

tidal flux) would be about 3 to 4 days.

Flushing rates for system volumes can be combined with freshwater discharge probabilities to obtain 

a likelihood of flushing the system at a given flux 

(fig. 7). Ignoring mixing and stratified flow, freshwater 

discharges are sufficient to flush the South Prong 

upstream from its confluence with the St. Sebastian 

River (SPStage) and to flush the St. Sebastian River 

upstream from its confluence with the Indian River 

(Sebcon2) in a single day up to 9 percent of the time. 

Freshwater discharges are sufficient to flush Canal 54 

upstream from site C54con in a single day up to 

19 percent of the time.

Table 4. Daily mean freshwater discharge summary for St. Sebastian River basin

[All discharge values are in cubic feet per second] 

Site 

Maximum 

daily 

discharge

Date

Minimum 

daily discharge

Date

Annual mean discharge 

Mean 

dischargeYears

1992 1993 1994 1995 1

1996

QSP 2,300 11-16-94 4.2 11-23-93 189.5 88.0 177 114 87.7 117 1992-1996

QFells 1,780 11-16-94 18 04-21-95 67.1 75.2 118 94.3 59.5 91.9 1992-1996

QC54 3,990 11-20-94 0 multiple days 24.1 30.3 110 64.1 20.0 42.2 1971-1996

QNP 1,740 11-16-94 6.3 06-20-89 39.3 39.9 89.5 70.9 45.9 44.1 1987-1996

Total 295.2

1

Based on partial year.
Specific-Conductance Characteristics 13

Annual mean tidal flux and annual mean freshwater discharge can be plotted on the same graph with 

freshwater discharge duration curves to obtain an estimate of the probability that freshwater discharges will 

exceed the annual means of tidal flux and freshwater 

discharge. Daily mean freshwater discharges will 

exceed the combined annual mean freshwater discharge, 295 ft3/s (represented by the discharge-duration curve for site Sebcon2), 28 percent of the time and 

daily mean freshwater discharge will exceed the annual 

mean tidal flux, 198 ft3/s, 45 percent of the time at 

Sebcon2.

SPECIFIC-CONDUCTANCE 

CHARACTERISTICS

Daily mean specific-conductance values for the 

upper and lower layers were calculated for each of the 

five conductance data-collection sites. Mean and standard deviation of differences between the upper layer 

and lower layer specific-conductance values were computed for each site by using instantaneous and daily 

mean values. The statistics were computed for the 

entire study period and for data divided into seasons 

(wet and dry); the resulting statistics are discussed later 

in this report.

Conductance data exhibit long-term serial correlation attributed to the gradual mixing of water 

upstream from day to day. Instantaneous and daily 

mean values of specific conductance were used as a 

comparison of the range of daily values. Standard deviations of daily mean specific conductance are smaller 

than the standard deviations of instantaneous specificconductance values. Daily mean specific-conductance 

values were used for comparison and analysis in order 

to reduce the size of the data set, smooth the data set by 

averaging in outliers, reduce serial correlation, and 

maintain trends and characteristics of relatively shortterm events. The use of daily mean specific-conductance values rather than instantaneous values did not 

cause any bias in the data analysis. The use of daily 

mean specific-conductance values may not reflect 

short-term extremes but better represents the general 

trend of the data.

In general, the daily range in instantaneous specific-conductance values is greater in the upper layer 

than in the lower layer at Sebcon2 (fig. 8). The daily 

range of instantaneous specific-conductance values 

tends to increase with increasing daily mean specific 

0.01 0.1 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99 99.9 99.99

DAYS DISCHARGE WAS EXCEEDED, IN PERCENT

10

10,000

20

30

40

50

70

100

200

300

400

500

700

1,000

2,000

3,000

4,000

5,000

7,000

DAILY MEAN FRESHWATER DISCHARGE, IN CUBIC FEET PER SECOND

SPStage (freshwater discharge from QSP)

C54con (combined freshwater discharges from QFells and QC54)

Sebcon2 (combined freshwater discharges from QNP, QC54,

Total flux (tidal flux plus freshwater discharge)

Annual mean tidal flux

Annual mean freshwater discharge

QFells, and QSP)

Figure 7. Discharge duration curves for South Prong St. Sebastian River at Roseland (SPStage), Canal 54 near 

North Prong St. Sebastian River (C54con), and St. Sebastian River at U.S. 1 at Sebastian (Sebcon2).
14 Probability Analysis of the Relation of Salinity to Freshwater Discharge in the St. Sebastian River, Florida

conductance to a maximum of about 50,000 μS/cm for 

daily mean specific conductances above about 

15,000 μS/cm. Instantaneous specific conductance of 

the lower layer generally remains within a narrow 

range over the period of a day, typically ranging less 

than 20,000 μS/cm.

Time series of daily mean specific conductance 

data indicate some evidence of long-term seasonal 

trends associated with periods of sustained or diminished freshwater inflows; large changes in specific conductance were observed for a limited number of large 

storms. Though much of the long-term variation in specific conductance can be related to discharge, much of 

the short-term variation is not strongly related to discharge and appears to be influenced by other factors.

Daily mean specific-conductance values generally increased with distance downstream; values at 

C54con typically exceed values at SPcon (table 5). 

Specific-conductance values upstream from Sebcon1 

ranged from near seawater values to freshwater values, 

less than 500 μS/cm, during the study. At the mouth of 

the estuary (site Sebcon2) specific-conductance values 

in the freshwater range were common in samples from 

the upper layer and values for upper and lower layers 

indicate greater stratification than at sites upstream 

from the mouth.

0

60,000

10,000

20,000

30,000

40,000

50,000

DAILY RANGE, IN MICROSIEMENS PER CENTIMETER

SPECIFIC CONDUCTANCE, UPPER LAYER

0 5,000 10,000 15,000 20,000 25,000 30,000 35,000 40,000 45,000 50,000 55,000 60,000 65,000 70,000

DAILY MEAN, IN MICROSIEMENS PER CENTIMETER

0

60,000

10,000

20,000

30,000

40,000

50,000

SPECIFIC CONDUCTANCE, LOWER LAYER

Figure 8. Daily range of specific-conductance values for St. Sebastian River at U.S. 1 at Sebastian (Sebcon2).
Specific-Conductance Characteristics 15

The range in observed specific-conductance values may be characterized by a family of duration 

curves. Daily mean specific-conductance duration 

curves indicate the number of days within a given 

period for which specific conductance is expected to be 

below a given value (Searcy, 1959; Riggs, 1968b). 

Duration curves presented in this report do not represent probabilities of specific-conductance values being 

less than a given specific conductance on any particular 

day; rather the curves represent the percent of time that 

the specific conductance can be expected to be less 

than a given value.

Duration curves for specific conductance at five 

conductance data-collection sites and for each layer are 

shown in figure 9. The change from lower conductance 

waters upstream to higher conductance waters downstream and comparison of upper layer to lower layer 

variations are apparent. The freshwater and saltwater 

Table 5. Daily mean specific-conductance summary for St. Sebastian River basin, May 1, 1992, to March 18, 1996

[Specific-conductance values are in microsiemens per centimeter at 25 degrees Celsius; --, no data available]

Site 

Maximum 

daily mean 

conductance

Date of 

sample

Minimum 

daily mean 

conductance

Date of 

sample

Annual mean conductance Mean

conductance

1992-1996

1992 1993 1994 1995 1996

C54con 55,600 05-14-93 270 10-19-95 26,400 28,800 17,500 24,500 27,100 24,000

SPcon 45,400 05-22-95 120 09-13-95 -- 9,940 2,780 8,730 6,630 6,620

Sebcon1-upper1 53,800 09-04-93 120 10-19-95 27,800 20,100 15,200 15,500 20,200 22,000

Sebcon1-lower2 58,400 06-21-92 140 10-25-95 36,000 35,900 27,800 27,800 30,700 33,600

Sebcon2-upper1 58,100 06-21-95 350 10-19-95 31,500 33,300 26,600 32,400 35,300 31,400

Sebcon2-lower2 67,200 11-06-93 1,650 10-18-95 43,700 48,400 45,700 45,300 46,200 46,700

IRcon-upper1 62,800 10-09-92 7,670 10-17-95 39,000 43,900 43,800 43,100 28,300 42,500

IRcon-lower2 67,200 04-11-94 14,200 10-17-95 42,600 45,700 48,400 46,500 40,500 45,800

1Upper layer in the vertical water column.

2 Lower layer in the vertical water column.

Figure 9. Specific-conductance duration curves for data-collection sites.

100

70,000

200

300

400

500

700

1,000

2,000

3,000

4,000

5,000

7,000

10,000

20,000

30,000

40,000

50,000

SPECIFIC CONDUCTANCE, IN MICROSIEMENS PER CENTIMETER

0.1 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99

DAYS SPECIFIC CONDUCTANCE WAS NOT EXCEEDED, IN PERCENT

C54con

SPcon

Sebcon1, upper layer

Sebcon1, lower layer

Sebcon2, upper layer

Sebcon2, lower layer

IRcon, upper layer

IRcon, lower layer

layering in the system can be conceptualized as a wedge of less 

saline water on top of a wedge of 

saltwater. The similarities in 

shape and distribution of the specific-conductance duration 

curves from the upper layer specific-conductance values at the 

mouth of the estuary, Sebcon2-

upper, and the lower layer specific-conductance values near the 

middle of the estuary, Sebcon1-

lower, illustrate the wedges of 

freshwater and saltwater.

The differences in specific 

conductance from lower to upper 

layers indicate an increase in vertical stratification as the water in 

the estuary is channeled in and 

out of the constriction at the 

U.S. 1 bridge (Sebcon2) (fig. 2 

and table 6). Stratification of 

flow in the St. Sebastian River 

has little effect on the daily mean 

specific-conductance values 

from lower to upper layers in the 
16 Probability Analysis of the Relation of Salinity to Freshwater Discharge in the St. Sebastian River, Florida

nearby Indian River, as indicated by the small vertical 

differences in means at IRcon. Overall, the difference 

in daily mean specific-conductance values from lower 

to upper layers increases with distance downstream 

within the St. Sebastian River and decreases in the 

Indian River, indicating greater layering in the St. 

Sebastian River than in the receiving water body, the 

Indian River.

Duration curves of specific conductance grouped 

by discharge ranges were computed for the conductance-monitoring sites (fig. 10). The curves indicate 

that, on average, low daily mean discharges produce 

substantially fewer days of low specific-conductance 

values at the monitoring sites. Departure of individual 

duration curves from the specific-conductance duration 

curve for all discharges for specific ranges of discharge 

Table 6. Seasonal and period of record lower-layer to 

upper-layer differences for instantaneous and daily mean 

values of specific conductance, May 1, 1992, to March 18, 

1996

[Specific-conductance values are in microsiemens per centimeter at 

25 degrees Celsius; inst., instantaneous]

Site 

Period of record Wet season1

1

June through October.

Dry season2

2

November through May.

Mean Standard 

deviation Mean Standard deviation Mean Standard deviation

Sebcon1 (inst.) 10.29 13.14 8.19 12.79 13.89 12.96

Sebcon1 (daily) 10.18 11.68 9.02 12.38 12.05 9.73

Sebcon2 (inst.) 15.30 13.32 16.81 14.05 14.11 12.59

Sebcon2 (daily) 15.41 10.33 16.96 11.13 14.19 9.49

IRcon (inst.) 3.39 7.07 3.38 7.44 3.39 6.60

IRcon (daily) 3.38 6.04 3.34 6.29 3.42 5.74

0.1 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99 99.9

100

100,000

300

500

700

1,000

3,000

5,000

7,000

10,000

30,000

50,000

70,000

SPECIFIC CONDUCTANCE, IN MICROSIEMENS PER CENTIMETER

Discharges <50 ft3/s

Discharges <100 ft3/s

Discharges <200 ft3/s

Discharges <500 ft3/s

All Discharges

Canal 54 near North Prong St. Sebastian River, FL

0.1 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99 99.9

DAYS SPECIFIC CONDUCTANCE WAS NOT EXCEEDED, IN PERCENT

Discharges <50 ft3/s

Discharges <100 ft3/s

Discharges <200 ft3/s

Discharges <500 ft3/s

All Discharges

South Prong St. Sebastian River near Roseland, FL

0.1 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99 99.9

100

100,000

300

500

700

1,000

3,000

5,000

7,000

10,000

30,000

50,000

70,000

Upper Layer

Discharges <100 ft3/s

Discharges <200 ft3/s

Discharges <500 ft3/s

All Discharges

St. Sebastian River near Railroad Bridge at Roseland, FL

0.1 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99 99.9

Lower Layer

Discharges <100 ft3/s

Discharges <200 ft3/s

Discharges <500 ft3/s

All Discharges

St. Sebastian River near Railroad Bridge at Roseland, FL

Figure 10. Specific-conductance duration curves for selected freshwater discharge ranges at conductance-monitoring sites.

(Sebcon1) (Sebcon1)

(C54con) (SPcon)
Specific-Conductance Characteristics 17

is an indication that, in highly stratified waters, the 

effect of discharge magnitude on specific conductance 

(and thus, salinity) is relatively small up to a certain 

threshold discharge; beyond this threshold value, the 

volume of freshwater flow is sufficient to overcome the 

gradient between the freshwater/saltwater layers, thus 

shifting the duration curve.

The duration curves shown in figure 10 also indicate spatial variability in the specific-conductance 

duration curves with respect to discharge. Spacing of 

the duration curve family is a general indication of the 

effect freshwater discharges have on specific-conductance values at each site. Where freshwater is a small 

part of the total volume, specific-conductance values 

do not change significantly even with relatively large 

increases in freshwater discharge. Where freshwater is 

a large part of the total volume, small increases in 

freshwater inflow cause relatively large changes in 

specific conductance. These sites have a family of 

duration curves for selected discharge ranges that are 

well spaced and evenly separated at the lower ranges of 

specific conductance, indicating a more even distribution of specific conductances throughout the range of 

flows at the selected site (for example, sites C54con 

and SPcon). At these sites, small changes in freshwater 

discharge have a greater effect on specific conductance 

than at sites where freshwater discharge is a smaller 

percentage of the total flow (for example, downstream 

sites). As discharge gradients become more prevalent 

in the middle and near the mouth of the estuary, 

specific-conductance values are less readily affected by 

small changes in freshwater discharge and the gradient 

0.1 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99 99.9

100

100,000

300

500

700

1,000

3,000

5,000

7,000

10,000

30,000

50,000

70,000

SPECIFIC CONDUCTANCE, IN MICROSIEMENS PER CENTIMETER

Upper Layer

Discharges <100 ft3/s

Discharges <200 ft3/s

Discharges <500 ft3/s

All Discharges

St. Sebastian River at U.S. 1 at Sebastian, FL

0.1 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99 99.9

Lower Layer

Discharges <100 ft3/s

Discharges <200 ft3/s

Discharges <500 ft3/s

All Discharges

St. Sebastian River at U.S. 1 at Sebastian, FL

0.1 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99 99.9

DAYS SPECIFIC CONDUCTANCE WAS NOT EXCEEDED, IN PERCENT

10,000

100,000

20,000

30,000

40,000

50,000

60,000

70,000

80,000

90,000

Upper Layer

Discharges <200 ft3/s

Discharges <500 ft3/s

All Discharges

Indian River near Sebastian, FL

0.1 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99 99.9

Lower Layer

Discharges <200 ft3/s

Discharges <500 ft3/s

All Discharges

Indian River near Sebastian, FL

(Sebcon2) (Sebcon2)

(IRcon) (IRcon)

Figure 10. Specific-conductance duration curves for selected freshwater discharge ranges at conductance-monitoring 

sites--Continued.
18 Probability Analysis of the Relation of Salinity to Freshwater Discharge in the St. Sebastian River, Florida

between freshwater and saltwater is overcome only by 

much larger inflows of freshwater (for example, 

Sebcon1 and Sebcon2-upper). Specific-conductance 

distributions for the lower layer at sites closer to the 

Indian River (for example, Sebcon2-lower and IRcon) 

are more highly affected by saltwater flux and less 

influenced by freshwater discharges.

THE PROBABILISTIC RELATION OF 

SALINITY TO FRESHWATER DISCHARGE

The relation of specific conductance to discharge 

was investigated in order to determine the effect of different flow regimes on salinity in the St. Sebastian 

River. Estimating the effect of changing flow regimes 

requires a predictive model that integrates both systematic and random components of salinity variation. 

Based on the specific-conductance-discharge relation, 

specific-conductance values were predicted for a series 

of discharges and the remaining variation in specific 

conductance was modeled as a random normal process. 

This approach required determination of a functional 

relation between discharge and specific conductance 

that minimized the standard deviation of residuals and 

resulted in a normal random independent distribution 

of residuals over a range of discharges. Then, the standard deviation of the residuals was modeled to represent the uncertainty in the estimated specific 

conductance for each successive discharge value in the 

time series. The probability of a particular salinity condition (as indicated by the salinity-specific-conductance relation) exceeding or failing to exceed a given 

limit could then be computed by summing the conditional probabilities that each predicted specificconductance value in a time series was above or below 

that limit.

Modeling Specific Conductance as a 

Function of Discharge

The basic form of the model applied for this 

study has been used by Miller and McPherson (1991) 

and is derived from a simple solute-solvent dilution 

computation. The equation presented by Miller and 

McPherson was adapted to the St. Sebastian River 

estuary by assuming that a volume of water sampled at 

some fixed point can be represented by a volume of 

water composed of aliquots contributed from upstream 

and downstream. Specific-conductance data at each 

site for two depths indicates stratified flow of freshwater and saltwater. Because of this layering, the equation is applied independently to each layer with the 

assumption that the water within the layer is well 

mixed. Over a period of 1 day, specific-conductance 

values are averaged to produce a reasonable representation of the specific conductance of a volume of water 

that moved past the sampling point during the 24-hour 

period as water in the river moves upstream and downstream with the varying tide.

The model presented in this report assumes that 

the ratio of saltwater to freshwater mixing can be 

described as a function of freshwater discharge. The 

relation of specific conductance at a point of interest in 

the river to that of inflowing water is given by the equation

, (3)

where CM is the specific conductance at a specific 

measurement site, in microsiemens per centimeter;

CF is the specific conductance of the freshwater inflow, in microsiemens per centimeter;

CS is the specific conductance of the saltwater 

inflow, in microsiemens per centimeter;

VF is the volume of freshwater inflow, in cubic 

feet; and

VS is the volume of the saltwater inflow, in 

cubic feet.

Over uniform time intervals, discharge rates can be 

substituted for volumes, and the equation can be written in the following form:

, (4)

where QF is the freshwater discharge at a specific 

measurement site, in cubic feet per second.

Empirical evaluation of freshwater discharge 

(QF) and tidal flux (QT) indicates that the ratio of tidal 

flux to freshwater discharge can be expressed as a function of freshwater discharge (fig. 11):

, (5)

where b0 is a discharge coefficient, and

b1 is a power of discharge.

CM

CFVF CSVS +

VF VS + = --------------------------------

CM

CFQF CSQT +

QF QT + = ----------------------------------

QT QF ⁄ b0QF

b1 =
The Probabilistic Relation of Salinity to Freshwater Discharge 19

Equation 4 may be expressed in terms of the ratio QT/QF

by dividing the numerator and denominator by QF:

. (6)

Substitution in equation 6 of the expression for the ratio 

QT/QF from equation 5 yields a statistical model of specific conductance as a function of discharge:

, (7)

where is the random error and specific-conductance values of saltwater and freshwater are 

held constant.

The statistical model of specific conductance represents the effect of freshwater discharge displacing saltwater from the system. This effect generally is shortterm and produces a highly random estimate of specific 

conductance for a given freshwater discharge.

In the statistical model of specific conductance 

(eq 7), serial correlation and dilution of the saltwater 

inflow from sustained freshwater flows were taken into 

account by applying a lagged smoothing technique to 

the estimated specific-conductance time series to 

reduce randomness in the specific-conductance estimate. The lagged smoothing technique is a weighting 

function that describes how specific-conductance estimates are affected by freshwater and saltwater mixing 

from previous days and is given as:

, (8)

where is a specific-conductance value, in 

microsiemens per centimeter, accounting 

for serial correlation and mixing across 

days; and

 is a weighting coefficient.

The lagged smoothing technique produces a less random estimate of specific conductance for a given freshwater discharge and accounts for long-term effects of 

sustained freshwater discharges.

Saltwater conductance values (CS) were 

replaced by estimates for in the statistical model 

of specific conductance (eq 7) to provide an optimal 

mathematical expression that fit observed data. This 

optimal mathematical expression accounts for the relatively instantaneous response of specific conductance 

to freshwater discharge, serial correlation, and the dilution of saltwater inflow by sustained freshwater inflow. 

The statistical model of specific conductance as a function of discharge then becomes:

, (9)

where is a discharge coefficient.

The salinity-response model (eq 9), based on 

daily mean discharge, was fit to observed daily mean 

specific-conductance data. This model includes two 

parts, each fit by least-squares nonlinear optimization. 

These two parts accounted for variation in specific conductance associated with two processes: long-term 

dilution as a result of sustained freshwater inflows 

(eq 8), and a short-term effect of freshwater discharge 

on saltwater inflow (eq 7).

The modeled fit was optimized using an iterative 

search routine available in the Quattro Pro version 5.0 

spreadsheet software package. Nonlinear regression of 

measured and fitted specific-conductance values was performed by an iterative process of optimization. The 

summed squares of the residuals were minimized using 

quadratic, forward, conjugate, and automatic scaling processes in the optimizer routine. Various initial model 

parameters were used to start the optimization process to 

verify the final output of model variables and confirm that 

the minimization process was completed and repeatable.

Figure 11. Freshwater discharge (QF) as a function of the 

ratio of tidal flux to freshwater discharge (QT/QF) at St. 

Sebastian River at U.S. 1 at Sebastian (Sebcon2).

10 10,000 100 1,000

FRESHWATER DISCHARGE, 

0.01

5

0.02

0.03

0.04

0.05

0.07

0.1

0.2

0.3

0.4

0.5

0.7

1

2

3

4

RATIO OF TIDAL FLUX

 TO FRESHWATER DISCHARGE (

QT /QF)

IN CUBIC FEET PER SECOND (QF)

CM

CFQF QF ⁄ CSQT QF + ⁄

QF QF ⁄ QT QF + ⁄ = --------------------------------------------------------

CM

CF b0QF

b1

CS ( ) +

1 + b0QF

b1 ( )

= ----------------------------------------- + ε

ε

CSMi

CF b0QF

b1

CS ( ) +

1 b0QF

b1 ( ) +

-------------------------------------- 1 b2 ( ) – b2CSMi – 1 = + + ε

CSMi

b2

CSMi

CM

CF b3QF

b1

CSMi ( ) +

1 b3QF

b1 ( ) +

= ------------------------------------------- + ε

b3
20 Probability Analysis of the Relation of Salinity to Freshwater Discharge in the St. Sebastian River, Florida

The salinity-response model (eq 9) consists of six 

variables and an initial condition (table 7) used to start 

the model and activate the exponential smoothing part 

of the model. The initial condition for each site was set 

to represent specific-conductance values within the 

range of expected daily values of specific conductance 

for the given site. The specific conductance of freshwater (CF) and saltwater (CS) within the model was set at 

250 and 50,000 μS/cm, respectively, and remain constant in the equation. These specific-conductance values 

represent average salinity conditions for freshwater 

inflow into the system and for tidally driven waters from 

the Indian River and were used as constants in the model 

for all sites and depths. Evaluation of the data indicated 

that the discharge power, b1, could be set to the inverse 

cube of discharge at all sites, reducing the degrees of 

freedom without significantly affecting the fit of the 

salinity-response model. A damping coefficient (b2), 

representing a percentage of the daily mean specificconductance value from the previous day, that influences the current day was optimized for and found to be 

nearly constant for all sites (0.97 or 0.98) except South 

Prong St. Sebastian River (SPcon), which was more 

affected by freshwater inflow (damping coefficient of 

0.87). Discharge coefficients (b0 and b3) were optimized 

for each site and generally decreased from upstream to 

downstream in proportion to system storage.

There were three goals in development of the 

salinity-response model: (1) describe as much of the 

variation in specific conductance as possible in terms 

of freshwater discharge, (2) define a basic form of the 

model that is descriptive of all sites with a minimum 

number of degrees of freedom, and (3) eliminate bias in 

the residuals. The solute-solvent model applied by 

Miller and McPherson (1991) gave an “s” shaped form 

that appeared to match the overall pattern of the empirical relation of specific conductance to freshwater discharge observed in the data. The “s-curve” shape of 

equation 7 provided a reasonable relation of specific 

conductance to freshwater discharge and was consistent with the expected conditions given the relative 

influxes of freshwater and saltwater (fig. 12). Although 

the river was consistently stratified throughout most of 

the study (except during high-flow events) and, therefore, not well mixed vertically, conductance profiles 

measured during the study indicate water within the 

upper and lower layers is generally well mixed. However, movement of the halocline past the locations of 

the conductance-monitoring probes in the upper and 

lower layers added randomness to the data. This randomness was countered to some extent by the use of 

daily mean values of specific conductance.

Based on examination of the relation of specific 

conductance to discharge (fig. 12), two principal 

effects are indicated: a short-term effect from stratification, and a long-term effect from downstream saltwater 

dilution. The short-term effect is brought about by 

stormwater discharges which move the halocline to 

lower depths and farther downstream in the St. Sebastian River, producing a reduction in specific conductance at discharges above a given value. Although this 

discharge value differed from site to site, the underlying effect was similar for all sites -- a proportional 

decrease in specific conductance with an increase in 

discharge. Given the similarity in the rate of this 

response, a form of the statistical model (eq 7) based on 

a single set of calibration coefficients for freshwater 

and saltwater specific-conductance values was found to 

fit well for all sites. Discharge coefficients, which also 

were fit empirically, varied inversely with the crosssectional area of the stream above the sampling point at 

each site (table 7). The largest discharge coefficient 

was observed at the upstream site on the South Prong 

St. Sebastian River, SPcon, where the cross-sectional 

area was the smallest; the smallest discharge coefficient was observed for the lower sampling point at the 

downstream site in the Indian River Lagoon, IRcon.

The mixing model (eq 7) alone accounts well for 

the short-term effects of discharge on specific conductance but does not account well for long-term dilution 

effects. When the mixing model is fit without accounting 

for long-term dilution, the resulting residuals are asymTable 7. Statistical parameters for the model of specific 

conductance as a function of discharge for St. Sebastian 

River basin, May 1, 1992, to March 18, 1996

[μS/cm, microsiemens per centimeter at 25 degrees Celsius. For this 

table, specific conductance of freshwater is 250 μS/cm or less and 

specific conductance of saltwater is 50,000 μS/cm or greater. Power of 

discharge (b1) is -3 (see equation 9)]

Site 

Discharge 

coefficients

(b0) (b3)

Weighting 

coefficient

(b2)

Initial 

condition

μS/cm

( )

Standard 

error

μS/cm

C54con 0.011357 0.003666 0.98 27,000 11,900

SPcon .040506 .004896 .87 5,000 4,300

Sebcon1-upper1

1

Upper layer in the vertical water column.

.006008 .001801 .97 10,000 12,400

Sebcon1-lower2

2

Lower layer in the vertical water column.

.003152 .001212 .98 30,000 11,900

Sebcon2-upper1 .003045 .001303 .98 15,000 10,300

Sebcon2-lower2 .000745 .000312 .98 30,000 7,290

IRcon-upper1 .001602 .000008 .98 20,000 8,100

IRcon-lower2 .001132 .000008 .98 35,000 8,000

CSMi – 1
The Probabilistic Relation of Salinity to Freshwater Discharge 21

1 10,000 10 100 1,000

0

60,000

10,000

20,000

30,000

40,000

50,000 SPcon

1 10,000 10 100 1,000

0

60,000

10,000

20,000

30,000

40,000

50,000

1 10,000 10 100 1,000

0

60,000

10,000

20,000

30,000

40,000

50,000

SPECIFIC CONDUCTANCE, IN MICROSIEMENS PER CENTIMETER

1 10,000 10 100 1,000

0

60,000

10,000

20,000

30,000

40,000

50,000

FRESHWATER DISCHARGE, IN CUBIC FEET PER SECOND

1 10,000 10 100 1,000

0

60,000

10,000

20,000

30,000

40,000

50,000

1 10,000 10 100 1,000

0

60,000

10,000

20,000

30,000

40,000

50,000

Sebcon2, lower layer

SPcon

 Sebcon2, upper layer

Sebcon2, lower layer

 Sebcon2, upper layer

Figure 12. Specific conductance as a function of freshwater discharge at South Prong St. Sebastian River near Roseland 

(SPcon) and St. Sebastian River at U.S. 1 at Sebastian (Sebcon2). Left graphs show regression curve of specific 

conductance as a function of freshwater discharge (eq 7) using ordered freshwater discharges and observed specific 

conductance data. Right graphs show regression curve of specific conductance as a function of freshwater discharge 

(eq 7) using ordered freshwater discharges and specific conductance predicted using the salinity response model (eq 9).
22 Probability Analysis of the Relation of Salinity to Freshwater Discharge in the St. Sebastian River, Florida

metrical. The complete statistical model of specific conductance (eq 9) predicts specific-conductance values at 

high discharges well with very small residuals, but generally over predicts specific-conductance values at low discharges. This indicates that the system responds strongly 

to discharge at low specific conductances and to dilution 

at higher specific-conductance values. Simple exponential 

smoothing usually improved the model fit in the midrange 

of specific conductances, but was not sufficient to fit both 

the low and high specific conductances. The inclusion of 

a second long-term model (eq 8, a smoothed version of 

the original model, eq 7), in place of the saltwater specific 

conductance (CS), resolved the asymmetry in residuals 

and allowed the model to be fit throughout the complete 

range of specific-conductance values. Although this 

model (eq 8) increased the degrees of freedom in the overall model (eq 9) and added a new layer of complexity, the 

probability analysis is predicated on a well-behaved set of 

residuals that can be estimated as a random process.

Variability in model residuals confirms the observation that the St. Sebastian River estuary is a system in 

which specific conductance responds strongly to 

increased freshwater discharge. Small increases in discharge above long-term or base-flow conditions can 

reduce specific conductance relatively quickly. Specificconductance values respond more to higher discharges, 

as indicated by a reduction in the variability of model 

residuals at higher discharges (fig. 13). Model residuals 

10 10,000 100 1,000

FRESHWATER DISCHARGE, IN CUBIC FEET PER SECOND

-40,000

40,000

-30,000

-20,000

-10,000

0

10,000

20,000

30,000

RESIDUAL, IN MICROSIEMENS PER CENTIMETER

C54con

1 10,000 10 100 1,000 -30,000

20,000

-20,000

-10,000

0

10,000

SPcon

10 10,000 100 1,000 -50,000

40,000

-40,000

-20,000

0

20,000

Sebcon1, upper layer

10 10,000 100 1,000 -50,000

40,000

-40,000

-20,000

0

20,000

Sebcon1, lower layer

Figure 13. Relation of model residuals to freshwater discharge.
The Probabilistic Relation of Salinity to Freshwater Discharge 23

for upstream sites, where specific-conductance values 

are more strongly influenced by freshwater discharges 

(for example, SPcon), are increasingly heteroscedastic. 

Specific-conductance values at downstream sites, such 

as Sebcon2 and IRcon, are much less influenced by 

freshwater discharges and variance in model residuals 

becomes increasingly homoscedastic with distance 

downstream. Variation in the residuals at high specificconductance values is related to other system properties 

that are not monitored and, therefore, cannot be 

accounted for in either the model or prospective management decisions. Neglecting minor deviations, the residuals describe a relatively random normal distribution 

throughout the range of discharges observed. Unexplained variation in the salinity-response model represents factors other than discharge that alter the 

probability function of specific conductance.

Time-series hydrographs of measured and simulated specific conductance show the fit of the salinity 

response model (fig. 14). Overall, the best model fit at all 

sites was for data representing periods of highest flows 

when discharge has the greatest effect on specific conductance. The standard error of the best-fit models for all sites 

ranged from 4,300 to 12,400 μS/cm.

Figure 13. Relation of model residuals to freshwater discharge--Continued.

10 10,000 100 1,000

FRESHWATER DISCHARGE, IN CUBIC FEET PER SECOND

-40,000

40,000

-30,000

-20,000

-10,000

0

10,000

20,000

30,000

RESIDUAL, MICROSIEMENS PER CENTIMETER

Sebcon2, upper layer

10 10,000 100 1,000 -40,000

40,000

-30,000

-20,000

-10,000

0

10,000

20,000

30,000 Sebcon2, lower layer

10 10,000 100 1,000 -30,000

30,000

-20,000

-10,000

0

10,000

20,000

IRcon, upper layer

10 10,000 100 1,000 -30,000

30,000

-20,000

-10,000

0

10,000

20,000

IRcon, lower layer
24 Probability Analysis of the Relation of Salinity to Freshwater Discharge in the St. Sebastian River, Florida

M J J A S O N D J F M A M J J A S O N D J F M A M J J A S O N D J F M A M J J A S O N D J F M

1992 1993 1994 1995 1996

0

60,000

10,000

20,000

30,000

40,000

50,000 measured

simulated

Canal 54 near North Prong St. Sebastian River, Fla. (C54con)

M J J A S O N D J F M A M J J A S O N D J F M A M J J A S O N D J F M A M J J A S O N D J F M

1992 1993 1994 1995 1996

0

50,000

10,000

20,000

30,000

40,000

SPECIFIC CONDUCTANCE, IN MICROSIEMENS PER CENTIMETER

measured

simulated

South Prong St. Sebastian River near Roseland, Fla. (SPcon)

M J J A S O N D J F M A M J J A S O N D J F M A M J J A S O N D J F M A M J J A S O N D J F M

1992 1993 1994 1995 1996

0

60,000

10,000

20,000

30,000

40,000

50,000 measured

simulated

St. Sebastian River near Railroad Bridge at Roseland, Fla. (Sebcon1, upper layer)

M J J A S O N D J F M A M J J A S O N D J F M A M J J A S O N D J F M A M J J A S O N D J F M

1992 1993 1994 1995 1996

0

60,000

10,000

20,000

30,000

40,000

50,000

measured

simulated

St. Sebastian River near Railroad Bridge at Roseland, Fla. (Sebcon1, lower layer)

no record

no record

no record

no record

Figure 14. Measured and simulated daily specific-conductance values.
The Probabilistic Relation of Salinity to Freshwater Discharge 25

M J J A S O N D J F M A M J J A S O N D J F M A M J J A S O N D J F M A M J J A S O N D J F M

1992 1993 1994 1995 1996

0

70,000

10,000

20,000

30,000

40,000

50,000

60,000 measured

simulated

St. Sebastian River at U.S. 1 at Sebastian, Fla. (Sebcon2, upper layer)

M J J A S O N D J F M A M J J A S O N D J F M A M J J A S O N D J F M A M J J A S O N D J F M

1992 1993 1994 1995 1996

0

70,000

10,000

20,000

30,000

40,000

50,000

60,000

SPECIFIC CONDUCTANCE, IN MICROSIEMENS PER CENTIMETER

measured

simulated

St. Sebastian River at U.S. 1 at Sebastian, Fla. (Sebcon2, lower layer)

M J J A S O N D J F M A M J J A S O N D J F M A M J J A S O N D J F M A M J J A S O N D J F M

1992 1993 1994 1995 1996

0

70,000

10,000

20,000

30,000

40,000

50,000

60,000

measured

simulated

Indian River near Sebastian, Fla. (IRcon, upper layer)

M J J A S O N D J F M A M J J A S O N D J F M A M J J A S O N D J F M A M J J A S O N D J F M

1992 1993 1994 1995 1996

0

70,000

10,000

20,000

30,000

40,000

50,000

60,000

measured

simulated

Indian River near Sebastian, Fla. (IRcon, lower layer)

no record

no record

no record

no record

Figure 14. Measured and simulated daily specific-conductance values--Continued.
26 Probability Analysis of the Relation of Salinity to Freshwater Discharge in the St. Sebastian River, Florida

The limited ability of this model to predict specific conductance at low flows illustrates the limited 

extent to which controls on discharge (detention or 

retention of freshwater inflows) can be expected to 

alter salinity conditions in the river. Although discharge is a controlling factor in the occurrence of the 

very lowest salinity conditions in the river, observed 

specific-conductance values indicate many instances 

when salinities are at moderate levels when discharge 

clearly is not a controlling factor. 

A plot of the fitted statistical models without 

lagged smoothing for all sites and layers is shown in 

figure 15. The order of the curves from left to right in 

the graph generally is from upstream to downstream 

and from upper to lower layers. Specific conductances 

for the South Prong site (SPcon), for example, tend to 

plot in the lower left area of the curve set and data for 

the Indian River (IRcon) tend to plot in the upper right 

area of the curve set. The effect of vertical stratification 

produces a similar movement from lower left to upper 

right, reflecting a greater volume of water upstream 

1 10 100 1,000 10,000

FRESHWATER DISCHARGE, IN CUBIC FEET PER SECOND (QF)

0

60,000

10,000

20,000

30,000

40,000

50,000

C54con

SPcon

Sebcon1, upper layer

Sebcon1, lower layer

Sebcon2, upper layer

Sebcon2, lower layer

IRcon, upper layer

IRcon, lower layer

(b0 = 0.011357)

(b0 = 0.040506)

(b0 = 0.006008)

(b0 = 0.003152)

(b0 = 0.000745)

(b0 = 0.001132)

(b0 = 0.001602)

(b0 = 0.003045)

Fitted statistical model without lagged smoothing given by equation 7:

,

where CF is specific conductance of freshwater (250 μS/cm),

CM is specific conductance of saltwater (50,000 μS/cm),

b0 is a coefficient of discharge,

b1 is, -3, a power of discharge, and

ε is random error.

Figure 15. Fitted statistical model without lagged smoothing for 

specific conductance as a function of freshwater discharge at sites in 

the St. Sebastian River. SPECIFIC CONDUCTANCE, IN MICROSIEMENS PER CENTIMETER (CM)

CM

CF b0QF

b1

CS ( ) +

1 + b0QF

b1 ( )

= ----------------------------------------- + ε
The Probabilistic Relation of Salinity to Freshwater Discharge 27

from and above the sampling point. The relative effect 

of moving from upstream to downstream in the river 

and from top to bottom in the vertical profile is best 

illustrated by the overlap in the models for the lower 

layer of the upstream St. Sebastian River site (Sebcon1, 

lower layer) and the upper layer of the downstream site 

(Sebcon2, upper layer). 

During the process of optimization, the model 

parameters were not constrained to a given range of 

numerical values. The model calibration, however, was 

naturally constrained by the range of hydrologic conditions observed over the period of data collection. Discharge data during the study were reasonably 

representative of average conditions in the river basin 

based on comparisons to earlier discharge record 

(Knowles, 1995). The combined total flow of all freshwater inflows to the St. Sebastian River ranged from 50 

to 6,400 ft3

/s and included several periods of sustained 

high and low flow. The distribution of flows between 

discharge events, which also can have a considerable 

effect on model response, was not compared to historical record but likely is representative of conditions in 

the basin. The application of the model to changes in 

freshwater inflow to the system is considered reasonable, because modeled estimates of effects of detention 

and retention were based on interpolations within the 

time and discharge domain of the original calibration 

data set.

Residual Uncertainty and Exceedance 

Probabilities

The residuals of the best-fit relation of the salinity-response model (eq 9) were modeled as a function 

of freshwater discharge to produce estimates of the 

uncertainty in daily mean specific-conductance values. 

These uncertainty estimates were combined with estimated specific-conductance values to produce an overall probability model. The standard deviation of 

residuals (Z) was modeled as the square root of the 

variance which is represented as a log relation of the 

squared residuals to discharge. The resulting equation 

is:

, (10)

where is the standard deviation of model residuals for a given discharge;

Q is the freshwater discharge, in cubic feet per 

second; and

b4 and b5 are fitting coefficients.

This form was used to predict the uncertainty in an estimated specific-conductance value for each successive 

discharge in a time series. Both fitting coefficients b4

and b5 were determined using nonlinear optimization 

for data at each site based on minimization of the difference between predicted and observed nonexceedance probabilities (X) by the objective function:

, (11)

where is the estimated specific conductance, in 

microsiemens per centimeter;

CL is a given specific-conductance limit, in 

microsiemens per centimeter;

C is observed daily mean specific conductance, in microsiemens per centimeter; and

j is a discrete value of CL.

The probability of the observed specific conductance 

not exceeding a defined limit was taken directly from 

empirical duration curves. Thirteen discrete values of 

CL were evaluated, and CL values ranged from 100 to 

65,000 μS/cm and varied for individual sites.

The total probability (Kim, 1992, p. 16) of a particular specific-conductance value in the time series 

exceeding or failing to exceed a given limit was computed by summing the product of the probability of the 

conditioning event for all observed values of freshwater discharges and the conditional probability that each 

value in the complete time series might fall above or 

below that discrete specific conductance limit, given 

the uncertainty in residuals of the specific conductance 

model. The probability of daily specific-conductance 

values not exceeding a given limit was calculated 

assuming a Gaussian probability distribution of model 

residuals, and is represented as:

, (12) SEZq b4Q

b5 =

SEZq

X P Cˆ CLj [ ] < PC CLj – [ ] <

2

j1

jn

= ∑

C

ˆ

PC CL [ ] < PC CL = ∑ [ ] ( ) < ( ) Q q = ⋅ PQ q [ ] =
28 Probability Analysis of the Relation of Salinity to Freshwater Discharge in the St. Sebastian River, Florida

where is the probability that specific conductance, C, is less than a given limit of specific conductance, ;

 is the probability that 

specific conductance, C, is less than a given 

limit of specific conductance, , given that 

Q=q;

q is a given value of freshwater discharge, in 

cubic feet per second; and

 is the probability that Q=q.

In terms of a deterministic statistical model, the 

probability of a specific-conductance value being less 

than a given limit for a given discharge is expressed by:

, (13)

where Z is the residual from the specific-conductance 

model (eq 10).

If the residuals have a random normal distribution, the 

probability of nonexceedance for a given value of freshwater discharge is:

, (14)

where N is the Gaussian cumulative probability function, and

 is the standard deviation of Z at Q=q.

Accurate evaluations of equations 12, 13, and 14 

are properly obtained by integration over the probability function of discharge (Q). Because that function of 

discharge is undetermined, the sample distribution of 

discharges observed over the study period was substituted for the unknown population distribution of discharges in the probability model evaluation. Total 

nonexceedance probabilities were calculated as the 

average of daily nonexceedance probabilities estimated 

for each day over the period of record based on estimates of specific conductance (CM) from equation 9 

and estimates of standard error of residuals (SEZq) from 

equation 10. The total nonexceedance probabilities 

calculated for each site and depth, assuming 

 for all observed values of freshwater 

discharge, are thus:

, (15)

where i is a day, and

n is the number of days.

The model parameters were optimized to fit the 

predicted and observed specific-conductance duration 

curves for the subset of days having valid specificconductance data during the study period. Then the 

fitted model was used to generate specific-conductance 

values during periods of missing data, and new duration 

curves were calculated.

Duration curves fitted by the probability model 

(eq 15) for estimated specific-conductance values and 

duration curves of observed specific conductance data 

are shown in figure 16. The values for parameters used 

in the computation of standard errors are given in 

table 8. Estimated duration curves generally fit empirical curves by a standard error factor of 1.28 which, 

when multiplied by or divided into a given duration 

estimate, gives a range of uncertainty. The standard 

error of the residuals model ranged from 4,400 μS/cm 

at SPcon to 12,000 μS/cm at Sebcon1, upper layer. The 

value of the fitting parameters in table 8 have little 

meaning other then to indicate the trend of the relation 

of residuals to discharge. The exponent of discharge 

(b5) is positive for lower-layer sites at the lower end of 

the system and for IRcon, upper and lower layers, indicating an increase in residuals at higher discharges. 

Exponents were negative for upper-layer sites, indicating a downward trend in residuals with discharge. The 

relation of residuals to discharge for all sites shows a 

tendency to be at a minimum at the two extremes of specific conductance—the highest specific conductances, 

associated with the lowest discharges at downstream 

sites, and the lowest specific conductances, associated 

with the highest discharges at upstream sites. This pattern is to be expected from the plot of data in figure 12 

and the relations shown in figure 15.

PC CL [ ] <

CL

PC CL [ ] ( ) < ( ) Q q =

CL

PQ q [ ] =

PC CL [ ] ( ) < ( ) Q q = PZCL C

ˆ < –   =   ( ) Q q =

PZ CL C

ˆ [ ] < ( ) – ( ) Q q = N

CL C

ˆ –

σZq

----------------

      =

σZq

PQ q [ ] = = 1 ⁄ n

PC CL [ ] < N

CL CMi –

SEZq ( )i

--------------------- 1

n

⋅ --

i = 1

n

= ∑
The Probabilistic Relation of Salinity to Freshwater Discharge 29

0.1 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99 99.9

observed

modeled

SPcon

100

60,000

200

300

400

500

700

1,000

2,000

3,000

4,000

5,000

7,000

10,000

20,000

30,000

40,000

50,000

1 2 5 10 20 30 40 50 60 70 80 90 95 98 99 99.9

observed

modeled

Sebcon1, upper layer

1 2 5 10 20 30 40 50 60 70 80 90 95 98 99 99.9

observed

modeled

Sebcon1, lower layer

100

60,000

200

300

400

500

700

1,000

2,000

3,000

4,000

5,000

7,000

10,000

20,000

30,000

40,000

50,000

SPECIFIC CONDUCTANCE, IN MICROSIEMENS PER CENTIMETER

0.1 1 2 5 10 20 30 4050 60 70 80 90 95 98 99 99.9

DAYS SPECIFIC CONDUCTANCE WAS NOT EXCEEDED, IN PERCENT

observed

modeled

C54con

Figure 16. Specific-conductance duration curves for modeled and observed values.
30 Probability Analysis of the Relation of Salinity to Freshwater Discharge in the St. Sebastian River, Florida

0.1 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99 99.9

observed

modeled

Sebcon2, lower layer

10,000

70,000

15,000

20,000

25,000

30,000

40,000

50,000

60,000

0.1 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99 99.9

observed

modeled

IRcon, upper layer

0.1 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99 99.9

observed

modeled

IRcon, lower layer

100

70,000

200

300

400

500

700

1,000

2,000

3,000

4,000

5,000

7,000

10,000

20,000

30,000

40,000

50,000

SPECIFIC CONDUCTANCE, IN MICROSIEMENS PER CENTIMETER

0.1 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99 99.9

DAYS SPECIFIC CONDUCTANCE WAS NOT EXCEEDED, IN PERCENT

observed

modeled

Sebcon2, upper layer

Figure 16. Specific conductance duration curves for modeled and observed values--Continued.
The Probabilistic Relation of Salinity to Freshwater Discharge 31

The relation of the absolute value of residuals 

and the models applied to compute daily standard 

errors are illustrated for three sites in figure 17. The 

difference in curves for the upper layer and lower layer 

sites at Sebcon2 illustrates the overall difference in 

residuals. Most of the curves produce relatively poor 

fits. However, the goal of curve fitting in this part of the 

model development was to account as well as possible 

for the extent and trends in errors, not to produce an 

exacting model of errors. The curves fit to most of the 

upper sites underestimate the variation in residuals in 

the mid-range of discharges. This was the result of fitting the model through optimization of duration curves 

rather than using a least-squares approach. The fit of 

duration curves tended to weight the fit of the model 

equally throughout the range of discharge and was 

insensitive to the density of data throughout that range. 

However, because the curves underestimate variability 

in the mid range of discharges, model-derived probabilities of nonexceedance in this range may be underestimated. This effect should not, however, bias the 

direction of the change predicted.

The Response of Salinity Conditions to 

Changing Flow Regimes

Discharge management methods of detention 

and retention were investigated through use of the 

salinity response model (eq 9). Detention and retention 

affect timing and magnitude of discharge peaks, as well 

as days of occurrence above or below a given conductance limit. 

Predicted duration curves for the modeled estuary under two flow regimes (detention and retention) 

were developed for each conductance data-collection 

site to show the potential change in the recurrence of 

predicted specific-conductance values for the period of 

data collection (fig. 18). These duration curves were 

Table 8. Statistical parameters for the model of residuals 

as a function of discharge for St. Sebastian River basin,

May 1, 1992, to March 18, 1996

[See equation 10]

Site Multiplier

b4

Expondent

b5

Standard 

error1

1Standard error of residual model fit. Each unit is a factor of the 

probability ratio of the observed to fitted probability curves. A factor of 

1.00 indicates a perfect fit, a factor of 2.00 indicates half or twice the probability.

C54con 3.162 x 1012 -2.600 1.751

SPcon 3.200 x 1012 -3.780 1.287

Sebcon1-upper2

2

Upper layer in the vertical water column.

2.358 x 1013 -2.450 1.174

Sebcon1-lower3 2.060 x 108 -0.070 1.254

Sebcon2-upper2 2.258 x 1013 -2.515 1.261

Sebcon2-lower3

3

Lower layer in the vertical water column.

1.862 x 106 0.599 1.147

IRcon-upper2 2.291 x 106 0.599 1.172

IRcon-lower3 2.239 x 106 0.599 1.244

Average 1.286

10 10,000 100 1,000

0

40,000

10,000

20,000

30,000

Sebcon2, lower layer

10 10,000 100 1,000

40,000

0

10,000

20,000

30,000

ABSOLUTE VALUE OF MODEL RESIDUAL, IN MICROSIEMENS PER CENTIMETER

Sebcon2, upper layer

1 10,000 10 100 1,000

DISCHARGE, IN CUBIC FEET PER SECOND

40,000

0

10,000

20,000

30,000

SPcon

Figure 17. Absolute value of statistical model residuals as 

a function of discharge and residuals model regression 

curve.
32 Probability Analysis of the Relation of Salinity to Freshwater Discharge in the St. Sebastian River, Florida

computed from the probability model in the same way 

as previously described, except that the discharge time 

series were altered to reflect discharge management 

changes. In the case of detention, discharges for the 

period were smoothed over a 30-day period. The 

smoothing effectively produced the same volume of 

water for the period of record but spread the discharge 

peaks out over time. Stretching the discharges out over 

time increased the number of days flow was in the 

moderate- to low-flow discharge ranges. In the case of 

retention, peak discharges were reduced by subtracting 

all discharge above 800 ft3/s to produce a 20 percent 

reduction in total flow. This simulated the effects of 

increased retention or diversion of freshwater out of the 

basin.

Modeled probabilities based on detention show 

that the probability of specific-conductance values falling below a given limit would increase for all sites 

except Sebcon2, lower layer, which shows a decrease 

in days of occurrence by a factor of 0.75 at a nonex100

60,000

300

500

700

1,000

3,000

5,000

7,000

10,000

20,000

40,000

SPECIFIC CONDUCTANCE, IN MICROSIEMENS PER CENTIMETER

0.1 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99

DAYS SPECIFIC CONDUCTANCE WAS NOT EXCEEDED, IN PERCENT

period of record

detention

retention

C54con

0.1 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99

period of record

detention

retention

SPcon

100

60,000

300

500

700

1,000

3,000

5,000

7,000

10,000

20,000

40,000

1 2 5 10 20 30 40 50 60 70 80 90 95 98

period of record

detention

retention

Sebcon1, upper layer

1 2 5 10 20 30 40 50 60 70 80 90 95 98

period of record

detention

retention

Sebcon1, lower layer

Figure 18. Specific-conductance duration curves comparing period of record to detention and retention for conductance 

data-collection sites.
The Probabilistic Relation of Salinity to Freshwater Discharge 33

ceedance limit of 20,000 μS/cm (table 9). Probability 

of nonexceedance increased by a factor of 1.17 at 

Sebcon1, lower layer, and IRcon, lower layer, and 1.38 

at SPcon. Conductance limits for these sites were set at 

either 1,000 μS/cm or 20,000 μS/cm based on the 

ranges of daily values of specific conductance 

observed at the sites. The number of days of occurrence 

above a conductance of 20,000 μS/cm at SPcon and 

40,000 μS/cm at C54con and Sebcon1 also were computed. As a result of using 30-day smoothed data, the 

number of days the specific conductance fell below the 

specified limit increased.

Detention of freshwater discharge changes the 

distribution of values in the specific conductancedischarge relation by decreasing the number of peakflow days and increasing the number of low-flow days. 

Although detention reduces the likelihood of a low 

specific-conductance value on a given peak-flow day, it 

also increases the number of moderate-flow days by 

extending flow over a longer period of time. When 

100

70,000

300

500

700

1,000

3,000

5,000

7,000

10,000

30,000

50,000

SPECIFIC CONDUCTANCE, IN MICROSIEMENS PER CENTIMETER

0.001 0.01 0.1 1 2 5 10 20 30405060 70 80 90 95 98 99

DAYS SPECIFIC CONDUCTANCE WAS NOT EXCEEDED, IN PERCENT

period of record

detention

retention

Sebcon2, upper layer

0.001 0.01 0.1 1 2 5 10 20 30405060 70 80 90 95 98 99

period of record

detention

retention

Sebcon2, lower layer

10,000

70,000

15,000

20,000

25,000

30,000

40,000

50,000

60,000

0.1 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99

period of record

detention

retention

IRcon, upper layer

0.1 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99

period of record

detention

retention

IRcon, lower layer

Figure 18. Specific-conductance duration curves comparing period of record to detention and retention for conductance 

data-collection sites--Continued.
34 Probability Analysis of the Relation of Salinity to Freshwater Discharge in the St. Sebastian River, Florida

random factors are taken into account, moderate-flow 

days may be only marginally less likely to produce a 

low specific-conductance value than high-flow days; 

for example, reduction of a single high-flow day may 

result in 10 or more moderate-flow days. Thus, detention may actually increase the total number of days of 

exposure to low salinity conditions.

Retention of freshwater discharge from the 

system, as applied here, simulated the elimination of 

peak discharge values above 800 ft3/s, resulting in a 

20 percent reduction in freshwater inflows. By changing the volume of water that passed through the 

system, the distribution of the specific conductancedischarge relation was altered and probability of 

specific-conductance values exceeding given limits 

changed. The elimination of freshwater input to the 

system decreased the likelihood of falling below a 

selected limit of specific conductance at all sites. For 

limits of specific conductance (1,000 μS/cm or 

20,000 μS/cm, depending on the site), the predicted 

days of occurrence below a limit decreased, ranging 

from 17 percent (Sebcon2, lower layer) to 68 percent 

(SPcon and IRcon, upper layer) (table 9) of the predicted days of occurrence for unregulated flow.

Retention of volumes of freshwater from the system changes the specific conductance-discharge distribution by limiting the magnitude of peaks but not 

increasing the number of days of moderate flows. This 

dual effect of retention allows for fewer occurrences 

below a lower limit with no change in occurrences 

above a higher limit. Removal of freshwater inflow 

from peak flows decreases the number of high-flow 

discharge events, thereby decreasing the likelihood that 

specific conductances will fall below a limit. Retention 

of flow has less effect on low flows, because factors 

other than discharge control specific conductance variability so that the distribution of specific-conductance 

values is similar to natural conditions.

The primary finding to be drawn from this 

analysis is that understanding physical response of the 

system to a simple stimulus (such as discharge), as 

might be characterized by a deterministic model (or an 

empirical-response model alone), does not provide 

adequate information to assess either the character of 

the system or the response of the system to stress. The 

overall character of the system is fundamentally probabilistic in nature and should be characterized in such 

terms. Whether a given level of discharge can produce 

Table 9. Predicted days of occurrence at a given threshold of specific conductance for the period of 

record, detention, and retention conditions, May 1, 1992, to March 18, 1996

[μS/cm, microsiemens per centimeter at 25 degrees Celsius; d/yr, days per year]

Site 

Conductivity 

limit

(μS/cm)

Nonexceedence

Detention Retention

Predicted 

occurrence 

(d/yr)

Predicted 

occurrence 

(d/yr)

Occurrence 

adjustment

Predicted 

occurrence 

(d/yr)

Occurrence 

adjustment

C54con 1,000 15 20 1.30 3.9 0.25

SPcon 1,000 78 108 1.38 53 .68

Sebcon1-upper1 1,000 34 44 1.27 21 .60

Sebcon1-lower2 1,000 16 19 1.17 9.9 .60

Sebcon2-upper1 1,000 6.2 7.4 1.18 2.2 .35

Sebcon2-lower2 20,000 3.8 2.9 0.75 0.7 .17

IRcon-upper1 20,000 12 16 1.34 8.4 .68

IRcon-lower2 20,000 4.9 5.8 1.17 2.2 .46

Site 

Conductivity 

limit

(μS/cm)

Exceedence

Detention Retention

Predicted 

occurrence 

(d/yr)

Predicted 

occurrence 

(d/yr)

Occurrence 

adjustment

Predicted 

occurrence 

(d/yr)

Occurrence 

adjustment

C54con 40,000 50 46 0.92 50 1.00

SPcon 20,000 19 14 .74 19 1.00

Sebcon1-upper1 40,000 42 38 .90 42 1.00

Sebcon1-lower2 40,000 122 115 .95 122 1.00

1Upper layer in the vertical water column.

2 Lower layer in the vertical water column.
Summary and Conclusions 35

a given response in the system is not as important as 

the probability of that response on a given day and 

over a period of many days. Although the many environmental conditions and forces acting upon the system, such as wind, tide, rain, and discharge patterns, 

can be monitored over time for calibration, they cannot be anticipated in a deterministic way for hypothetical simulations of effects. By treating these factors as 

random terms in the response of the system, a degree 

of specificity is lost, but a degree of reliability is 

gained.

A deterministic model of this system based only 

on discharge (as any hypothetical simulation must be, 

because other factors are uncontrolled and can only be 

assumed) would predict that a smoothing of discharges 

produced by increased retention (and a corresponding 

reduction in peaks) should increase average salinity 

conditions in the St. Sebastian River. This follows from 

the simple relation of specific conductance to discharge 

which indicates higher specific conductance values at 

lower peak discharges. The probabilistic model produces a very different response because it addresses the 

likelihood that other uncontrolled factors will combine 

to create low-salinity events. The “s-curve” shaped 

relation of the response model demonstrated that specific conductance, and thus, salinity, can decrease by a 

power of three as discharge increases. The analysis of 

residuals showed further that, once storm discharges 

exceed a low threshold, random factors can predominate and control specific-conductance values (and 

salinity) until discharges increase sufficiently to flush 

the entire system of saltwater.

SUMMARY AND CONCLUSIONS

The St. Sebastian River lies in the southern part 

of the Indian River basin on the east coast of Florida. 

Increases in freshwater discharge due to urbanization 

and changes in land use have reduced salinity (dissolved solids) in the St. Sebastian River and consequently, salinity in the Indian River, affecting the 

commercial fishing industry. Wind, water temperature, 

tidal flux, freshwater discharge, and downstream salinity all affect salinity in the St. Sebastian River estuary, 

but freshwater discharge is the only one of these hydrologic factors which might be affected by water-management practices.

A specific-conductance monitoring network was 

established in May 1992 to provide data for calibration 

and verification of a two-dimensional deterministic 

hydrodynamic model of the St. Sebastian River. However, because of limitations of the deterministic model 

to simulate constituent diffusion due to vertical stratification of specific conductance, a statistical model was 

developed to predict specific conductance, and thus, 

salinity, from discharge. A probability analysis of salinity conditions in the St. Sebastian River estuary, taking 

into account the effects of freshwater discharge, was 

used to determine the likelihood (probability) that 

salinities will fall below a given threshold. The relation 

of specific conductance to discharge and a probability 

analysis of salinity conditions were used to determine 

the effect of different flow regimes on salinity in the St. 

Sebastian River.

Long-term prevailing winds can build up water 

levels within the estuary, but tidal changes and freshwater discharges are the primary factors affecting salinity within the system. Tidal flux, propagated by tidal 

changes, ranged from 45.3 to 878 ft3/s at the mouth of 

the St. Sebastian River during the study. Daily mean 

freshwater discharges at the mouth of the estuary 

ranged from 50 to 6,400 ft3/s (during tropical storm 

Gordon, November 1994). The annual mean freshwater discharge of the system was 295 ft3/s, and freshwater flux equivalent to the annual mean discharge would 

flush the system in 3.8 days. At minimum discharge, 

the freshwater flux would flush the system in 22.4 days 

and at maximum, the freshwater flux would flush the 

system in less than 5 hours, or 0.2 day. Based on the 

extreme vertical stratification observed in this system, 

it is estimated that, during average conditions, the 

flushing period for the upper freshwater layer would be 

considerably less than 2.5 days, whereas the flushing 

time for the lower layer (due to tidal flux) would be 

about 3 to 4 days.

Flushing rates for system volumes can be combined with freshwater discharge probabilities to obtain 

a likelihood of flushing the system at a given flux. 

Ignoring mixing and stratified flow, freshwater discharges are sufficient to flush the South Prong 

upstream from its confluence with the St. Sebastian 

River and to flush the St. Sebastian River upstream 

from its confluence with the Indian River in a single 

day up to 9 percent of the time. Freshwater discharges 

alone are sufficient to flush Canal 54 upstream from 

site C54con in 24 hours up to 19 percent of the time.

Specific-conductance values at monitored sites 

ranged from near seawater values to less than 

500 μS/cm and exhibit long-term serial correlation 

attributed to the gradual mixing of water from day to 
36 Probability Analysis of the Relation of Salinity to Freshwater Discharge in the St. Sebastian River, Florida

day. Overall, the difference in daily mean specific-conductance values from lower to upper layers increases 

with distance downstream in the St. Sebastian River 

and decreases in the Indian River, indicating greater 

layering up to the mouth of the St. Sebastian River than 

in the receiving water body, the Indian River.

At sites where freshwater is a large part of the 

total volume, small increases in freshwater inflow 

cause relatively large changes in specific conductance. 

These sites have a family of duration curves for 

selected discharge ranges that are well spaced and 

evenly separated at the lower ranges of specific conductance, indicating a more even distribution of specific-conductance values throughout the range of flows 

at the selected site. At these sites, small changes in 

freshwater discharge have a greater effect on specific 

conductance than at sites where freshwater discharge is 

a smaller percentage of the total flow. As discharge gradients become more prevalent in the middle and near 

the mouth of the estuary, specific conductance values 

are less readily affected by small changes in freshwater 

discharge and the gradient between freshwater and saltwater is overcome only by much larger inflows of 

freshwater. Specific-conductance distributions for the 

lower layers at sites closer to the Indian River are more 

highly affected by saltwater flux and less influenced by 

freshwater discharges.

Estimating the effects of changing flows on 

salinity requires a predictive model that integrates both 

systematic and random components of salinity variation. The best salinity-response model fit at all sites 

was for data representing periods of highest flows, 

when discharge has the greatest effect on specific conductance. The standard error of the best-fit models for 

all sites ranged from 4,300 to 12,400 μS/cm.

The residuals from the best-fit relation of the 

salinity-response model were modeled as a function of 

freshwater discharge to produce estimates of the uncertainty in daily mean specific-conductance values. 

Duration curves based on simulated values generally fit 

empirical curves by a standard error factor of 1.28, and 

the standard error of the residuals model ranged from 

4,400 μS/cm at SPcon to 12,000 μS/cm at Sebcon1-

upper. The application of the model to changes in freshwater inflow to the system is considered reasonable, 

because modeled estimates of effects of detention and 

retention were based on interpolations within the time 

and discharge domain of the original calibration data 

set.

Discharge management methods of detention 

and retention were investigated through use of the 

salinity-response model. Detention of freshwater discharges increased the probability of specific conductance values falling below a given limit (20,000 μS/cm) 

for all sites but one. The retention of freshwater input 

to the system decreased the likelihood of falling below 

a selected limit of specific conductance at all sites. For 

limits of specific conductance (1,000 μS/cm or 

20,000 μS/cm, depending on the site) the predicted 

days of occurrence below a limit decreased ranging 

from 17 percent (Sebcon2-lower) to 68 percent (SPcon 

and IRcon-upper) of the predicted days of occurrence 

for unregulated flow.

The primary finding to be drawn from the discharge-salinity analysis is that an empirical-response 

model alone does not provide adequate information to 

assess the response of the system to changes in flow 

regime. Whether a given level of discharge can produce 

a given response on a given day is not as important as 

the probability of that response on a given day and over 

a period of many days. A deterministic model of the St. 

Sebastian River estuary based only on discharge would 

predict that retention of discharge peaks should 

increase the average salinity conditions in the St. 

Sebastian River estuary. The probabilistic model produces a very different response indicating that salinity 

can decrease by a power of three as discharges 

increase, and that random factors can predominate and 

control salinity until discharges increase sufficiently to 

flush the entire system of saltwater.
Selected References 37

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No. 28, 104 p.

Crain, L.J., Hughes, G.H., and Snell, L.J., 1975, Water 

resources of Indian River County, Florida: Florida 

Bureau of Geology Report of Investigations No. 80, 

75 p.

Doering, P.H., 1996, Temporal variability of water quality in 

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Bulletin, American Water Resources Association, v. 32, 

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Indian River Lagoon National Estuary Program, 1993, The 

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Kim, S.H., 1992, Statistics and decisions, an introduction to 

foundations: New York, Van Nostrand Reinhold, 336 p.

Knowles, Leel, Jr., 1995, Rainfall and freshwater discharge 

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and discharge; and Volume 2. Computation of Discharge: U.S. Geological Survey Water-Supply Paper 

2175, 631 p.

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U.S. Geological Survey Techniques of Water Resources 

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———1968b, Frequency curves: U.S. Geological Survey 

Techniques of Water Resources Investigations, book 4, 

chap. A2, 15 p.

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book 4, chap. B1, 18 p.

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Smith, N.P., 1990, Wind domination of residual tidal transport in a coastal lagoon: Residual currents and longterm transport, in Coastal and estuarine studies: v. 38, 

p. 123-133.

Steward, J.S., and VanArman, J.A., 1987, Indian River 

Lagoon joint reconnaissance report: St. Johns Water 

Management District and South Florida Water Management District, Final Report, 301 p.
38 Probability Analysis of the Relation of Salinity to Freshwater Discharge in the St. Sebastian River, Florida