REGIONAL AND TEMPORAL PATTERNS OF TOTAL SOLUTES IN THE SASKATCHEWAN RWER BASIN1

ABSTRACT: In the Saskatchewan River Basin (365,000 km²), which drains the Canadian prairie from the Rocky Mountains east to Manitoba, concentrations of total solutes are usually within the range of 100 to 1000 mg/L. Total solutes levels in tributaries increase markedly from west to east across the basin, as mountain snowmelt and dilute surface runoff are replaced by ion-rich ground water and concentrated prairie runoff as the major influences on solute concentrations. In contrast, total solutes concentrations in main-stem rivers are nearly constant, ranging 200–300 mg/L, with only a small increase across the basin. Dilute mountain runoff dominates solute concentrations in main-stem rivers, despite the influx of increasingly ion-rich water from tributaries. The principal long-term trends in total solute concentrations across the basin, as revealed by linear and sine-curve regressions, were due to the construction of reservoirs, which depress the natural winter maximum in solute concentrations and disrupt the sinusoidal annual pattern, while sharply reducing seasonal variation. These regression methods did not show anticipated anthropogenic increases in salt load in the Red Deer or South Saskatchewan Rivers, but a trend of slowly increasing solutes concentrations (2 mg/L/yr) was detected for autumn flows in the lower Bow River. Municipal wastes from the City of Calgary or irrigation return flows are probably responsible for this increase.     

HYDROLOGIC IMPACTS OF A LARGE-SCALE MOUNTAIN PINE BEETLE (DENDROCTONUS PONDEROSAE HOPKINS) EPIDEMIC1

The Jack Creek watershed, a 133 km² (51.5 mi²) drainage in southwestern Montana, was impacted by a mountain pine beetle (Dendroctonus ponderosae Hopkins) epidemic in 1975–1977 which killed an estimated 35 percent of its total timber. Analyses of USGS streamflow data for four years prior to and five years after mortality suggest a 15 percent post-epidemic increase in annual water yield, a two-to three-week advance in the annual hydrograph, a 10 percent increase in low flows and little increase of peak runoff.     

Spatial Convergence in Major Dissolved Ion Concentrations and Implications of Headwater Mining for Downstream Water Quality

Spatial patterns in major dissolved solute concentrations were examined to better understand impact of surface coal mining in headwaters on downstream water chemistry. Sixty sites were sampled seasonally from 2012 to 2014 in an eastern Kentucky watershed. Watershed areas (WA) ranged from 1.6 to 400.5 km² and were mostly forested (58%–95%), but some drained as much as 31% surface mining. Measures of total dissolved solutes and most component ions were positively correlated with mining. Analytes showed strong convergent spatial patterns with high variability in headwaters (<15 km² WA) that stabilized downstream (WA > 75 km²), indicating hydrologic mixing primarily controls downstream values. Mean headwater solute concentrations were a good predictor of downstream values, with % differences ranging from 0.55% (Na⁺) to 28.78% (Mg²⁺). In a mined scenario where all headwaters had impacts, downstream solute concentrations roughly doubled. Alternatively, if mining impacts to headwaters were minimized, downstream solute concentrations better approximated the 300 μS/cm conductivity criterion deemed protective of aquatic life. Temporal variability also had convergent spatial patterns and mined streams were less variable due to unnaturally stable hydrology. The highly conserved nature of dissolved solutes from mining activities and lack of viable treatment options suggest forested, unmined watersheds would provide dilution that would be protective of downstream aquatic life.     

MODELING THE DISTRIBUTION OF DIFFUSE NITROGEN SOURCES AND SINKS IN THE NEUSE RIVER BASIN OF NORTH CAROLINA,USA1

This study quantified nonpoint source nitrogen (NPS‐N) sources and sinks across the 14,582 km² Neuse River Basin (NRB) located in North Carolina, to provide tabular data summaries and graphic overlay products to support the development of management approaches to best achieve established N reduction goals. First, a remote sensor derived, land cover classification was performed to support modeling needs. Modeling efforts included the development of a mass balance model to quantify potential N sources and sinks, followed by a precipitation event driven hydrologic model to effectively transport excess N across the landscape to individual stream reaches to support subsequent labeling of transported N values corresponding to source origin. Results indicated that agricultural land contributed 55 percent of the total annual NPS‐N loadings, followed by forested land at 23 percent (background), and urban areas at 21 percent. Average annual N source contributions were quantified for agricultural (1.4 kg/ha), urban (1.2 kg/ha), and forested cover types (0.5 kg/ha). Nonpoint source‐N contributions were greatest during the winter (40 percent), followed by spring (32 percent), summer (28 percent), and fall (0.3 percent). Seasonal total N loadings shifted from urban dominated and forest dominated sources during the winter, to agricultural sources in the spring and summer. A quantitative assessment of the significant NRB land use activities indicated that high (greater than 70 percent impervious) and medium (greater than 35 percent impervious) density urban development were the greatest contributors of NPS‐N on a unit area basis (1.9 and 1.6 kg/ha/yr, respectively), followed by row crops and pasture/hay cover types (1.4 kg/ha/yr).     

Modeled Sources,Transport, and Accumulation of Dissolved Solids in Water Resources of the Southwestern United States1

Anning, David W., 2011. Modeled Sources, Transport, and Accumulation of Dissolved Solids in Water Resources of the Southwestern United States. Journal of the American Water Resources Association (JAWRA) 47(5):1087‐1109. DOI: 10.1111/j.1752‐1688.2011.00579.x Abstract: Information on important source areas for dissolved solids in streams of the southwestern United States, the relative share of deliveries of dissolved solids to streams from natural and human sources, and the potential for salt accumulation in soil or groundwater was developed using a SPAtially Referenced Regressions On Watershed attributes model. Predicted area‐normalized reach‐catchment delivery rates of dissolved solids to streams ranged from <10 (kg/year)/km² for catchments with little or no natural or human‐related solute sources in them to 563,000 (kg/year)/km² for catchments that were almost entirely cultivated land. For the region as a whole, geologic units contributed 44% of the dissolved‐solids deliveries to streams and the remaining 56% of the deliveries came from the release of solutes through irrigation of cultivated and pasture lands, which comprise only 2.5% of the land area. Dissolved‐solids accumulation is manifested as precipitated salts in the soil or underlying sediments, and (or) dissolved salts in soil‐pore or sediment‐pore water, or groundwater, and therefore represents a potential for aquifer contamination. Accumulation rates were <10,000 (kg/year)/km² for many hydrologic accounting units (large river basins), but were more than 40,000 (kg/year)/km² for the Middle Gila, Lower Gila‐Agua Fria, Lower Gila, Lower Bear, Great Salt Lake accounting units, and 247,000 (kg/year)/km² for the Salton Sea accounting unit.     

PREDICTION OF PEAK FLOWS ON SMALL WATERSHEDS IN OREGON FOR USE IN CULVERT DESIGN1

ABSTRACT: Using data from 80 Oregon watersheds that ranged in size from 0.54 km² to 27.45 km², equations were developed to predict peak flows for use in culvert design on forest roads. Oregon was divided into six physiographic regions based on previous studies of flood frequency. In each region, data on annual peak flow from gaging stations with more than 20 years of record were analyzed using four flood frequency distributions: type 1 extremal, two parameter-log normal, three parameter-log normal, and log-Pearson type III. The log-Pearson type III distribution was found to be suitable for use in all regions of the State, based on the chi-square goodness-of-fit-test. Flood magnitudes having recurrence intervals of 10, 25, 50, and 100 years were related to physical and climatic characteristics of drainage basins by multiple regression. Drainage basin size was the most important variable in explaining the variation of flood peaks in all regions. Mean basin elevation and mean annual precipitation were also significantly related to flood peaks in two regions of western Oregon. The standard error of the estimate for the regression relationships ranged from 26 to 84 percent.     

STREAM NITRATE-N LOADS IN RELATION TO VARIATIONS IN ANNUAL AND SEASONAL RUNOFF REGIMES1

ABSTRACT: The calculation of stream nutrient loads from a sampling period of one year or, at most, a few years may provide an inaccurate estimate of average seasonal or annual loads due to considerable year-to-year variations in hydrological regime. The number of years of record required to give a reliable estimate of long-term average NO₃-N loads was analyzed for E. Duffin Creek and the Nottawasaga River in Ontario, Canada. Nitrate load rating relationships were used in combination with a continuous stream discharge record for 22 years (E. Duffin Creek) and 34 years (Nottawasaga River) to simulate long-term seasonal and annual variation in NO₃.N loads. The errors involved in calculating average loads were examined by comparing the loads derived from sampling periods of one or more consecutive years duration with the estimated long-term average load for the two rivers. Annual NO₃-N loads for a single year deviated from the long-term average load by ± 20 to 53 percent in 8 out of 22 years in E. Duffin Creek and in 13 of 34 years in the Nottawasaga River. Six consecutive years of record would be required for both rivers to ensure that an error of > ± 20 percent would occur in only 5 percent of these observation periods. February-April NO₃-N loads for a single year could deviate by up to +90 percent or -61 percent from the long-term average spring period load for the two rivers. A sampling period of at least 6–7 years would be needed to estimate average NO₃-N loads for the spring runoff season with an error <± 20 percent.     

FACTORS AFFECTING PESTICIDE OCCURRENCE AND TRANSPORT IN A LARGE MIDWESTERN RIVER BASIN1

ABSTRACT: Several factors affect the occurrence and transport of pesticides in surface waters of the 29,400 km² White River Basin in Indiana. A relationship was found between pesticide use and the average annual concentration of that pesticide in the White River, although this relationship varies for different classes of pesticides. About one percent of the mass applied of each of the commonly used agricultural herbicides was transported from the basin via the White River. Peak pesticide concentrations were typically highest in late spring or early summer and were associated with periods of runoff following application. Concentrations of diazinon were higher in an urban basin than in two agricultural basins, corresponding to the common use of this insecticide on lawns and gardens in urban areas. Concentrations of atrazine, a corn herbicide widely used in the White River Basin, were higher in an agricultural basin with permeable, well‐drained soils, than in an agricultural basin with less permeable, more poorly drained soils. Although use of butylate and cyanazine was comparable in the White River Basin between 1992 and 1994, concentrations in the White River of butylate, which is incorporated into soil, were substantially less than for cyanazine, which is typically applied to the soil surface.     

SPRING RUNOFF AND DRAINAGE N AND P LOSSES FROM HOG-MANURED CORN1

ABSTRACT: A two-year study was conducted to assess the effect of hog manure on the losses of nitrogen and phosphorus in runoff and drainage from grain-corn (Zea mays L.) plots, and the importance of spring versus annual loads. Treatments consisted of mineral N-P-K fertilizer applied at rates of 152 kg N ha^-1, 35 kg P ha^-1, and 86 kg K ha^-1; and hog (Sus scrofa domestica L.) manure applied preplant or post-emergence (six-to-eight leaf stage), at 152 kg N ha^-1, 39 kg P ha^-1, and 112 kg K ha^-1. The plots were rototilled (7 cm depth) in spring to incorporate fertilizer and preplant hog manure, and fall chisel-plowed (15 cm depth) to incorporate chopped corn residues. They were arranged in a completely randomized plot design. Flow volumes and nutrient levels in runoff and drainage waters were monitored year round but occurred mainly during the snowmelt (March 25-April 9), and post.snowmelt (April 10-May 13) periods. Of the total amount of water lost during snowmelt, 90 percent was in runoff, while 92 percent occurred as drainage in the post-snowmelt period. Sixty-five percent of the total annual volume of water lost was lost during these two periods as runoff and drainage. Treatments did not affect the annual snowmelt or post-snowmelt N and P loads. Total annual loads averaged 8.0 kg TKN ha^-1, 1.8 kg NH₄-N ha^-1, 43 kg NO₃-N ha^-1, 0.4 kg TP ha^-1, and 0.15 kg PO₄-P ha^-1. Spring (snowmelt and ost-snowmelt) runoff and drainage loads averaged 2.9 kg TKN ha^-1, 1.2 kg NH₄-N ha^-1, 18 kg NO₃-N ha^-1, 0.25 kg TP ha^-1, and 0.04 kg PO₄-P ha^-1, which were 40 percent to 70 percent of the yearly nutrient loads. Therefore, the hog manure management systems examined were of no greater threat to the environment than mineral fertilizers. However, spring N and P losses do represent an important part of the annual nutrient loss budget, even with conservation practices.     

Modeling Changes to Streamflow,Sediment, and Nutrient Loading from Land Use Changes Due to Potential Natural Gas Development

Natural gas development using hydraulic fracturing has many potential environmental impacts, but among the most certain is the land disturbance required to build the well pads and other infrastructure required to drill and extract the gas. We used the Soil and Water Assessment Tool (SWAT) model to investigate how natural gas development could impact streamflow and sediment, total nitrogen (TN), and total phosphorous (TP) loadings in the upper Delaware River Basin (DRB), a relatively undeveloped watershed of 7,950 km² that lies above the Marcellus Shale formation. If gas development was permitted, our projections show the DRB could experience development of over 600 well pads to extract natural gas at build out, which, with supporting infrastructure (roads, gathering pipelines), could convert over 5,000 ha from existing land uses in the study area. In subbasins with development activity we found sediment, TN, and TP yields could increase by an average of 15, 0.08, and 0.03 kg/ha/yr, respectively (an increase of 2, 3, and 15%, respectively) for each one percent of subbasin land area converted into natural gas infrastructure. At the study area outlet on the Delaware River at Port Jervis, New York, we found increases in the annual average streamflow and sediment, nitrogen, and phosphorus loads of up to 0.01, 0.2, 0.2, and 1%, respectively, for a rapid development year, and 0.08, 1.3, 2.0, and 11%, respectively, for the full development scenario. Editor's note : This paper is part of the featured series on SWAT Applications for Emerging Hydrologic and Water Quality Challenges. See the February 2017 issue for the introduction and background to the series.     

THE ATMOSPHERIC WATER BALANCE AND THE HYDROLOGY OF LARGE RIVER BASINS1

The concept of using the atmospheric water balance technique in the study of the hydrology of large (greater than 10⁵km²) river basins is described. The atmospheric water balance technique consists of determining the spacial and time distributions and fluxes of water vapor through the atmospheric volume overlying the basin. The quantity precipitation minus evaporation at the earth's surface is determined as a residual of the computation. A review of the results of various experiments employing this technique is provided. The incorporation of the technique in a study of the hydrology of a large river basin is demonstrated by showing the results of a study of the hydrometeorology of the Upper Colorado River Basin. The example covers the study of eleven winter seasons, November through April, 1957–1968. The seasonal accumulation of water over the basin as determined by the atmospheric water balance is highly correlated with annual runoff. Correlation coefficient r = .8. The daily evaporation rate during dry days varies by a factor of two over the winter season, and is shown to be related to the incoming solar radiation intensity.     

NUTRIENT LOADS TO WISCONSIN LAKES: PART I. NITROGEN AND PHOSPHORUS EXPORT COEFFICIENTS1

ABSTRACT: Export coefficients (kg/km²/yr) for dissolved ortho-phosphate (OP), total phosphorus (TP), total inorganic nitrogen (TIN), and total nitrogen (TN) were derived for watersheds in Wisconsin using data bases available for 17 basins from the U.S. Environmental Protection Agency — National Eutrophication Survey, U.S. Geological Survey, and the Wisconsin Department of Natural Resources. Three general land use categories, representative of most regions in Wisconsin, were established: forest, mixed, and agricultural. Data for the 17 basins indicated greater exports of OP. TP, TIN, and TN as the percentage of forest decreased and agriculture increased. These region-specific coefficients are compared to the values reported in the literature representing much broader areas of the U.S.     

SUSPENDED SEDIMENT PRODUCTION FROM FORESTED WATERSHEDS ON OAHU,HAWAII1

Suspended sediment from forested and agricultural watersheds was sampled over a five-year period on the island of Oahu. A variety of storm conditions were sampled, giving a measure of the extreme variability in suspended sediment production. Total annual suspended sediment from all watersheds sampled ranged from 8400 kg/km² to 617,000 kg/km². Normally, about 90 percent of the total suspended sediment was produced during less than 2 percent of the time. Suspended sediment concentrations rapidly increased during rising stream flow resulting from rain storms. Time to peak of less than two hours is common, with a similarly rapid return to prestorm conditions. The data presented indicate the great variability of suspended sediment yields, making establishment of effective standards difficult.     

Use of SWAT to Estimate Spatial Scaling of Phosphorus Export Coefficients and Load Reductions Due to Agricultural BMPS

Phosphorus export coefficients (kg/ha/yr) from selected land covers, also called phosphorus yields, tend to get smaller as contributing areas get larger because some of the phosphorus mobilized on local fields gets trapped during transport to regional watershed outlets. Phosphorus traps include floodplains, wetlands, and lakes, which can then become impaired by eutrophication. The Sunrise River watershed in east central Minnesota, United States, has numerous lakes impaired by excess phosphorus. The Sunrise is tributary to the St. Croix River, whose much larger watershed is terminated by Lake St. Croix, also impaired by excess phosphorus. To support management of these impairments at both local and regional scales, a Soil and Water Assessment Tool (SWAT) model of the Sunrise watershed was constructed to estimate load reductions due to selected best management practices (BMPs) and to determine how phosphorus export coefficients scaled with contributing area. In this study, agricultural BMPs, including vegetated filter strips, grassed waterways, and reduction of soil‐phosphorus concentrations reduced phosphorus loads by 4‐20%, with similar percentage reductions at field and watershed spatial scales. Phosphorus export coefficients from cropland in rotation with corn, soybeans, and alfalfa decreased as a negative power function of contributing area, from an average of 2.12 kg/ha/yr at the upland field scale (~0.6 km²) to 0.63 kg/ha/yr at the major river basin scale (20,000 km²). Editor's note : This paper is part of the featured series on SWAT Applications for Emerging Hydrologic and Water Quality Challenges. See the February 2017 issue for the introduction and background to the series.     

GENERALIZED WATERSHED LOADING FUNCTIONS FOR STREAM FLOW NUTRIENTS1

ABSTRACT: Loading functions are proposed as a general model for estimating monthly nitrogen and phosphorus fluxes in stream flow. The functions have a simple mathematical structure, describe a wide range of rural and urban nonpoint sources, and couple surface runoff and ground water discharge. Rural runoff loads are computed from daily runoff and erosion and monthly sediment yield calculations. Urban runoff loads are based on daily nutrient accumulation rates and exponential wash off functions. Ground water discharge is determined by lumped parameter unsaturated and saturated zone soil moisture balances. Default values for model chemical parameters were estimated from literature values. Validation studies over a three-year period for an 850 km² watershed showed that the loading functions explained at least 90 percent of the observed monthly variation in dissolved and total nitrogen and phosphorus fluxes in stream flow. Errors in model predictions of mean monthly fluxes were: dissolved phosphorus - 4 percent; total phosphorus - 2 percent; dissolved nitrogen - 18 percent; and total nitrogen - 28 percent. These results were obtained without model calibration.     

EFFECTIVE USE OF LANDSAT DATA IN HYDROLOGIC MODELS1

ABSTRACT: In a cooperative demonstration project, NASA and the U.S. Army Corps of Engineers (Corps) compared conventional and Landsat-derived land-use data for use in hydrologic models, and the resulting discharge frequency curves were analyzed. When a grid-based data-management system was used on a cell-by-cell basis (size about 1.1 acres or 0.45 hectare), Landsat classification accuracy was only 64 percent, but, when the grid cells were aggregated into watersheds, the classification accuracy increased to about 95 percent. When both conventional and Landsat land-use data were input to the HEC-1 model for generating discharge frequency curves, the differences in calculated discharge were judged insignificant for subbasins as small as 1.0mi² (2.59 km²). For basins larger than 10mi² (25.9km²), use of the Landsat approach is more cost-effective than use of conventional methods. Digital Landsat data can also be used effectively by local and regional agencies for hydrologic analysis by incorporating the data into grid-based data-management systems. The transfer of this new technology is well under way through inclusion in some Corps training courses and through use by both county government personnel and private consultants.     

“R1-R4” AND “BOISED” SEDIMENT PREDICTION MODEL TESTS USING FOREST ROADS IN GRANITICS1

ABSTRACT: Erosion and sedimentation data from research watersheds in the Silver Creek Study Area in central Idaho were used to test the prediction of logging road erosion using the R1-R4 sediment yield model, and sediment delivery using the “BOISED” sediment yield prediction model. Three small watersheds were instrumented and monitored such that erosion from newly constructed roads and sediment delivery to the mouths of the watersheds could be measured for four years following road construction. The errors for annual surface erosion predictions for the two standard road tests ranged from +31.2 t/ha/yr (+15 percent) to -30.3 t/ha/yr (-63 percent) with an average of zero t/ha/yr and a standard deviation of the differences of 18.7 t/ha/yr. The annual prediction errors for the three watershed scale tests had a greater range from -40.8 t/ha/yr (-70 percent) to +65.3 t/ha/yr (+38 percent) with a mean of -1.9 t/ha/yr and a standard deviation of the differences of 25.2 t/ha/yr. Sediment yields predicted by BOISED (watershed scale tests) were consistently greater (average of 2.5 times) than measured sediment yields. Hillslope sediment delivery coefficients in BOISED appear to be overly conservative to account for average site conditions and road locations, and thus over-predict sediment delivery. Mass erosion predictions from BOISED appear to predict volume well (465 tonnes actual versus 710 tonnes predicted, or a 35 percent difference) over 15 to 20 years, however mass wasting is more episodic than the model predicts.     

BEDLOAD TRANSPORT IN AN OREGON COAST RANGE STREAM1

Bedload transport was measured with two sampler types (vortex tube and Helley-Smith pressure differential) for three major storms at Flynn Creek, which drains a 2.2-km² forested watershed in the Oregon Coast Range. The largest flow during two winters of monitoring had a peak discharge of 0.79 m³ s^-1 km^-2, with an associated recurrence interval of ? 1.3 yr. The median particle diameter of sediment in transport was generally < 1 mm. The vortex tube and its associated sample box were relatively inefficient at trapping particles < 10 mm in diameter; however, even after transport rates were adjusted to account for sampling deficiencies of the sample box, they still averaged 42–47 percent of those obtained with the Helley-Smith sampler. Organic matter and sand sized sediments in transport also were observed to partially plug the 0.2-mm-mesh bag of the Helley-Smith sampler. Large temporal variability in bedload transport rates was measured during periods of high flow.     

MODELING THE EFFECTS OF VARIABLE ANNUAL FLOW ON RIVER CHANNEL MEANDER MIGRATION PATTERNS,SACRAMENTO RIVER,CALIFORNIA, USA1

ABSTRACT: Flow regulation impacts the ecology of major rivers in various ways, including altering river channel migration patterns. Many current meander migration models employ a constant annual flow or dominant discharge value. To assess how flow regulation alters river function, variable annual flows ‐ based on an empirical relationship between bank erosion rates and cumulative effective stream power ‐ were added into an existing migration model. This enhanced model was used to evaluate the potential geomorphic and ecological consequences of four regulated flow scenarios (i.e., different hydrographs) currently being proposed on the Sacramento River in California. The observed rate of land reworked correlated significantly with observed cumulative effective stream power during seven time increments from 1956 to 1975 (r²= 0.74, p = 0.02). The river was observed to rework 3.0 ha/yr of land (a mean channel migration rate of 7.7 m/yr) with rates ranging from 0.8 ha/yr to 5.1 ha/yr (2.0 to 13.3 m/yr), during the analyzed time periods. Modeled rates of land reworked correlated significantly with observed rates of land reworked for the variable flow model (r²= 0.78, p = 0.009). The meander migration scenario modeling predicted a difference of 1 to 8 percent between the four flow management scenarios and the base scenario.     

EFFECTS OF LAND USE AND TOPOGRAPHY ON SOME WATER QUALITY VARIABLES IN FORESTED EAST TEXAS1

ABSTRACT: Spatial variation of five water quality variables were analyzed using composite water samples collected periodically from eight small watersheds (11.4–71.6 km²) in forested East Texas during 1977 through 1980. Based on 31 observations during the four-year period the average yield of nitrate-nitrite nitrogen (NNN), total kjeldahl nitrogen (TKN), total phosphorus (PO4), chloride (CHL), and total suspended sediment (TSS) were 1.43, 21.96, 3.09, 50.11, and 90.39 ka/ha/yr, respectively. Compared to the water quality standards of the U.S. Environmental Protection Agency (1976) and the Texas Department of Water Resources (1976) for CHL, TSS, and NNN, none of the observations exceeded the limits for public water supplies. The study showed that forested watersheds normally yielded stream flow with better quality than that from agricultural watersheds. Watersheds of greater percent of pasture area, mean slope, stream segment frequency, and drainage density produced greater concentrations for these five chemical parameters in water samples. Meaningful equations were developed for estimating mean average yields for each chemical parameter for each watershed with R² ranging from 0.77 to 0.96 and standard error of estimates from 17 to 33 percent of the observed means.