2.3 Freshwater influx indices

Sixty-one days is the average freshwater residence time of the four major rivers flowing into PS [30–32], accounting for the temporal lag between the upriver

gauging of freshwater and the delivery of that water to S. Therefore, we defined the long-term metric 2moFWI\_rt for river r and time period t where r ¼ 1, … , 4, and t ¼ 1, … , T ¼ 40 as the average daily discharge rate in the 61 days prior to mt, the first day of the survey in time period t. Because Ramus et al. [33] calculated a seven-day residence time for the Neuse and Pamlico Rivers after Hurricanes Dennis and Floyd deposited 1 m of rainfall in eastern NC less than 2 weeks before the September 1999 survey, we defined the short-term metric 1wkFWI\_rt, by averaging daily discharge rates in the 7 days prior to mt .

Since freshwater from river r in time period t should have more of an effect on salit the closer site i is to the river, a unique measure of the influence of 1wkFWI\_rt and 2moFWI\_rt for each site was formed by dividing each by dist\_rit, r ¼ 1, … , 4, the distance separating the gauge on river r from site i within time period t:

$$1\,\text{wokFWM\\_}r\_{it} = \frac{1\,\text{wokFWM\\_}r\_{t}}{dist\\_r\_{it}}, \text{ and } 2\,\text{noFWM\\_}r\_{it} = \frac{2\,\text{noFMI\\_}r\_{t}}{dist\\_r\_{it}} \tag{1}$$

The coordinates of each gauge station were used to calculate distance because the gauge was the location of the 1wkFWI\_rt and 2moFWI\_rt observations. Like all distances in this study, dist\_rit represents distance "as the crow flies" as opposed to water-path distance. Though the superiority of using water-path distance when modeling water-quality variables in stream and estuarine systems seems intuitive, results from studies that compare these two distance metrics are inconclusive. For example, Gardner et al. [34] found more accurate predictions of stream temperatures when models incorporated water-path distance, but only when this distance was modified and weighted by stream order. Peterson and Urquhart [35] predicted various nutrient concentrations in 17 Maryland rivers and concluded that using water-path distance works well when modeling certain nutrients, but not others, and that the crow-flies distance appeared to be the most suitable distance measure overall. Comparing the accuracy of predictions of water quality parameters generated from two different multiple linear regression models containing the explanatory variable "distance to inlet mouth", Little et al. [36] found that predictions from models using water-path distance were no more accurate than those from models using crow-flies distance. None of these studies demonstrated marked predictive improvement using water-path distance, therefore we used crow-flies distance from each of the four river gauges to each of 2100 sample stations and over 6000 prediction locations.

(A fortieth indicator variable was not used because it would create a non-fullrank design matrix, and the effect for the fortieth time period can be derived using the intercept.) This latter consideration led to the creation of two distinct mean function models: the process and time models. The first has process variables only, and the second has process variables in addition to the time-period indicator variables to address the possibility that salinity is affected by some aspect of physical

<sup>1</sup> km<sup>1</sup> from 1987 to 2006. Groups of values within the same time period exhibit

Although salinity on the inner-continental shelf of the U.S. Southeast Atlantic coast exhibits some spatial variability near PS [37], we follow Xie et al. [38] and assume constant open ocean salinity. This assumption allows for modeling the effect of ocean water mixing as a function of only the distance to inlet, as opposed to distance interacting with the salinity of the ocean water, from each spatial location in the sound to each of the major PS inlets: Oregon, Hatteras, and Ocracoke. Exploratory analyses reveal that models using a single variable (distance to the nearest inlet) rather than three variables (distances to each of the three inlets), explains the same amount of variability in salinity when other explanatory variables are also included. Therefore, we consider for inclusion in subsequent models the variable closest\_inlet\_distit, defined to be the distance separating site i, sampled in

phenomena that is not accounted for by any other variable in the model.

Observed bottom salinity (psu) vs. the Roanoke River two-month relative freshwater influx index

Process-Based Statistical Models Predict Dynamic Estuarine Salinity

DOI: http://dx.doi.org/10.5772/intechopen.89911

time period t, from the center of the most proximate inlet.

2.4 Saltwater mixing and tidal signal

Figure 2.

165

(2moFWII\_rit) in m3 s

relationships with different slopes.

The plot in Figure 2 of salit against Roanoke River 2moFWII\_rit typifies the relationships between salinity and each of the eight 1wkFWII\_rit and 2moFWII\_rit variables. Larger values of the metric are associated with smaller values of salinity, but groups of observations have different slopes. Closer examination revealed that the different groups corresponded to different time periods. We attempted to account for the different slopes in two ways, first by considering the 28 pair-wise interactions among the 1wkFWII\_rit and2moFWII\_rit and second by considering 39 time-period indicator variables defined as

$$\begin{aligned} \text{timer\\_1}\_{t} &= \begin{cases} 1 & \text{if } t = 1 \\ 0 & \text{otherwise} \end{cases}, \dots, \text{timeout\\_\tau}\_{t} = \begin{cases} 1 & \text{if } t = \tau \\ 0 & \text{otherwise} \end{cases}, \dots, \text{timeout\\_3\mathfrak{g}\_t}\_{t} \\ &= \begin{cases} 1 & \text{if } t = 3\mathfrak{g} \\ 0 & \text{otherwise} \end{cases} \end{aligned}$$

Figure 2.

gauging of freshwater and the delivery of that water to S. Therefore, we defined the long-term metric 2moFWI\_rt for river r and time period t where r ¼ 1, … , 4, and t ¼ 1, … , T ¼ 40 as the average daily discharge rate in the 61 days prior to mt, the first day of the survey in time period t. Because Ramus et al. [33] calculated a seven-day residence time for the Neuse and Pamlico Rivers after Hurricanes Dennis and Floyd deposited 1 m of rainfall in eastern NC less than 2 weeks before the September 1999 survey, we defined the short-term metric 1wkFWI\_rt, by averaging daily discharge

Since freshwater from river r in time period t should have more of an effect

The coordinates of each gauge station were used to calculate distance because the gauge was the location of the 1wkFWI\_rt and 2moFWI\_rt observations. Like all distances in this study, dist\_rit represents distance "as the crow flies" as opposed to water-path distance. Though the superiority of using water-path distance when modeling water-quality variables in stream and estuarine systems seems intuitive, results from studies that compare these two distance metrics are inconclusive. For example, Gardner et al. [34] found more accurate predictions of stream temperatures when models incorporated water-path distance, but only when this distance was modified and weighted by stream order. Peterson and Urquhart [35] predicted various nutrient concentrations in 17 Maryland rivers and concluded that using water-path distance works well when modeling certain nutrients, but not others, and that the crow-flies distance appeared to be the most suitable distance measure overall. Comparing the accuracy of predictions of water quality parameters generated from two different multiple linear regression models containing the explanatory variable "distance to inlet mouth", Little et al. [36] found that predictions from models using water-path distance were no more accurate than those from models using crow-flies distance. None of these studies demonstrated marked predictive improvement using water-path distance, therefore we used crow-flies distance from each of the four river gauges to each of 2100 sample stations and over 6000

The plot in Figure 2 of salit against Roanoke River 2moFWII\_rit typifies the relationships between salinity and each of the eight 1wkFWII\_rit and 2moFWII\_rit variables. Larger values of the metric are associated with smaller values of salinity, but groups of observations have different slopes. Closer examination revealed that the different groups corresponded to different time periods. We attempted to account for the different slopes in two ways, first by considering the 28 pair-wise interactions among the 1wkFWII\_rit and2moFWII\_rit and second by considering

, … , timeper\_τ<sup>t</sup> <sup>¼</sup> 1 if <sup>t</sup> <sup>¼</sup> <sup>τ</sup>

(

0 otherwise

, … , timeper\_39<sup>t</sup>

, and 2moFWII\_rit <sup>¼</sup> <sup>2</sup>moFWI\_rt

dist\_rit

(1)

on salit the closer site i is to the river, a unique measure of the influence of 1wkFWI\_rt and 2moFWI\_rt for each site was formed by dividing each by dist\_rit, r ¼ 1, … , 4, the distance separating the gauge on river r from site i within

dist\_rit

rates in the 7 days prior to mt .

<sup>1</sup>wkFWII\_rit <sup>¼</sup> <sup>1</sup>wkFWI\_rt

Lagoon Environments Around the World - A Scientific Perspective

time period t:

prediction locations.

39 time-period indicator variables defined as

0 otherwise

<sup>¼</sup> 1 if <sup>t</sup> <sup>¼</sup> <sup>39</sup> 0 otherwise

timeper\_1<sup>t</sup> <sup>¼</sup> 1 if <sup>t</sup> <sup>¼</sup> <sup>1</sup>

(

(

164

Observed bottom salinity (psu) vs. the Roanoke River two-month relative freshwater influx index (2moFWII\_rit) in m3 s <sup>1</sup> km<sup>1</sup> from 1987 to 2006. Groups of values within the same time period exhibit relationships with different slopes.

(A fortieth indicator variable was not used because it would create a non-fullrank design matrix, and the effect for the fortieth time period can be derived using the intercept.) This latter consideration led to the creation of two distinct mean function models: the process and time models. The first has process variables only, and the second has process variables in addition to the time-period indicator variables to address the possibility that salinity is affected by some aspect of physical phenomena that is not accounted for by any other variable in the model.
