Developing Numeric Nutrient Criteria for Lakes and

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Paul, M., A. Herlihy, D. Bressler, L. Zheng and A. Roseberry-Lincoln (2014). Methodologies ..... Analysis of the network topology was accomplished using the ...... modeling, and scientific literature) in developing nutrient criteria. ..... 5 http://www.adem.state.al.us/alEnviroRegLaws/files/Division6Vol1.pdf (Accessed June 2013).
EPA/600/R-15/262

Science Supporting Numeric Nutrient Criteria for Lakes and Their Watersheds: A Synopsis of Research Completed for the US Environmental Protection Agency

EPA/600/R-15/262

Science Supporting Numeric Nutrient Criteria for Lakes and Their Watersheds: A Synopsis of Research Completed for the US Environmental Protection Agency

James D. Hagy III Research Ecologist U.S. Environmental Protection Agency Office of Research and Development National Health and Environmental Effects Research Laboratory, Gulf Ecology Division

U.S. Environmental Protection Agency Office of Research and Development National Health and Environmental Effects Research Laboratory, Gulf Ecology Division 1 Sabine Island Drive, Gulf Breeze, FL 32561

Disclaimer This document has been reviewed in accordance with U.S. Environmental Protection Agency policy and approved for publication. The views expressed are those of the author. The results presented are a synopsis of research completed by Tetra Tech, Inc, under Task Order 0005 under Contract EP-C-11-037 and Work Assignment 02-12 under Contract EP-C-12-060. This report is intended to be distributed with the Final Reports resulting from those research projects. The U.S. EPA made comments and suggestions on the Final Reports intended to improve the scientific analysis and technical accuracy of the documents. However the author did not contribute otherwise to the production of the cited Final Reports. The Final Reports and views expressed are those of Tetra Tech, Inc. or its employees.

Citation: Hagy, J. D. III. 2015. Developing Numeric Nutrient Criteria for Lakes and Their Watersheds: A Synopsis of Research Completed for the US Environmental Protection Agency. US Environmental Protection Agency, Office of Research and Development, National Health and Environmental Effects Laboratory, Research Triangle Park, NC. EPA/600/R-15/262. To be distributed with: Paul, M., J. Butcher, D. Allen, L. Zheng, and T. Zi (2015). Methods for Computing Downstrewam Use Protection Criteria for Lakes and Reservoirs. Prepared for US Environmental Protection Agency by Tetra Tech, Inc, Research Triangle Park, NC and Tetra Tech, Inc, Center for Ecological Sciences, Research Triangle Park, NC.: 144 pp. Paul, M., A. Herlihy, D. Bressler, L. Zheng and A. Roseberry-Lincoln (2014). Methodologies for Development of Numeric Nutrient Criteria for Freshwaters. Prepared by Tetra Tech, Inc., Research Triangle Park, NC and Tetra Tech, Inc, Center for Ecological Science, Research Triangle Park, NC.: 345 pp. (includes Appendices 1-25)

Cover Photo Interfalls Lake as viewed from Pattison State Park, Wisconsin. Pattison State Park is located on the Black River and contains Big Manitou Falls, the highest waterfall in Wisconsin at 165 feet. Photo by Jessica Aukamp.

List of Figures Figure 1. Relationship between Log Average total nitrogen (TN) in lakes from the North Temperate Lakes Long-Term Ecosystem Research (LTER) site and a 6-taxa metric based on fish species with high nutrient tolerance values. Reprinted from Figure 15 in Paul, Herlihy et al. (2014). Figure 2. A frequency histogram and cumulative distribution function for total phosphorus concentrations in the French Broad River, TN computed on the base of an inverted LOADEST model adjusted to ensure attainment of the downstream TP loading target. Across the long term distribution of flow levels, this TP distribution results in attainment of a flow-weighted average concentration of 0.39 mg/L at discharge to Douglas Reservoir. Reprinted from Paul, Butcher et al. (2015).

Abstract Nutrient pollution remains one of the most prevalent causes of water quality impairment in the United States. The U.S. Environmental Protection Agency’s (EPA) approach to addressing the challenge of managing nutrient pollution has included supporting development of numeric nutrient criteria for the Nation’s waters. To create scientific information that could assist the Agency and its partners work toward this goal, EPA’s Office of Research and Development funded a two-year extramural research project focused on criteria development for lakes and reservoirs, and in particular the challenge of relating criteria for streams to protection of downstream lakes and reservoirs. Research focused on two areas of the US with abundant water resources, namely the upper Midwest and the Southeast. Study areas in Wisconsin and Tennessee were selected based on the availability of extensive long-term data sets quantifying both water quality and aquatic life variables. The resulting research was documented in considerable detail in two final project reports (Paul, M., A. Herlihy, D. Bressler, L. Zheng and A. Roseberry-Lincoln (2014). Methodologies for Development of Numeric Nutrient Criteria for Freshwaters. Prepared by Tetra Tech, Inc., Research Triangle Park, NC and Tetra Tech, Inc, Center for Ecological Science, Research Triangle Park, NC.: 345 pp.; Paul, M., J. Butcher, D. Allen, L. Zheng, and T. Zi (2015). Methods for Computing Downstrewam Use Protection Criteria for Lakes and Reservoirs. Prepared for US Environmental Protection Agency by Tetra Tech, Inc, Research Triangle Park, NC and Tetra Tech, Inc, Center for Ecological Sciences, Research Triangle Park, NC: 144 pp) This report provides an accessible overview of the research with interpretation of its possible significance from a scientific perspective. The research illustrates the fact that relating aquatic life condition to nutrients and water quality is challenging but tractable and that the application of optima and tolerance models is a useful approach. Exploration of theoretical considerations related to developing downstream use protection criteria identified several separate, yet related challenges. Novel methods that were developed and applied illustrated that empirical approaches of intermediate complexity may offer a viable way to improve explicit consideration of downstream use protection in water quality management. These methods are, nonetheless challenging to understand. Further development, application and explanation will be needed to make widespread application a greater possibility.

Introduction Excess loading of nitrogen (N) and phosphorus (P) is one of the most prevalent causes of water quality impairment in the United States, affecting nearly 7,000 surface water bodies for nutrients and nearly as many for organic enrichment or oxygen depletion (2010 CWA Sec. 303(d) List). Excess N and P in aquatic systems comes from many point and nonpoint sources, including urban and suburban storm-water runoff, municipal and industrial waste water discharges, fertilizer use, livestock production, atmospheric deposition, and legacy groundwater nutrient pollution. Land use alterations in watersheds across the U.S. increase the delivery of N and P applied to the landscape into surface and groundwater, impacting aquatic life uses, human health and economic prosperity (Compton, Harrison et al. 2011). Because of the complexity of the problem, the large number and diversity of stakeholders, and the inherent integration with 1

social and economic systems, many recognize that nutrient pollution is a “wicked” problem, and that solutions will not necessarily be simple or permanent. Such problems are “at best re-solved - over and over again” (Rittel and Webber 1973). One way to advance the prospects for long-term success is to identify quantifiable environmental goals or targets. These could include numeric nutrient criteria (NNC), loading limits, biocriteria, or other objectives based on environmental outcomes (e.g., recovery of seagrass habitat, fish populations, etc.). Clear goals help define the current status relative to the goal, and inform efforts needed to restore and protect the environment, even when policy allows an extended period of time to fully attain the goal (e.g., Montana Department of Environmental Quality 2014). Absent clear, quantifiable goals that could serve as a fixed reference point, the “shifting baselines syndrome” (Pauly and Christensen 1995, Papworth, Rist et al. 2009) can erode public awareness of degraded environmental conditions. As one approach to addressing these challenges in the context of the nutrient pollution, the US Environmental Protection Agency (EPA) in 1998 called for “accelerating development of scientific information concerning the levels of nutrients that cause water quality problems”, and working with states and tribes to adopt nutrient criteria as part of enforceable state water quality standards (Environmental Protection Agency 1998). EPA (2011) reaffirmed the Agency’s focus on partnering with states to develop nutrient criteria and reduce nutrient loading. Development of NNC for freshwaters continues to present technical and policy challenges despite the fact that freshwater ecology has developed over a significant period of time and has addressed the effects of nutrient pollution in a variety of ways (Environmental Protection Agency 2014). A major complicating factor is that many stressors that co-occur with nutrient pollution also impact biotic condition. For example, conversion of natural land uses to agriculture or developed land uses usually increases nutrient concentrations in streams (Beaulac and Reckhow 1982), but can also change hydrology, stream channel morphology, temperature regimes, and sediment loading, among other effects that can impact biotic condition. Whereas a decline in biotic condition may be readily observed in relation to developed landscapes, condition may not relate strongly to nutrient concentrations specifically. Karr (1981) noted the difficulty of characterizing stream condition via water quality proxies such as nutrients and chlorophyll-a, and instead proposed biotic indices as a more direct approach to condition assessment. While this approach is viable, it leaves open the question of causes and possible solutions. A lack of relationship between a stream condition index (i.e., a biotic index) and nutrient concentrations in Florida streams led EPA and the State to develop NNC via a reference stream approach, rather than via the explicit linkage between nutrients and biotic condition preferred by stakeholders (Environmental Protection Agency 2010a). Another technical challenge associated with development of NNC is protection of downstream waters. The strategy EPA outlined in 1998 recognized that “the finally developed criteria must limit not only the unacceptable enrichment of a given water body or watercourse, but also must factor in the effects of that enrichment on downstream receiving waters” (Environmental Protection Agency 1998). However, the technical guidance and recommended criteria that EPA published during the next several years (e.g., Environmental Protection Agency 2000) utilized a reference approach that does not explicitly consider protection of downstream waters. Downstream use protection received much greater attention with EPA’s 2010 proposed 2

nutrient criteria for the State of Florida, which explicitly addressed stream water quality required to protect downstream lakes and estuaries (Environmental Protection Agency 2010b). Development of these criteria proved both complex and controversial, as the proposed total nitrogen (TN) concentration limits required to protect downstream estuaries were often more stringent than required to protect the streams. By later in 2010 when EPA’s proposed rule was finalized, the lake DPVs had been revised in response to public comment and the proposed DPVs for estuaries were removed and slated for consideration in a later rulemaking (Environmental Protection Agency 2010c). When the State of Florida ultimately adopted NNC, the State noted that limits for nutrients in streams to protect downstream estuaries was to be addressed via the narrative nutrient criterion, but provided no indication of how this was to be done (Florida Department of Environmental Protection 2013). These developments suggest that additional scientific research addressing key areas of uncertainty are needed to encourage implementation of NNC that explicitly address downstream protection. Given the ongoing need for science to support development of NNC in freshwaters, and in particular science to support development of numeric criteria to protect downstream uses, EPA’s Office of Research and Development funded an extramural research project which was conducted in two phases by researchers at Tetra Tech, Inc. The overarching research concept was that nutrient thresholds for streams and could be explored principally by considering protection of downstream uses, particularly lakes and reservoirs. Moreover, with possibly fewer mitigating factors, quantifying nutrient limits for protection of nutrient-sensitive aquatic life in lakes and reservoirs may be more tractable than for streams. Phase I, which concluded in 2014, focused on identifying nutrient-sensitive aquatic life uses in freshwater lakes and reservoirs, specifically lakes in the upper Midwest and reservoirs in the Southeast US. Phase II, which was conducted during 2014 and 2015 examined how NNC could be developed to support attainment of the identified nutrient-sensitive aquatic life uses within the same regional lakes and reservoirs. Phase II went on to explore how, given NNC for a lake or reservoir, one could approach development of numeric criteria for the associated network of streams in the contributing watershed that would ensure attainment of the downstream NNC. These criteria have been referred to as “downstream protection values” or “DPVs.” Phase II aimed to develop solutions that were intermediate in complexity and data requirements, such that they might be applied broadly to support nutrient management. This report examines the results of these extramural research projects, which are presented in considerable detail in a Final Report for each phase (Paul, Herlihy et al. 2014, Paul, Butcher et al. 2015) and associated supplemental information. These Final Reports are intended to be distributed as attachments to this report. Key findings are evaluated to identify potential applications in nutrient management as well as questions in need of further investigation.

Research Approaches, Data and Study Sites Study sites within the broad target regions were identified based on both the availability of suitable data. For the upper Midwest region, the study used a data set from the North Temperate Lakes Long-Term Ecosystem Research (LTER) Project, a National Science Foundation funded research project focused on the ecology of lakes and sustained since 1981. Data selected for use in the study include water quality and nutrient concentrations, phytoplankton community composition, macrobenthic invertebrates, zooplankton, and fish 3

communities. Depending on the dataset, the period covered is a subset of the overall period of record, which is 1981 through 2013. For the Southeast region, data collected by the Tennessee Valley Authority covered as many as 32 lakes and included water quality data collected from 1960 to 2006 and fish community data collected from 1993 to 2012. Research addressing on quantifying DPVs focused on two watersheds, one in each of the target regions. For the upper Midwest, the study focused on Holcombe Flowage, Wisconsin, located on the Chippewa River. Although the intent for the upper Midwest region was to study natural lakes, this reservoir site was considered a good site for developing analysis for downstream protection because of the availability of data in both the target waterbody and its watershed. For the Southeast, analysis was focused on Douglas Reservoir, located in eastern Tennessee on the French Broad River. During research Phase I, relationships between water quality variables or abundances of biota and nutrients were explored using several variations on indicator value approaches characterizing either optimal nutrient concentrations, nutrient tolerance limits, or both (Yuan 2006). In Phase II, relationships among biotic and water quality variables were explored using scatterplots, locally-weighted scatterplot smoothing (LOESS) regressions, and change-point analysis, the objective being to identify water quality conditions at which nutrient-sensitive biotic endpoints may be impaired (i.e., such as identified in Phase I). Given nutrient targets for Holcombe Flowage and Douglas Reservoir (either hypothetical or based on state water quality standards), research in Phase II explored methods that could be used to calculate DPVs. The analysis examined key theoretical concerns associated with development of DPV criteria, an important task that has not received sufficient attention, and also reviewed prior work in the State of Florida that relates to calculation of DPVs. Phase II sought specifically to identify approaches of intermediate complexity; that is, methods that lie between the simplest possible approaches, which may result in erroneous or perhaps unnecessarily restrictive criteria, and detailed mechanistic modeling of both watershed and receiving water. The latter may be sufficient, but generally comes at a high cost and may not actually offer the expected increase in accuracy that might be assumed on the basis of complexity (Reckhow 1994). Several alternative approaches were considered. One approach involved empirical analysis of data from a stream network using statistical methods designed to account for spatial correlations among monitoring sites distributed in a stream network. Analysis of the network topology was accomplished using the Spatial Tools for the Analysis of River Systems (STARS) and the Functional Linkage of Waterbasins and Streams (FLoWS) toolsets, which generate data objects that can be analyzed using spatial linear models using the SSN package in R (Ver Hoef, Peterson et al. 2014). The resulting estimates of the spatial structure of water quality in the stream network were used to generate complete realizations of water quality data in the stream network. These were then evaluated to quantify how sample data similar to a real assessment could be interpreted relative to the downstream or “pour point” requirement associated with attainment of water quality and aquatic life goals in the receiving water lake.

Overview of Results Phase I Research in Phase I included an evaluation of biotic metrics that have been used in nutrient management by states in the Midwest and Southeast. A number of states were found to have collected data on aspects of biotic condition in their lakes and reservoirs and several were 4

developing new indicators and methods involving macrophytes and fish metrics. Regardless, most states continue to use water quality measures as indirect indicators of aquatic life use attainment, rather than direct measures of biotic condition. Florida was the only state found to use biotic measures directly in regulatory assessments. A number of states had noteworthy approaches, however, which are examined state by state in the Final Reports. For the upper Midwest region, nutrient optima and tolerance analysis showed that inference models performed best (in decreasing order) for fish, zooplankton, benthic macroinvertebrates, and phytoplankton. This result was counterintuitive, as one might expect strongest inference among ecosystem components where causation was most direct (e.g., nutrients to phytoplankton). The generality of these observations is unclear, as it its dependence on unique aspects of the datasets involved. For example, the investigators noted that macroinvertebrate inference models may have been poor because the gradient in nutrient concentrations was relatively small within the dataset that included macroinvertebrate data. The relatively weak relationships between phytoplankton species and nutrient data could also reflect a mismatch between observation scale and scales of variability for these potentially very dynamic ecosystem components. Because the strongest results were obtained for fish optima models, use of multispecies metrics considering either nutrient-tolerant or nutrient-sensitive fish species could be worthy of further consideration (e.g., Fig 1). Based on the analysis of upper Midwest vs. Southeast region lakes and reservoirs, it is clear that regional models are needed to determine regionally applicable thresholds based on fish metrics. An additional benefit of an approach based on fish-indicators is that there is a relatively direct conceptual linkage between fish abundance and most definitions of aquatic life use attainment. Phase II Phase II research explored linkages between nutrient inputs to Midwest and Southeast region lakes and reservoirs and support for nutrient-sensitive aquatic life uses such as those identified in Phase I. The objective was to establish useful new approaches that could be applied or adapted by those tasked with determining criteria, rather than to develop or recommend criteria values. Thus, applications were developed even if a state had already adopted regulatory criteria based on another approach. Where applicable, any state-adopted values were also noted and considered in later analyses. This was the case for lakes in the State of Wisconsin (Wisconsin Administrative Code, NR 102.06, accessed via http://docs.legis.wisconsin.gov/code/). In the conceptual model applied, an important effects pathway was identified in which nutrients were related to phytoplankton chlorophyll-a and planktonic chlorophyll-a was related to hypolimnetic dissolved oxygen (DO). Neither chlorophyll-a nor hypolimnetic DO were closely related to most biotic measures in the lakes, however, and none provided a relationship sufficient to quantify a target nutrient concentration. As a result, nutrient thresholds were derived based on existing DO thresholds (e.g., 2.0 mg/L and 5.0 mg/L, based on Wisconsin statute) and relationships between nutrients, chlorophyll-a, and DO. Specifically, the proportion of DO observations greater than 2 and 5 mg/L decreased in relation to growing season average chlorophyll-a (p14 µg/L and, via a nutrient-chlorophyll-a regression relationships, TP 0.03, Adjusted

513188

Marsh Creek at Little Rapids Road

0.081

100%

82%

513189

Wolf Creek at Wolf Creek Road

0.061

100%

55%

553003

Chippewa River at USH8 Bridge

0.040

71%

15%

553042

Jump River at CTH C, at Sheldon, WI

0.053

80%

40%

553063

Thornapple River - CTH A NW Ladysmith

0.053

83%

58%

553097

Devils Creek - Low Site at Hwy 40 Brg

0.066

100%

50%

553126

Main Creek at Broken Arrow Road

0.081

100%

58%

553131

Deer Tail Creek at Broken Arrow Rd

0.114

100%

100%

553137

McDermott Creek at CTH F, near Weyerhauser, WI

0.160

100%

100%

553138

McDermott Creek at Horseshoe Lk Rd

0.169

100%

100%

553149

Flambeau River at USGS Station, Hwy E

0.042

85%

8%

553156

Flambeau River - Near Bruce, WI

0.044

62%

8%

553158

Little Soft Maple Creek at Kief Rd, near Weyerhauser

0.064

96%

46%

553167

Jump River - CTH G Bridge about 4 Miles Downstream Of Sheldo

0.053

85%

46%

553169

Deer Tail Creek at CTH B

0.065

100%

64%

553170

Main Creek, North Fork at Cutoff Road

0.143

100%

90%

553171

Bear Creek at STH 73

0.055

100%

50%

553172

Alder Creek at STH 73

0.107

100%

90%

553173

Unnamed Creek at STH 73

0.089

100%

100%

613199

Levitt Creek at CTH D

0.077

100%

100%

10029123

MEADOW BROOK at STH 27

0.116

100%

100%

10029539

Big Weirgor Creek--downstream of Short Cut Road

0.044

58%

25%

10030672

Jump River at Highway 73

0.072

100%

75%

10030673

South Fork Jump River along CTH I

0.080

100%

92%

10031838

Mud Creek at CTH D

0.151

100%

100%

10031884

Chippewa River at boat landing near CTH/H and STH 40

0.046

92%

25%

Note: The “Adjusted” column shows results obtained with all observations adjusted by the ratio needed to meet the target concentration at the downstream pour point. Station locations are shown in Figure 3-12

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The median CV across all stations is 0.047. Under assumptions of a lognormal distribution, the corresponding UCL on the downstream criterion is 0.056 mg/L. The average observed concentration at some, but not all stations in the watershed exceed this level, and 53 percent of individual observations are greater than 0.056 mg/L. Even after the adjustment to guarantee that all three of the downstream stations achieve a sample average of 0.030 mg/L, rates of excursion of up to 100 percent of samples occur, and only two stations have no excursions (Table 3-2). This demonstrates that frequent observations in the watershed in excess of the downstream criterion do not necessarily indicate failure to meet that downstream criterion.

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April 2015

Table 3-2. Comparison of Total Phosphorus Data to UCL on Downstream Criterion, Holcombe Flowage Watershed Station

Station Name

% >0.056, Current

% > 0.056, Adjusted

513188

Marsh Creek at Little Rapids Road

82%

36%

513189

Wolf Creek at Wolf Creek Road

55%

9%

553003

Chippewa River at USH8 Bridge

10%

5%

553042

Jump River at CTH C, at Sheldon, WI

40%

10%

553063

Thornapple River - CTH A NW Ladysmith

50%

0%

553097

Devils Creek - Low Site at Hwy 40 Brg

50%

21%

553126

Main Creek at Broken Arrow Road

58%

25%

553131

Deer Tail Creek at Broken Arrow Rd

100%

25%

553137

McDermott Creek at CTH F, near Weyerhauser, WI

100%

100%

553138

McDermott Creek at Horseshoe Lk Rd

100%

85%

553149

Flambeau River at USGS Station, Hwy E

8%

0%

553156

Flambeau River - Near Bruce, WI

8%

8%

553158

Little Soft Maple Creek at Kief Rd, near Weyerhauser

46%

13%

553167

Jump River - CTH G Bridge about 4 Miles Downstream Of Sheldo

46%

8%

553169

Deer Tail Creek at CTH B

64%

9%

553170

Main Creek, North Fork at Cutoff Road

90%

50%

553171

Bear Creek at STH 73

30%

0%

553172

Alder Creek at STH 73

90%

30%

553173

Unnamed Creek at STH 73

80%

40%

613199

Levitt Creek at CTH D

100%

9%

10029123

MEADOW BROOK at STH 27

100%

50%

10029539

Big Weirgor Creek--downstream of Short Cut Road

25%

8%

10030672

Jump River at Highway 73

67%

17%

10030673

South Fork Jump River along CTH I

75%

17%

10031838

Mud Creek at CTH D

100%

73%

10031884

Chippewa River at boat landing near CTH/H and STH 40

92%

25%

Note: The “Adjusted” column shows results obtained with all observations adjusted by the ratio needed to meet the target concentration at the downstream pour point. Station locations are shown in Figure 3-12

84

Downstream Use Protection

April 2015

3.5.2 Simplified Assignment of Load, Holcombe Flowage The significance of some excursions of the downstream criterion within the watershed network may be mitigated by transit losses. The regional SPARROW model of Robinson and Saad (2011), developed for the Upper Midwest, including the drainages to the Great Lakes, the Upper Mississippi River Basin, and the Ohio River Basin, estimates exponential attenuation coefficients for TP (day-1) as a function of travel time. For average flows less than 1.416 m3/s the calibrated attenuation coefficient is 0.198 day-1, while for flows between 1.416 and 2.265 m3/s the attenuation coefficient is 0.298 day-1. No attenuation is assigned when average flows are above 2.265 m3/s. A GIS analysis was conducted using NHDPlus Version 2 coverages to generate river miles and time of travel estimates from the flowage to the upstream 26 stations, respectively. The NHDPlus data included reach lengths, estimated annual average flow, and estimated average velocity. Travel time was calculated in days. Application of the resulting total estimated attenuation yielded only minor changes in results as travel times from the stations examined to Holcombe Flowage are short and many of the stream segments exceed the average annual flow cutoff of 2.265 m3/s (80 cfs) above which no attenuation occurs. The targets at individual stations adjusted for attenuation (Table 3-3) range up to 0.053 mg/L, but are all less than the UCL analysis target presented above. Table 3-3. Revision to Upstream Targets based on SPARROW Attenuation Station

Adjusted Target TP (mg/L)

513188 513189 553003 553042 553063 553097 553126 553131 553137 553138 553149 553156 553158 553167 553169 553170 553171 553172 553173 613199 10029123 10029539 10030672 10030673

0.030 0.030 0.030 0.030 0.030 0.031 0.030 0.032 0.034 0.036 0.030 0.030 0.036 0.030 0.053 0.040 0.036 0.035 0.042 0.032 0.030 0.031 0.030 0.030

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A load partitioning analysis was undertaken using the stratified regression approach. The stratified regression approach provided a better fit to the observed data than simple LOADEST approach (although it should be noted that LOADEST contains a whole array of more complex models that would likely provide a better fit, but which are difficult to rearrange for the purpose of predicting concentration as a function of flow). The stratified regression was applied to develop a relationship between the natural log of load and the natural log of flow at the three downstream USGS stations (Chippewa River near Bruce, 05356500; Flambeau River near Bruce, 05360500; and Jump River at Sheldon, 05362000), which together account for 87 percent of the upstream drainage area. The resulting R2values were 0.49, 0.44, and 0.16 for the Chippewa, Flambeau, and Jump River stations, respectively. We used the stratified regression relationships to develop a complete time series of predicted loads and concentrations for water years 1951 – 2013. Application of the rating curve (Table 3-4) suggests that the Chippewa River contributes a somewhat larger percentage of the TP load than is accounted for by its drainage area. Table 3-4. Stratified Regression Analysis of Loads at Holcombe Flowage Gages USGS ID (WQID)

Station

Drainage Area (mi2, %)

TP Load (%)

05356000 (553003)

Chippewa R nr Bruce

1,650 (40.4%)

46.1%

05360500 (553149)

Flambeau R nr Bruce

1,860 (45.5%)

43.8%

05362000 (553042)

Jump R at Sheldon

576 (14.1%)

10.1%

For the Chippewa River, the long-term flow-weighted concentration predicted by the stratified regression is 0.055, or nearly twice the downstream target of 0.030 mg/L. A compliance scenario was developed by iteratively reducing the concentrations in the upper stratum only (above 1,552 cfs) until the flow-weighted concentration met the desired target of 0.030. The analysis includes a random error term based on the root mean squared error in the estimates of ln(P). Achieving the 0.030 target required a reduction of 67.7 percent in the upper stratum. After the target flow-weighted concentration was achieved, 46.1 percent of the individual observations were still greater than 0.03 mg/L.

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April 2015

4,000

100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0%

3,500

Bin Count

3,000 2,500 2,000 1,500 1,000 500

0.070

0.065

0.060

0.055

0.050

0.045

0.040

0.035

0.030

0.025

0.020

0.015

0.010

0.005

0.000

0

Percentile

Downstream Use Protection

Total P (mg/l)

Figure 3-13. Distribution of TP Concentrations after Achieving Flow-Weighted Concentration of 0.030 mg/L, Chippewa River

8,000

100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0%

7,000

Bin Count

6,000 5,000 4,000 3,000 2,000 1,000

0.070

0.065

0.060

0.055

0.050

0.045

0.040

0.035

0.030

0.025

0.020

0.015

0.010

0.005

0.000

0

Percentile

A similar analysis for the Flambeau River yields somewhat different results. For the Flambeau, a reduction in the upper stratum (greater than 804 cfs) of 40.5 percent was needed to achieve the flowweighted concentration target of 0.030 mg/L, resulting in 38 percent of individual observations remaining greater than the target (Figure 3-14).

Total P (mg/l)

Figure 3-14. Distribution of TP Concentrations after Achieving Flow-Weighted Concentration of 0.030 mg/L, Flambeau River

A load-based analysis can also be done using the USGS SPARROW model. Holcombe Flowage and its watershed fall within the SPARROW total phosphorus model for the Great Lakes, Ohio, Upper Mississippi, and Souris-Red-Rainy region (Robertson and Saad, 2011). Reaches in the SPARROW

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model are based on Enhanced River Reach File 2.0, which is defined at a much coarser scale than NHDPlus. The SPARROW model results are available on a companion decision support website and include incremental and cumulative SPARROW inputs and outputs for each model reach. Outputs include total delivered model load and estimated annual average flow, from which an annual average flow-weighted concentration was calculated. The SPARROW model predicts a flow-weighted concentration of total phosphorus within Holcombe Flowage of 0.0484 mg/L, suggesting a need for a 38 percent reduction in loads (much less than the load reduction suggested by the stratified regression analysis). In addition, the flow-weighted concentrations from SPARROW differ in relative ordering and magnitude from those obtained with the site-specific regression analysis (Table 3-5). This likely primarily reflects the uncertainty in SPARROW load estimates at the local watershed scale. Table 3-5. Comparison of SPARROW Flow-weighted Concentrations to Results of Stratified Regression Analysis, Holcombe Flowage Gage Location

SPARROW flow-weighted TP concentration (mg/L)

Stratified regression flowweighted TP concentration (mg/L)

0535600 (Chippewa)

0.048

0.055

05360500 (Flambeau)

0.040

0.048

05362000 (Jump)

0.068

0.037

3.5.3 Network Spatial Correlation Analysis 3.5.3.1

Model Estimation

Twenty-seven observation sites within Holcombe Flowage watershed stream network were used to derive the spatial regression model. For each site, the accumulated upstream area of different land uses and the area of local land uses (both in km2) were calculated and considered as regression factors. A variety of different spatial covariance models were tested with the SSN package (Table 3-6). Table 3-6. Spatial Model Components Tested for the Holcombe Flowage Watershed Model Regression factor

Regression factor acronyms

Spatial covariance models

Kernel function type

Total drainage area

acc_area

Tail.up

Linear with sill

Accumulation area of water body

acc_wb

Tail.down

Spherical

Accumulation area of agriculture

acc_agri

Euclidean

exponential

Accumulation area of wetland

acc_wet

Cauchy

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Regression factor

Regression factor acronyms

Accumulation area of agriculture

acc_graze

Upstream distance

upDist

Spatial covariance models

Kernel function type

Mariah

To evaluate the potential role of different regressors the potential models were first tested without spatial covariance. Fitted results for the complete model are shown in Table 3-7. Table 3-7. Evaluation of Regression Model Evaluation with All the Regression Factors, Holcombe Flowage Coefficients

Estimate Parameter

Std. Error

t-value

Pr(>|t|)

Significance code

Model: TP ~ acc_area + acc_wb + acc_agri +acc_urban+ acc_wet + upDist (Intercept)

1.09E-01

2.44E-02

4.488

0.00023 ***

acc_wet

1.34E-05

3.50E-05

0.383

0.70607

-3.27E-05

6.90E-05

-0.474

0.64061

acc_wb

2.28E-04

2.00E-04

1.14

0.26759

acc_agri

4.99E-05

3.56E-05

1.401

0.17666

acc_area

-1.25E-05

9.46E-06

-1.32

0.20189

upDist

-3.10E-02

4.05E-02

-0.766

0.4528

acc_graze

Note: Significance codes: 0.09, Current

% > 0.09, Adjusted

03448800

Swannanoa R at I-40 at Black Mtn, NC

0.100

36%

18%

03455000

French Broad R nr Newport, TN

0.208

50%

11%

E0150000

French Broad R at NC-178 at Rosman

0.042

3%

0%

E1490000

Mills R at end of SR-1337 nr Mills R

0.037

4%

4%

E2120000

Mud Cr at SR-1508 nr Balfour

0.046

10%

0%

E2730000

French Broad R at SR-3495 Glenn Bridge Rd nr Skyland

0.073

14%

5%

E3520000

Hominy Cr at SR-3413 nr Asheville

0.067

17%

8%

E4280000

French Broad R at SR-1348 at Asheville X Ref E3420000

0.079

25%

6%

E4770000

French Broad R at SR-1634 at Alexander

0.151

74%

17%

FRBLK1

Lake Kenilworth

0.048

13%

0%

FRBLK2

Lake Kenilworth

0.032

0%

0%

E5495000

Pigeon R at NC-215 nr Canton

0.074

15%

15%

E5600000

Pigeon R at SR-1642 at Clyde

0.195

72%

31%

E6450000

Cataloochee Cr at SR-1395 nr Cataloochee

0.036

6%

3%

06010105 Upper French Broad

06010106 Pigeon

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06010108 Nolichucky

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WQ Sta. ID

Name

Average TP (mg/L)

% > 0.09, Current

% > 0.09, Adjusted

E6480000

Pigeon R at SR-1338 nr Hepco

0.168

49%

10%

FRB047A

Lake Junaluska

0.029

0%

0%

FRB047B

Lake Junaluska

0.023

0%

0%

FRB047C

Lake Junaluska

0.025

0%

0%

03465500

Nolichucky R at Embreeville, TN

0.046

7%

7%

03466208

Big Limestone Cr nr Limestone, TN

0.151

58%

13%

0.151

58%

13%

03467609

Nolichucky R nr Lowland

The median CV across all stations is 0.91, much higher than for Holcombe Flowage. Assuming a lognormal distribution, the corresponding UCL on the downstream criterion is 0.239 mg/L. Only 6.8 percent of individual observations are greater than 0.239 mg/L. The percentage of observations greater than the UCL in individual samples range from 0 to 24 percent. Even after the adjustment to guarantee that all three of the downstream stations achieve a sample average of 0.090 mg/L, some excursions are still present (Table 3-15). Table 3-15. Comparison of Total Phosphorus Data to UCL on Downstream Criterion, Douglas Reservoir Watershed HUC8

06010105 Upper French Broad

WQ Sta. ID

Name

% > 0.239, Current

% > 0.239, Adjusted

03448800

Swannanoa R at I-40 at Black Mtn, NC

18%

0%

03455000

French Broad R nr Newport, TN

11%

4%

E0150000

French Broad R at NC-178 at Rosman

0%

0%

E1490000

Mills R at end of SR-1337 nr Mills R

4%

0%

E2120000

Mud Cr at SR-1508 nr Balfour

0%

0%

E2730000

French Broad R at SR-3495 Glenn Bridge Rd nr Skyland

5%

0%

E3520000

Hominy Cr at SR-3413 nr Asheville

5%

0%

E4280000

French Broad R at SR-1348 at Asheville X Ref E3420000

6%

0%

E4770000

French Broad R at SR-1634 at Alexander

12%

2%

FRBLK1

Lake Kenilworth

0%

0%

FRBLK2

Lake Kenilworth

0%

0%

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WQ Sta. ID

Name

% > 0.239, Current

% > 0.239, Adjusted

E5495000

Pigeon R at NC-215 nr Canton

15%

0%

E5600000

Pigeon R at SR-1642 at Clyde

24%

4%

E6450000

Cataloochee Cr at SR-1395 nr Cataloochee

3%

0%

E6480000

Pigeon R at SR-1338 nr Hepco

10%

5%

FRB047A

Lake Junaluska

0%

0%

FRB047B

Lake Junaluska

0%

0%

FRB047C

Lake Junaluska

0%

0%

03465500

Nolichucky R at Embreeville, TN

0%

0%

03466208

Big Limestone Cr nr Limestone, TN

9%

5%

5%

1%

06010106 Pigeon

06010108 Nolichucky

03467609

Nolichucky R nr Lowland

3.6.2 Simplified Assignment of Load, Douglas Reservoir Douglas Reservoir has a large watershed with relatively long travel times from more distant locations. We used the southeastern regional SPARROW phosphorus model (Garcia et al., 2011) to calculate attenuation. In contrast with SPARROW models in which attenuation is solely a function of travel time (but sorted into different flow range bins), this model calculates attenuation relative to the product of travel time divided by mean water depth, with a coefficient of 0.048 m/day. Unfortunately, average depths were not readily available for the reach network. For example, if a depth of 1 m is assumed, the adjusted targets at the monitoring stations would increase from 0.090 to a maximum of 0.119 mg/L. A stratified regression loading analysis was performed at two downstream stations, French Broad River near Newport (03455000) and Nolichucky River near Lowland (03467609), which together account for 83 percent of the total drainage area of Douglas Reservoir. The analysis was restricted to water years 1997-2013 due to changes in point source loads and land use in the watershed relative to earlier time periods. Gaps in the flow record for Nolichucky River were filled by a linear regression against Nolichucky River near Embreeville (03465500), with an R2 of 0.59. The correlation is lower than would be expected given the locations of the stations on the stream network because of the influence of the Nolichucky Dam, which lies between the stations. . As with the analysis for Holcombe Flowage, the stratified regression approach provided a better fit to the loads derived from observed data than the simple LOADEST approach. The stratified regression was applied to develop a relationship between the natural log of load and the natural log of flow at the two downstream stations. The resulting R2values were 0.66 for the French Broad River and 0.21 for the Nolichucky River. We used the stratified regression relationships to develop a complete time series of predicted loads and concentrations for water years 1997 – 2013. For the French Broad River, the long-term flow-weighted total phosphorus concentration estimate is 0.147 mg/L. For the Nolichucky River, the long-term flowweighted total phosphorus concentration estimate is 0.089 mg/L, which is just below the downstream

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1,600

100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0%

1,400

Bin Count

1,200 1,000 800 600 400 200

0.360

0.330

0.310

0.280

0.260

0.230

0.210

0.180

0.150

0.130

0.100

0.080

0.050

0.030

0.000

0

Percentile

target of 0.090 mg/L. A compliance scenario was developed for the French Broad by iteratively reducing the concentrations in the upper stratum only (above 3,562 cfs) until the flow-weighted concentration met the desired target of 0.0390. The analysis includes a random error term based on the root mean squared error in the estimates of ln(P). Achieving the 0.090 target required a reduction of 64.7 percent in the upper stratum. After the target flow-weighted concentration was achieved, 31.1 percent of the individual observations were still greater than 0.09 mg/L (Figure 3-24). The distribution is strongly skewed, so some individual observations are much greater than 0.09 mg/L.

Total P (mg/l)

Figure 3-24. Distribution of TP Concentrations after Achieving Flow-Weighted Concentration of 0.090 mg/L, French Broad River (TN)

For the Nolichucky River, direct estimates with the stratified regression have a flow-weighted mean concentration that is below the target; however, the series created with random perturbations in ln(P) exceeds the target by a small amount at 0.11 mg/L. For the Nolichucky, a reduction in the upper stratum (greater than 1,429 cfs) of 20.0 percent was needed to achieve the flow-weighted concentration target of 0.090 mg/L, resulting in 18 percent of individual observations remaining greater than the target (Figure 3-14).

108

0.270

0.250

0.230

0.210

0.190

0.170

0.150

0.140

0.120

0.100

0.080

0.060

0.040

0.020

100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0%

Percentile

April 2015

2,000 1,800 1,600 1,400 1,200 1,000 800 600 400 200 0

0.000

Bin Count

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Total P (mg/l)

Figure 3-25. Distribution of TP Concentrations after Achieving Flow-Weighted Concentration of 0.090 mg/L, Nolichucky River (TN)

A load-based analysis was also be done using the USGS SPARROW model. Douglas Reservoir and its watershed fall within the SPARROW total phosphorus model for the Southeastern United States region (Garcia et al., 2011). The SPARROW model predicts a flow-weighted concentration within Douglas Reservoir of 0.102 mg/L, suggesting a need for an 11.5 percent reduction in loads (much less than the LOADEST analysis). In addition, the flow-weighted concentrations from SPARROW differ in relative ordering and magnitude from those obtained with the site-specific regression analysis (Table 3-5). As with Holcombe Flowage, the SPARROW load estimates differ substantially from the estimates obtained from analysis of data from an individual station. Table 3-16. Comparison of SPARROW Flow-weighted Concentrations to Results of Stratified Regression Analysis, Douglas Reservoir Gage Location

SPARROW flow-weighted TP concentration (mg/L)

Stratified regression flowweighted TP concentration (mg/L)

03455000 (French Broad River near Newport)

0.042

0.147

03467609 (Nolichucky River near Lowland)

0.154

0.089

3.6.3 Network Spatial Correlation Analysis 3.6.3.1 Model Estimation Twenty-six observation sites within the Douglas Reservoir stream network were used to derive the spatial regression model. For each site, the accumulated upstream area of different land uses and the area of local land uses (both in km2) were calculated and considered as regression factors. A variety of different spatial covariance models were tested with the SSN package (Table 3-17). Table 3-17. Spatial Model Components Tested for the Douglas Reservoir Watershed Model

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Regression factor

Regression factor acronyms

Spatial covariance models

Kernel function type

Total drainage area

acc_area

Tail.up

Linear with sill

Accumulation area of water body

acc_wb

Tail.down

Spherical

Accumulation area of agriculture and pasture

acc_agrigr

Euclidean

exponential

Accumulation area of urban area

acc_urban

Mariah

area of local RCA

rca_area

Cauchy

Agriculture and pasture area of local RCA

rca_agrigr

Urban area of local RCA

rca_urban

Urban area ratio of local RCA

urbanratio

Agriculture and pasture area ratio of local RCA

agrigrrati

Water body area ratio of local RCA

wbratio

Upstream distance

upDist

To evaluate the potential role of different regressors the potential models were first tested without spatial covariance. Fitted results for the complete model are shown in Table 3-18. Table 3-18. Evaluation of Regression Models for Douglas Reservoir with all the Regression Factors Coefficients

Estimate Parameter

Std. Error

t-value

Pr(>|t|)

Significance code

Model: TP ~ acc_area + acc_wb + acc_agrigr +acc_urban+ rca_agrigr + urban_ratio+agrigrrati + wbratio+upDist+rca_area (Intercept)

1.10E-01

8.50E-02

1.294

0.2165

acc_area

-1.73E-04

7.83E-05

-2.214

0.0439

acc_wb

-1.90E-02

1.98E-02

-0.957

0.3547

acc_agrigr

1.92E-03

8.05E-04

2.378

0.0322

acc_urban

9.07E-08

2.49E-03

0

1

rca_agrigr

5.91E-02

1.70E-01

0.349

0.7326

*

*

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Estimate Parameter

Std. Error

t-value

Pr(>|t|)

Urbanratio

-2.52E-02

2.20E-01

-0.115

0.9104

Agrigrrati

-4.24E-01

4.83E-01

-0.878

0.3948

Wbratio

-2.70E-01

1.93E-01

-1.4

0.1834

upDist

-3.39E-03

2.68E-02

-0.126

0.9012

rca_area

5.41E-03

2.07E-02

0.261

0.798

rca_urban

2.06E-02

2.04E-01

0.101

0.921

Significance code

Note: Significance codes: |t|)

Significance code

Intercept

5.40E-02

1.30E-02

4.15

0.00036

***

acc_agrigr

2.36E-04

6.48E-05

3.638

0.00131

**

We then investigated different spatial covariance model combinations together with the regression function (Table 3-20). The exponential kernel function was used to compare different spatial covariance components. Table 3-20. Spatial Covariance Models Evaluated, Douglas Reservoir Number

Formula

Variance Components

1

TP ~ acc_agrigr

Exponential.tailup + Exponential.taildown + Exponential.Euclid + Nugget

2

TP ~ acc_agrigr

Exponential.tailup + Exponential.taildown+ Nugget

3

TP ~ acc_agrigr

Exponential.tailup + Exponential.Euclid+ Nugget

4

TP ~ acc_agrigr

Exponential.taildown + Exponential.Euclid+ Nugget

5

TP ~ acc_agrigr

Exponential.tailup+ Nugget

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Formula

Variance Components

6

TP ~ acc_agrigr

Exponential.taildown+ Nugget

7

TP ~ acc_agrigr

Exponential.Euclid+ Nugget

As was presented above in the discussion for the Holcombe Flowage example, spatial covariance models are selected first for lowest negative log-likelihood (neg2LogL), with ties resolved by the standardized mean-squared prediction error (std.MSPE). On the basis of neg2LogL, the spatial variance model with taildown and Euclidean distsance components (No.4) was determined to provide the best fit (Table 3-21). Choice of a Euclidean distance model makes some sense in a watershed where agricultural land is a major source of load, but is distributed in small patches throughout the watershed. The observation points along the stream network may not represent the spatial impact of agriculture very accurately if the sampling density is low. The Euclidean distance thus may serve as a surrogate for upstream network influences. It also may reflect spatial patterns of native soil phosphorus concentrations. Table 3-21. Performance of Spatial Covariance Models, Douglas Reservoir Number

Variance Components

neg2LogL

std.MSPE

1

Exponential.tailup + Exponential.taildown + Exponential.Euclid + Nugget

-68.8421

0.976457

2

Exponential.tailup + Exponential.taildown+ Nugget

-67.8262

0.94896

3

Exponential.tailup + Exponential.Euclid+ Nugget

-68.8428

0.976362

4

Exponential.taildown + Exponential.Euclid+ Nugget

-69.5757

1.011669

5

Exponential.tailup+ Nugget

-67.8258

0.94914

6

Exponential.taildown+ Nugget

-69.0542

1.015237

7

Exponential.Euclid+ Nugget

-67.8686

0.987441

The taildown plus Euclidean spatial covariance components were combined with the spatial regression model based on the results of Table 3-21. Different kernel functions for the selected two spatial covariance were then tested. Using the same performance indicators (neg2LogL, std.MSPE), the best-fit spatial covariance components are Exponential Taildown and Cauchy Euclidean. The final spatial regression model has the following form: 𝑇𝑇𝑃𝑃 = 0.0565 + 2.27 ∗ 10−4 𝐴𝐴𝑎𝑎𝑎𝑎 + 𝑆𝑆𝑐𝑐𝑐𝑐𝑐𝑐 (𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸. 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 + 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶ℎ𝑦𝑦. 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸),

where TP is the concentration of total phosphorus (mg/L), Aag is the accumulated agriculture and pasture area (km2), and Scov is the spatial covariance corrector vector. This model has a generalized R2 of 0.34 and a median residual of -0.017 mg/L.

3.6.3.2 Population Level Results for Douglas Reservoir Watershed

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The selected spatial regression model was used to estimate the average TP concentration of stream segments through the reach network (Figure 3-26). The histogram of TP concentrations is shown in Figure 3-27. The mean of the TP concentration for all segments is 0.065 mg/L and the standard deviation is 0.034 mg/L. The minimum predicted TP concentration within the stream network is 0.02 mg/L and the maximum predicted TP concentration is 0.6mg/L. The 75th percentile predicted TP concentration is less than 0.06mg/L, and more than 50% of predicted TP concentrations were within a range from 0.05 mg/L to 0.06 mg/L. This implies that the structure of spatial variation of TP concentrations within the stream network has a limited impact, in which case random sampling of stream sites would be a simple and direct way to assess the population level TP concentrations.

Figure 3-26. TP Concentrations Estimated by Spatial Regression Model, Douglas Reservoir

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Figure 3-27 Histogram of Predicted TP Concentrations, Douglas Reservoir

3.6.4 Sampling Results on Spatially Correlated Network 3.6.4.1 Prior Information on TP Targets The Douglas Reservoir watershed lies with Level IV ecoregion 67g. The Tennessee nutrient criterion study (Denton et al., 2001) proposes a criterion TP concentration for this ecoregion as 0.09 mg/L, the 90th percentile of the reference TP concentrations in the ecoregion. The reference values have a mean concentration of 0.052 mg/L and a standard deviation of 0.085 mg/L. Section 2.3.4 derived a target TP concentration for Douglas Reservoir of 0.04 mg/L. An alternative calculation was done based on the Level IV stream criterion. Level IV results are not available for lakes, but quantile mapping can be used to project the stream criterion into the lake based on quantile mapping (Panofsky and Brier, 1968) from the Level III lake criterion to Level IV results for the sub-ecoregion: −1 𝑇𝑇𝑃𝑃𝑠𝑠𝑠𝑠𝑠𝑠_𝑙𝑙𝑙𝑙𝑙𝑙 = 𝐹𝐹𝑇𝑇𝑃𝑃 (𝐹𝐹𝑇𝑇𝑃𝑃𝑒𝑒𝑒𝑒𝑒𝑒_𝑠𝑠𝑠𝑠𝑠𝑠 (𝑇𝑇𝑃𝑃𝑠𝑠𝑠𝑠𝑠𝑠_𝑠𝑠𝑠𝑠𝑠𝑠 )) 𝑒𝑒𝑒𝑒𝑒𝑒_𝑙𝑙𝑙𝑙𝑙𝑙

where F indicates the cumulative distribution function and F-1 is the inverse cumulative distribution function. In the nutrient criterion study, the critical value of TP in stream is defined as the 90th percentile TP value. Given the median and 95th percentile value of TP concentration, a log-normal distribution was fit to the TP concentration in both streams and lakes. Then the percentile of critical TP value (0.09 mg/L) in the level III ecoregion streams was retrieved and was used to project the lake critical TP concentration. In this case, the projected critical lake TP concentration in sub-ecoregion 67g is 0.05 mg/L and projected mean lake TP concentration is 0.04 mg/L. Based on this prior information, two downstream endpoint TP concentration scenarios were created for the upper and lower boundaries. The first downstream endpoint TP concentration scenario is a reference scenario, based on the 90th percentile value of 0.05 mg/L. The critical scenario has a mean concentration of 0.05 mg/L (as opposed to a 90th percentile), and the observed concentrations have a CV of 0.695. Lognormal probability density functions were fit to both distributions (Figure 3-28).

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Figure 3-28. PDFs of Reference and Critical Stream TP Concentration Scenarios at the Downstream Pour Point, Douglas Reservoir

3.6.4.2 Thresholds for the Number of Exceedances The weighted mean exceedance ratio and one-sided 95% confidence intervals were calculated for both endpoint TP concentration scenarios (Figure 3-29, Figure 3-30). Only cases in which the number of sampling sites is greater than or equal to 10 are presented. The upper bound of critical scenario and the lower bound of reference scenario were used to calculate the “safe” and “danger” thresholds for number of sample means exceeding the target.

Figure 3-29. Exceedance Ratio and One-sided 95% Confidence Interval for Reference Scenario, Douglas Reservoir

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Figure 3-30. Exceedance Ratio and One-sided 95% Confidence Interval for Critical Scenario, Douglas Reservoir

We define two thresholds for the number of sample means exceeding the target. The lower bound exceedance ratio was used to calculate the “safe” threshold. This threshold implies that if the number of exceedance sites is smaller than the given threshold, there is only a small chance (less than 5%) that the downstream endpoint TP concentration is exceeding the criterion TP concentration. In this case, we could conclude with a high level of confidence that the TP concentration at the downstream pour point is below the target. Similarly, the upper bound exceedance ratio was used to calculate the “danger” threshold. If the number of sites with concentrations above this boundary is larger than the “danger” threshold, there is a high probability that the TP concentration at the downstream pour point is higher than the criterion TP concentration. Table 3-22 lists the thresholds for different sample sizes. Table 3-22. Exceedance Thresholds (Count) for Different Sample Sizes, Douglas Reservoir Sample Size

Safe Threshold

Danger Threshold

10

1

6

11

1

7

12

1

7

13

1

7

14

1

8

15

2

8

16

2

9

17

2

9

18

2

9

19

2

10

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Sample Size

Safe Threshold

Danger Threshold

20

2

10

3.6.4.3 Mean TP Concentration at Sampling Sites For 12 different downstream endpoint TP concentrations, the mean TP concentration and first and third quantiles for sampling sites were calculated for different sampling sizes and downstream concentration (DC) targets (Table 3-23). The mean sample TP concentration differs little across DC when sample size is larger than 10. The sample TP concentrations were fitted with log-normal probability distribution functions for different downstream endpoint TP concentrations to derive the 25th and 75th percentile values. Table 3-23. TP Concentrations for Different Sample Sizes, Douglas Reservoir Watershed Sample Size

DC=0.005

DC=0.045

DC=0.063

DC=0.081

DC=0.099

DC=0.12

10

0.063, [0.042,0.060]

0.057, [0.040,0.057]

0.058, [0.028,0.057]

0.063, [0.019,0.062]

0.065, [0.008,0.064]

0.067, [0.005,0.066]

11

0.063, [0.043,0.061]

0.057, [0.041,0.057]

0.058, [0.029,0.057]

0.063, [0.020,0.062]

0.065, [0.014,0.064]

0.067, [0.010,0.066]

12

0.063, [0.044,0.061]

0.057, [0.043,0.057]

0.058, [0.028,0.057]

0.063, [0.024,0.062]

0.065, [0.016,0.064]

0.067, [0.010,0.066]

13

0.063, [0.045,0.061]

0.057, [0.041,0.057]

0.057, [0.031,0.057]

0.063, [0.022,0.062]

0.065, [0.014,0.064]

0.067, [0.004,0.066]

14

0.063, [0.044,0.061]

0.057, [0.041,0.057]

0.057, [0.031,0.057]

0.063, [0.024,0.062]

0.065, [0.014,0.064]

0.067, [0.012,0.066]

15

0.063, [0.045,0.061]

0.057, [0.044,0.057]

0.058, [0.032,0.057]

0.063, [0.023,0.062]

0.065, [0.017,0.064]

0.067, [0.016,0.066]

16

0.063, [0.045,0.061]

0.057, [0.044,0.057]

0.057, [0.031,0.057]

0.063, [0.025,0.062]

0.065, [0.023,0.064]

0.067, [0.016,0.066]

17

0.063, [0.044,0.062]

0.057, [0.044,0.057]

0.057, [0.034,0.057]

0.063, [0.025,0.063]

0.065, [0.019,0.064]

0.067, [0.012,0.066]

18

0.063, [0.046,0.062]

0.057, [0.045,0.057]

0.057, [0.035,0.057]

0.063, [0.027,0.063]

0.065, [0.018,0.064]

0.067, [0.017,0.066]

19

0.063, [0.047,0.062]

0.057, [0.045,0.057]

0.057, [0.035,0.057]

0.063, [0.029,0.063]

0.065, [0.023,0.064]

0.067, [0.017,0.067]

20

0.063, [0.046,0.062]

0.057, [0.045,0.057]

0.057, [0.035,0.057]

0.063, [0.028,0.063]

0.065, [0.025,0.064]

0.067, [0.018,0.067]

10

0.070, [0.002,0.069]

0.074, [0.001,0.073]

0.078, [0.003,0.077]

0.077, [0.001,0.075]

0.081, [0.000,0.079]

0.119, [0.000,0.120]

11

0.071, [0.005,0.069]

0.074, [0.002,0.073]

0.078, [0.001,0.077]

0.077, [0.003,0.076]

0.081, [0.001,0.079]

0.119, [0.000,0.120]

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DC=0.005

DC=0.045

DC=0.063

DC=0.081

DC=0.099

DC=0.12

12

0.070, [0.008,0.069]

0.074, [0.008,0.073]

0.078, [0.007,0.077]

0.077, [0.000,0.076]

0.081, [0.001,0.079]

0.119, [0.000,0.120]

13

0.070, [0.010,0.069]

0.074, [0.008,0.073]

0.078, [0.005,0.077]

0.077, [0.008,0.076]

0.081, [0.004,0.080]

0.119, [0.001,0.120]

14

0.071, [0.011,0.070]

0.074, [0.008,0.073]

0.078, [0.006,0.077]

0.077, [0.003,0.076]

0.081, [0.002,0.080]

0.119, [0.002,0.120]

15

0.071, [0.009,0.070]

0.074, [0.010,0.073]

0.078, [0.010,0.077]

0.077, [0.007,0.076]

0.081, [0.005,0.080]

0.119, [0.006,0.120]

16

0.071, [0.007,0.070]

0.074, [0.010,0.073]

0.078, [0.007,0.077]

0.077, [0.003,0.076]

0.081, [0.007,0.080]

0.119, [0.008,0.120]

17

0.070, [0.016,0.070]

0.074, [0.010,0.073]

0.078, [0.008,0.077]

0.077, [0.009,0.076]

0.081, [0.010,0.080]

0.119, [0.002,0.120]

18

0.070, [0.015,0.070]

0.074, [0.012,0.073]

0.078, [0.010,0.077]

0.077, [0.010,0.076]

0.081, [0.008,0.080]

0.119, [0.008,0.120]

19

0.071, [0.011,0.070]

0.074, [0.011,0.073]

0.078, [0.012,0.077]

0.077, [0.010,0.076]

0.081, [0.009,0.080]

0.119, [0.008,0.120]

20

0.071, [0.017,0.070]

0.074, [0.015,0.073]

0.078, [0.010,0.077]

0.077, [0.013,0.076]

0.081, [0.012,0.080]

0.119, [0.016,0.120]

Note: Results show mean; first and third quantile values shown in parentheses.

For Douglas Reservoir, all stations have at least 10 samples and Monte Carlo simulation results with sample size equal to 10 are presented for the mean TP concentration analysis. The exceedance probability at the downstream pour point for different sampled mean TP concentrations was calculated using the methods described in Section 3.4.5.4. The reference and critical scenarios were also applied to the mean TP concentration analysis. The exceedance probability range is derived from these two scenarios (Figure 3-31, Table 3-24). For different sampled mean TP concentrations, the exceedance probability calculated with the reference situation is considered the lower bound and upper bound for the value calculated with the critical situation.

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Figure 3-31. Exceedance Probability for Different Sample TP Concentrations, Douglas Reservoir

Table 3-24. Exceedance Probability for Different Sample TP Concentrations, Douglas Reservoir TP (mg/L)

Exceedance Probability (%)

TP (mg/L)

Exceedance Probability (%)

0.055

1.5-16.2

0.18

97.9-99.7

0.06

1.1-14.9

0.185

98.4-99.8

0.065

1.8-18.6

0.19

98.8-99.8

0.07

4.0-25.6

0.195

99.1-99.9

0.075

7.8-33.2

0.2

99.4-99.9

0.08

12.5-40.5

0.205

99.5-99.9

0.085

17.5-47.2

0.21

99.7-100.0

0.09

22.6-53.7

0.215

99.8-100.0

0.095

27.9-60.0

0.22

99.8-100.0

0.1

33.5-66.2

0.225

99.9-100.0

0.105

39.5-72.1

0.23

99.9-100.0

0.11

46.0-77.5

0.235

99.9-100.0

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TP (mg/L)

Exceedance Probability (%)

TP (mg/L)

Exceedance Probability (%)

0.115

52.7-82.2

0.24

100.0-100.0

0.12

59.4-86.1

0.245

100.0-100.0

0.125

65.8-89.4

0.25

100.0-100.0

0.13

71.7-92.0

0.255

100.0-100.0

0.135

77.0-94.1

0.26

100.0-100.0

0.14

81.6-95.6

0.265

100.0-100.0

0.145

85.4-96.8

0.27

100.0-100.0

0.15

88.6-97.7

0.275

100.0-100.0

0.155

91.2-98.3

0.28

100.0-100.0

0.16

93.3-98.8

0.285

100.0-100.0

0.165

94.9-99.1

0.29

100.0-100.0

0.17

96.1-99.4

0.295

100.0-100.0

0.175

97.1-99.6

0.3

100.0-100.0

3.7 Conclusions Results presented in this section address only two example watersheds and the analysis of Monte Carlo simulation analyses of those two watersheds is open to further data exploration. Nonetheless, it is clear that the problem of assessing instream concentrations relative to a downstream receiving water target is amenable to a statistical analysis that evaluates whether or not sample means observed upstream in the watershed are consistent with achieving the downstream target. When land uses and other sources of nutrient loading are relatively homogenous and distributed evenly throughout the watershed an analysis of evidence from watershed sampling sites could be made based solely on the measured or estimated lognormal distribution CDF at the downstream pour point. This approach breaks down when there are different types and sources of loads in the watershed, in which case measurements at different observation sites are likely to exhibit strong spatial correlation. This situation can be addressed through the development of spatial covariance representations (SSN/STARS) coupled with either a regression analysis of loads based on landscape features or a simple mean representation for homogenous source distributions. The regression analyses reported herein met with only moderate success, but could likely be improved through better accounting of point source discharges in particular. Taking into account stream network-based autocorrelation enables estimation of the expected value of concentrations throughout the stream network. This distribution is dependent on the concentration and load present at the downstream pour point. A relatively straightforward solution to the assessment problem can be achieved by estimating a lower confidence limit on the distribution across all watershed segments, based on the assumption that the downstream is at relatively undisturbed reference concentration levels, and an upper confidence limit calculated under the assumption that the target

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concentration is just met. This leads to a three-tiered approach: Site concentrations below the lower confidence limit can be deemed “safe” in that they do not present statistically significant results that the downstream pour point concentration will exceed targets. In contrast, sites where the flow-weighted mean concentration is above the upper confidence limit are not consistent with the distribution of site loads that would be expected to achieve the downstream criterion. In the preceding sections we propose methods for determining the appropriate confidence limits when network-based spatial correlation is present. This requires estimation of the spatial correlation structure and underlying regression on watershed characteristics (if appropriate), from which an empirical distribution of the relationship between the desired downstream pour point concentration and concentrations at sites distributed throughout the watershed can be obtained. Additional research is needed to determine simpler methods for evaluating appropriate adjustment factors to relate the distribution of the downstream pour point concentration to concentrations in individual segments throughout the watershed.

4 REFERENCES Agresti, A., and B.A. Coull. 1998. Approximate is better than 'exact' for interval estimation of binomial proportions. The American Statistician, 52: 119–126. Baker, L.A., P. L. Brezonik, E.S. Edgerton, and R. W. OGBURN III. 1985. Sediment acid neutralization in softwater lakes. Water Air Soil Pollut., 25: 2 15-230. Barbour, M. T., J. Gerritsen, B. D. Snyder, and J. B. Stribling. 1999. Rapid Bioassessment Protocols for Use in Wadeable Streams and Rivers: Periphyton, Benthic Macroinvertebrates, and Fish. 2nd ed. U.S. Environmental Protection Agency, Office of Water, Washington, DC. EPA 841-B-99-002. Canfield, D.E., Jr., and R.W. Bachmann. 1981. Prediction of total phosphorus concentrations, chlorophyll a and Secchi depths in natural and artificial lakes. Canadian Journal of Fisheries and Aquatic Sciences, 38(4):414-423. Carlson RE. 1977. A trophic state index for lakes. Limnology and Oceanography 22:361-369. Denton, G.M., D.H. Arnwine, and S.H. Wang. 2001. Development of Regionally-based Interpretations of Tennessee’s Narrative Nutrient Criterion. Tennessee Dept. of Environment and Conservation, Nashville, TN. http://www.tennessee.gov/assets/entities/environment/attachments/nutrient_final.pdf Garcia, A.M., A.B. Hoos, and S. Terziotti. 2011. A regional modeling framework of phosphorus sources and transport in streams of the southeastern United States. Journal of the American Water Resources Association, 47(5): 991-1010. Hirsch, R. M., D.L. Moyer, and S.A. Archfield, 2010. Weighted Regressions on Time, Discharge, and Season (WRTDS), with an Application to Chesapeake Bay River Inputs. Journal of the American Water Resources Association, 46: 857–880. doi: 10.1111/j.1752-1688.2010.00482.x Hoos, A.B. and G. McMahon. 2009. Spatial analysis of instream nitrogen loads and factors controlling nitrogen delivery to stream in the southeastern United States using spatially referenced regression on watershed attributes (SPARROW) and regional classification frameworks. Hydrological Processes, doi:10.1002/hyp.7323. Isaak D., C. Luce, B. Rieman, D. Nagel, E. Peterson, D. Horan, S. Parkes, and G. Chandler. 2010. Effects of recent climate and fire on thermal habitats within a mountain stream network. Ecological Applications, 20(5):1350-1371. Kalff, J. 2002. Limnology: Inland Water Ecosystems. Prentice-Hall, Upper Saddle River, NJ

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Larsen, D.P., and H.T. Mercier. 1976. Phosphorus retention capacity of lakes. Journal of the Fisheries Research Board of Canada, 33:1742-1750. Novotny, V. 2004. Simplified databased Total Maximum Daily Loads, or the world is log-normal. Journal of Environmental Engineering, 130:674–683. Organization for Economic Cooperation and Development (OECD). 1982. Eutrophication of Waters: Monitoring, Assessment and Control. Organization for Economic Cooperation and Development, Paris, France. Panofsky, H.A., and G.W. Brier. 1968. Some Applications of Statistics to Meteorology. Pennsylvania State University, University Park, PA Paul, M.J., A. Herlihy, D. Bressler, L. Zheng, and A.Roseberry-Lincoln. 2014. Methodologies for Development of Numeric Nutrient Criteria for Freshwaters. Prepared by Tetra Tech, Inc., Research Triangle Park, NC and Tetra Tech, Inc., Center for Ecological Sciences, Research Triangle Park, NC Peterson E.E., and J.M. Ver Hoef. 2010. A mixed-model moving-average approach to geostatistical modeling in stream networks. Ecology, 91(3): 644-651. Peterson E.E., and J.M. Ver Hoef. 2014. STARS: An ArcGIS toolset used to calculate the spatial information needed to fit spatial statistical models to stream network data. Journal of Statistical Software, 56(2). Press, W.H., B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling. 1992. Numerical Recipes, the Art of Scientific Computing. Cambridge University Press, Cambridge. Preston, S.D., R.B. Alexander, G.E. Schwarz, and C.G. Crawford. 2011. Factors affecting stream nutrient loads: A synthesis of regional SPARROW model results from the continental United States. Journal of the American Water Resources Association, 47(5): 891-915. Qian, S.S., R.S. King, and C.J. Richardson. 2003. Two statistical methods for the detection of environmental thresholds. Ecological Modeling 166: 87–97. Qian, S.S., K.H. Reckhow, J. Zhai, and G. McMahon. 2005. Nonlinear regression modeling of nutrient loads in streams: A Bayesian approach. Water Resources Research, vol. 41, W07012, doi:10.1029/2005WR003986. R Core Team (2014). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria.URL http://www.R-project.org/. Robertson, D.M., and D.A. Saad. 2011. Nutrient inputs to the Laurentian Great Lakes by source and watershed estimated using SPARROW watershed models. Journal of the American Water Resources Association, 47(5): 1011-1033. Runkel, R.A., C.G. Crawford, and T.A. Cohn. 2004. Load Estimator (LOADEST): A FORTRAN Program for Estimating Constituent Loads in Streams and Rivers. Techniques and Methods Book 4, Chapter A5. U.S. Geological Survey, Reston, VA. Robertson, D.M., and D.A. Saad. 2011. Nutrient inputs to the Laurentian Great Lakes by source and watershed estimated using SPARROW watershed models. Journal of the American Water Resources Association, 47(5): 1011-1033. Runkel, R.L., C.G. Crawford, and T.A. Cohn. 2004. Load Estimator (LOADEST): A FORTRAN Program for Estimating Constituent Loads in Streams and Rivers. U.S. Geological Survey Techniques and Methods Book 4, Chapter A5. U.S. Geological Survey, Reston, VA.

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Saad, D.A., G.E. Schwarz, D.M. Robertson, and N.L. Booth. 2011. A multi-agency nutrient dataset used to estimate loads, improve monitoring design, and calibration regional nutrient SPARROW models. Journal of the American Water Resources Association, 47(5): 933-949. Shannon, E.E., and P.L. Brezonik. 1972. Limnological characteristics of north and central Florida lakes. Limnology and Oceanography, 17:97–110. Steward, J.S., and E.F. Lowe. 2010. General empirical models for estimating nutrient load limits for Florida’s estuaries and inland waters. Limnology and Oceanography, 55(1): 433–445 Theobald, D., J. Norman, E.E. Peterson, S. Feraz, A. Wade, and M.R. Sherburne. 2006. Functional Linkage of Waterbasins and Streams (FLoWS) v1 User’s Guide: ArcGIS Tools to Analyze Freshwater Ecosystems. Colorado State University, Fort Collins, CO TDEC. 2001. Development of Regionally Based Interpretations of Tennessee’s Narrative Nutrient Criterion. Tennessee Department of Environment and Conservation, Tennessee Water Quality Control Board. Nashville, TN. Theobald, D., J. Norman, E.E. Peterson, S. Feraz, A. Wade, and M.R. Sherburne. 2006. Functional Linkage of Waterbasins and Streams (FLoWS) v1 User’s Guide: ArcGIS Tools to Analyze Freshwater Ecosystems. Colorado State University, Fort Collins, CO. United States Environmental Protection Agency (USEPA). 1991. Technical Support Document for Water Quality-based Toxics Control. EPA/505/2-90-001. Office of Water, U.S. Environmental Protection Agency, Washington, DC. USEPA. 1998. Lake and reservoir bioassessment and biocriteria. Technical guidance document. United States Environmental Protection Agency, Office of Water, Washington, DC. EPA 841-B-98-007. USEPA. 2000. Nutrient Criteria Technical Guidance Manual, Lakes and Reservoirs. United States Environmental Protection Agency, Office of Water, Washington, DC. EPA-822-B00-001. USEPA. 2001. Nutrient Criteria Technical Guidance Manual, Estuarine and Coastal Marine Waters. United States Environmental Protection Agency, Office of Water, Washington, DC. EPA-822-B-01-003. USEPA. 2009. National Lakes Assessment: A Collaborative Survey of the Nation’s Lakes. U.S. Environmental Protection Agency, Office of Water and Office of Research and Development, Washington, D.C. EPA 841-R-09-001. USEPA. 2010. Technical Support Document for U.S. EPA’s Final Rule for Numeric Criteria for Nitrogen/Phosphorus Pollution in Florida’s Inland Surface Fresh Waters. Van Buren, M.A., W.E. Watt, and J. Marsalek. 1997. Application of the log-normal and normal distributions to stormwater quality parameters. Water Research, 31(1): 95-104. . Ver Hoef, J.M., and E.E. Peterson. 2010. A moving average approach to spatial statistical models of stream networks. The Journal of the American Statistical Association, 489: 6-18. Ver Hoef, J.M., E.E. Peterson, D. Clifford, and R. Shah. 2014. SSN: An R package for spatial statistical modeling on stream networks. Journal of Statistical Software, 56(3). Virginia Water Resources Research Center (VWRRC). 2005. Issues Related to Freshwater Nutrient Criteria for Lakes and Reservoir in Virginia. Virginia Water Resources Research Center, Blacksburg, VA. VWRRC Special Report SR27-2005. Vollenweider, R.A. 1975. Input-output models with special reference to the phosphorus loading concept in limnology. Schweizerische Zeitschrift fur Hydrologie (Swiss Journal of

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Vollenweider, R.A. 1976. Advances in defining critical loading levels for phosphorus in lake eutrophication. Memorie dell’Istituto Italiano di Idrobiologia, 33:53–83. Walker, W.W. 1987. Empirical Methods for Predicting Eutrophication in Impoundments; Report 4, Phase III: Applications Manual. Technical Report E-81-9. U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS. Walker, W.W., 1999. Simplified Procedures for Eutrophication Assessment and Prediction: User Manual. Instruction Report W-96-2. U.S. Army Corps of Engineers Waterways Experiment Station, Vicksburg, MS. Wellen, C., G.B. Arhonditsis, T. Labencki, and D. Boyd. 2014. Application of the SPARROW model in watersheds with limited information: A Bayesian assessment of the model uncertainty and the value of additional monitoring. Hydrological Processes, 28: 1260-1283, doi:10.1002/hyp.9614. Yuan, L. 2006. Estimation and Application of Macroinvertebrate Tolerance Values. United States Environmental Protection Agency, National Center for Environmental Assessment, Office of Research and Development, U.S. Environmental Protection Agency, Washington, D.C. EPA/600/P-04/116F.

APPENDIX A METHODS FOR CORRELATED RANDOM VARIABLE GENERATION This appendix describes the mathematical methods used to generate multiple random variables that preserve a correlation structure. Derivation of the generating equations is most readily illustrated in vector-matrix notation, but, for simplicity, we focus on the bivariate case in algebraic notation. Suppose Gx is a column vector of length two, containing realizations (at location x) of the variables ax and bx, M is a column vector of length two containing the means of the variables, μa and μb, and Ex is a disturbance vector of length two containing the deviations of each realization from its mean value at location x, εa,x, and εb,x. Then Gx = M + Ex or, in algebraic notation,

a x = µ a + ε a,x b x = µ b + ε b,x The two variables are correlated, so the disturbance terms are not independent of one another. The variance-covariance matrix, S, of E can be written as

 s11 s12  σ 2a CV a,b  S= =  2  s 21 s 22  CV b,a σ b  where σ2a is the variance of series a, CVa,b is the covariance of series a and b, and so on. Now suppose S is decomposed into two identical matrices R, such that RRT=S, where “T” stands for the transpose of the matrix. Algebraically, this means

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 r 11 r 12   r 11 r 21  R • RT =  = S • r 21 r 22  r 12 r 22  2 2 2 r 11 + r 12 = s11 = σ a r 11 r 21 + r 12 r 22 = s12 = CV a,b r 21 r 11 + r 22 r 12 = s 21 = CV a,b = CV b,a r 21 + r 22 = s 22 = σ b 2

2

2

If the random variables are generated by

G x = M + R W x , or a x = µ a + r 11 w1 + r 12 w2 = µ a + ε a,x b x = µ b + r 21 w2 + r 22 w2 = µ b + ε b,x where Wx is a vector of standard normal variates with mean 0 and standard deviation of 1, then the proper variance and covariance of the correlated variables will be reproduced. This is easily shown, as the variance-covariance matrix of Gt is

Var (Gx ) = (Gx − M )(Gx − M ) = (M + RW T − M )(M + RW T − M ) T

T

T 2 2 T T RW x W x R = σ w • RR = σ w • S = S

It remains to find the matrix R such that R RT = S. For any covariance matrix, R can be expressed as a lower triangular matrix, such that all entries above the diagonal are zero (the Cholesky decomposition). For the two variable case,

R R T = S , or  r 11 0  r 11 r 12   s11 s12   • =   r 12 r 22   0 r 22   s 21 s 22  For the general case, this may be solved by using the Cholesky decomposition procedure, found in most texts on linear algebra or numerical analysis (e.g., Press et al., 1992). In this project, Gx is the TP concentration at location x, M is the linear regression component based on the sources/sinks terms (land uses) and the Ex is the spatial correlation matrix of all the prediction sites. Monte Carlo simulations were used to explore the relationship of upstream exceedances to downstream conditions under conditions of network spatial correlation. We assumed the outlet (pour point) of the stream network is at the receiving waterbody of interest. To explore cases in which we wish to hold the concentration at this point to a specified target value and allow the remainder of the network to vary randomly in accordance with the spatial correlation structure, we set the row of the spatial covariance matrix that corresponds to the pour point to the first row. After Cholesky decomposition, the spatial covariance value associated with this point is determined by the single top left entry (r11) in the decomposed lower triangle matrix. If the first random variate in the vector Wx is set to zero, the generated concentration at the pout point will be equal to the desired mean or regression result. The standard

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W

𝑛𝑛

1/2

deviation of the remaining normal deviates in x was then inflated by � � 𝑛𝑛−1 one degree of freedom caused by fixing the value at the pour point.

to reflect the reduction of

126