David J. Holtschlag

This report describes an investigation of visible light reflectances from Torch Lake in Antrim County, Michigan. The oligotrophic lake is the largest inland lake in Michigan by volume and the second largest by surface area. Local residents have expressed concern that a recent, on-going proliferation of golden-brown algae may be impacting the water quality and aesthetics of the lake. This report examines the seasonal and long-term variations in reflectance intensities of visible blue, green, and red light bands measured by the Landsat series of satellites from 1984 through 2023. To provide a basis for assessing possible changes, a comparable analysis was conducted for two regions having water depths greater than 100 m in Grand Traverse Bay, which lie just west of Torch Lake. Results for Torch Lake indicate that the seasonal response for the more photosynthetically active blue and green bands were similar to each other, both having a secondary peak in late August. The amplitudes of these seasonal responses were greater for water depths greater than 50 feet. In contrast, the red band for Torch Lake had a minimum in late August, which is comparable with the late August minimums in all three bands in Grand Traverse Bay. Minor seasonal differences were detected between water depths in the red band. The local levels for the blue and green bands of Torch Lake generally follow a sigmoid curve, which maintain a plateau of positive values until about 2010. After 2010, levels decline precipitously, especially in the blue band, before leveling off in 2020. Amplitudes of these trends vary more at shallower water depths. Local levels in the red band of Torch Lake and all bands of Grand Traverse Bay tend to decrease sinusodially over time. Differences between seasonal and trend responses between bands and regions of interest may indicate differences in underlying physical, chemical, or biological processes.

A generalized additive model (GAM) and three generalized additive mixed-effect models (GAMMs) were developed and compared to describe variations of surface reflectances of visible light from Torch Lake in the blue, green, and red bands from Sentinel 2 satellite imagery. All models included fixed effects associated with water-depth intervals, satellite orbits, trend, and smooth effects associated with season by water depth, and a measure of cloudiness. The GAMM models also included random effects of strata or timing, and spatial or temporal correlation effects. The analysis was based on 287 European Space Agency’s Sentinel 2A, 2B, and 2C satellite images of Torch Lake in Antrim County, Michigan, obtained from 2019 to 2025. The approach is motivated by the recognition of multiple sources of randomness that affect model uncertainty, where, at least one of which, is separable. Primary sources of randomness include day-to-day variations in surface reflectances from the water surface and sensor measurement uncertainties. A separable source of randomness is the uncorrected atmospheric distortions through which light passes from the sun to the lake surface and back to the satellite sensors. Results indicated that the GAMM models with random effects were more accurate than the GAM model without them. The preferred GAMM model accounted for error correlation of median reflectances in 60 strata spanning the surface of Torch Lake and a random temporal intercept component associated with the uncertainties in corrections for daily atmospheric conditions. Seasonal changes in reflectances within the 0-5 ft water-depth interval had lower magnitudes and were out of phase with seasonal variability in the three deeper water-depth intervals. Changes in reflectances in shallow areas may be more indicative of changes in reflectance within benthic areas, whereas changes in deeper areas may be related to dissolved and suspended materials in the water column. The uncertainty of shallow-water depths degrades the interpretability of surface reflections from strata in shallow waters.

Relations between stage (water level) and discharge of streamflow through a natural channel are the result of time-varying processes, which are commonly described by time-varying stage-discharge ratings. Hydrographers with the U.S. Geological Survey successfully maintain the accuracy of streamflow data by manually applying time-tested approaches to adapt ratings to temporal changes in hydraulic conditions. The difficulty with the manual approach is that it is a subjective, time-consuming process that requires considerable skill and experience to implement. In addition, manual adjustments of ratings make quantification of resulting streamflow data uncertainties problematic. A computer-aided adaptive stage-discharge estimation approach is proposed to track sequential changes in the relation between stage and discharge at continuous-record streamgages. In this report, adaptations are based strictly on discrete measurement data that are then used to compute the magnitudes and uncertainties of streamflow. The approach entails the parameterization of a cubic regression spline (CRS) for the stage-discharge relation based on an existing rating or on a set of discrete measurements. A state-space model is then parameterized to track temporal changes in stage-discharge relations beginning with the initial CRS parameterization using discrete measurements. Finally, Kalman estimation is used with the state-space model to estimate the magnitude and uncertainty of flows. In a case study using data from streamgage U.S. Geological Survey 04122500 Marquette River at Scottville, Michigan, a five-parameter CRS model was estimated from data in an existing stage-discharge rating to provide an initial CRS parameter set for a state-space model. The initial CRS parameters were updated sequentially in a state-space model based on periodic discrete measurements of stage and discharge that spanned a 30-year period for this analysis. Additional analysis is needed to determine the timing of rapidly varying shifts more precisely in stage-discharge relations than the relatively infrequent discrete measurements currently enabled. Unit streamflow estimates based on flow in a local streamgaging network may provide a basis for adapting a stage-discharge rating at unit time intervals by augmenting discrete measurement data within the state-space model.

Median summer water yields and resultant flows for streams are used in Michigan to regulate large water withdrawals to help prevent negative effects on characteristic fish populations. Large water withdrawals commonly are associated with irrigation in rural areas. In an earlier statewide report, an index-flow statistic for the period of record, defined as the median flow during the summer month of lowest flow, was used to characterize median summer flows and associated water yields. In this report, the annual series of median summer water yields for the period July 1 through September 30 within the period of record is used to characterize median summer water yields. For 27 streamgages included in both reports, the average index water yield was at the 37th percentile of the distribution of median summer water yields. In contrast to an index statistic, an annual time series provides a basis for detecting trends in median summer water yields and for determining basin, climatic, and irrigation factors affecting spatial and temporal variations in summer water yields. Daily flow data from 40 selected U.S. Geological Survey streamgages in southwestern Michigan were used in this analysis. Two mixed models were identified to estimate median summer water yields based on fixed basin characteristics and temporally varying climatic factors for 1945–2015. No irrigation data were available prior to 1970, so no irrigation variables were included in the mixed models for 1945–2015. Then, two mixed models were developed for 1970–2015, a period in which a partial annual series of county-level irrigation data also were available. One of the 1970–2015 mixed models provides a basis for estimating median summer water yields at sites in southwestern Michigan using an estimated trend component, and selected basin, climatic, and irrigation factors. Re-estimation of model parameters in this mixed model with more spatially precise information on irrigation withdrawals may improve model accuracy.

First posted August 6, 2018 Director, Michigan Water Science Center U.S. Geological Survey 6520 Mercantile Way, Suite 5 Lansing, MI 48911 An exploratory discrete Bayesian network (BN) was developed to assess the potential of this type of model for estimating the magnitudes and uncertainties of an arbitrary subset of unmeasured water-quality parameters given the measured complement of parameters historically measured at a U.S. Geological Survey streamgage. Water-quality data for 27 water-quality parameters from 596 discrete measurements at U.S. Geological Survey streamgage 03374100 White River at Hazleton, Indiana, were used to develop this BN. Data for each of the water-quality parameters were discretized into five intervals based on the quintiles of the measured values. The 596 discrete measurements were randomly partitioned into a training set with 80 percent of the data and a testing set with 20 percent of the data to identify, estimate, and assess the training and testing accuracy of the Bayesian network. A BN with 28 nodes was formed from the 27 water-quality parameters and the month of sample collection. Based on data in the training set, a network with 53 directed edges and month as the target node was identified by minimizing the negative log-likelihood function for all nodes treated, in turn, as the target variable. The edge structure determines the number and magnitude of elements in conditional probability tables associated with all nodes. The effectiveness of the BN was assessed on the basis of correct classification rates to one of the five discrete intervals, which were computed separately for the training and testing datasets and for two conditioning variable sets. The selected sets of conditioning variables represent two of many possible sets of measured parameters on which to base estimates of unmeasured parameters. The first set includes only the month of sample collection (month), and an expanded set includes month and six other continuously measurable parameters, referred to as the ContMeasSet, all of which were obtained from the discrete data. Results indicated that the training dataset had average correct classification rates of 41.7- and 61.2-percent rates conditioned on the month and ContMeasSet sets, respectively. The testing dataset had somewhat lower average correct classification rates of 40.8 and 56.5 percent for the two conditioning variable sets. When conditioned on month only, the average correct classification rate for the testing dataset was only slightly lower than the average correct classification rate in the training dataset, indicating little model overfitting. When using the ContMeasSet, however, the average decrease in accuracy between training and testing sets was 4.9 percent. The training and testing datasets and both sets of conditioning variables, however, indicate that the BN would substantially outperform a random assignment model, which would be expected to have a 20-percent correct classification rate. In addition, the edge structure of the BN depicts how information can flow through the network, which may help prioritize parameters for measurement to facilitate estimation of unmeasured parameters. Finally, extension of a static BN, like the one developed in this report, to a dynamic BN may provide a basis for using high-frequency or continuous water-quality data to extend information in time between discrete water-quality samples, and this integration could mitigate some of the limitations of high-frequency and discrete water-quality sampling methods.

First posted July 26, 2018 Upper Midwest Water Science Center U.S. Geological Survey 6520 Mercantile Way Suite 5 Lansing, MI 48911 The UFINCH (Unit Flows In Networks of Channels) computer application can be used to simulate daily and unit flows in networks of streams based on geospatial data in the National Hydrography Dataset NHDPlus (with value added attributes), and U.S. Geoogical Survey daily streamflow data from a downstream (or base) streamgage. Among streamflow augmentation methods, UFINCH has the unique capability to estimate time series of flows from a single base (downstream) streamgage to many upstream reaches, while conserving flows within the basin. UFINCH also provides a simple statistical model to adjust simulated flows to better match continuous flows from data at an upstream streamgage. Parameters of the statistical model are estimated using overlapping periods of record at the two streamgages, but the adjustment can be applied to all years of record available at the base streamgage. This report describes the main features of UFINCH and presents results from a sample application. Interactive graphical user interfaces and automated geographical information processing facilitate flow-data retrievals provide an intuitive environment for efficient and effective generation of flow information in a network. UFINCH is coded in the Matlab programming language and can be run in the Matlab programming environment, with supporting statistical, optimization, and mapping toolboxes, or from compiled code on a Microsoft Windows computer.

Two-dimensional hydrodynamic and transport models were applied to a 34-mile reach of the Ohio River from Cincinnati, Ohio, upstream to Meldahl Dam near Neville, Ohio. The hydrodynamic model was based on the generalized finite-element hydrodynamic code RMA2 to simulate depth-averaged velocities and flow depths. The generalized water-quality transport code RMA4 was applied to simulate the transport of vertically mixed, water-soluble constituents that have a density similar to that of water. Boundary conditions for hydrodynamic simulations included water levels at the U.S. Geological Survey water-level gaging station near Cincinnati, Ohio, and flow estimates based on a gate rating at Meldahl Dam. Flows estimated on the basis of the gate rating were adjusted with limited flow-measurement data to more nearly reflect current conditions. An initial calibration of the hydrodynamic model was based on data from acoustic Doppler current profiler surveys and water-level information. These data provided flows, horizontal water velocities, water levels, and flow depths needed to estimate hydrodynamic parameters related to channel resistance to flow and eddy viscosity. Similarly, dye concentration measurements from two dye-injection sites on each side of the river were used to develop initial estimates of transport parameters describing mixing and dye-decay characteristics needed for the transport model. A nonlinear regression-based approach was used to estimate parameters in the hydrodynamic and transport models. Parameters describing channel resistance to flow (Manning’s “n”) were estimated in areas of deep and shallow flows as 0.0234, and 0.0275, respectively. The estimated RMA2 Peclet number, which is used to dynamically compute eddy-viscosity coefficients, was 38.3, which is in the range of 15 to 40 that is typically considered appropriate. Resulting hydrodynamic simulations explained 98.8 percent of the variability in depth-averaged flows, 90.0 percent of the variability in water levels, 93.5 percent of the variability in flow depths, and 92.5 percent of the variability in velocities. Estimates of the water-quality-transport-model parameters describing turbulent mixing characteristics converged to different values for the two dye-injection reaches. For the Big Indian Creek dye-injection study, an RMA4 Peclet number of 37.2 was estimated, which was within the recommended range of 15 to 40, and similar to the RMA2 Peclet number. The estimated dye-decay coefficient was 0.323. Simulated dye concentrations explained 90.2 percent of the variations in measured dye concentrations for the Big Indian Creek injection study. For the dye-injection reach starting downstream from Twelvemile Creek, however, an RMA4 Peclet number of 173 was estimated, which is far outside the recommended range. Simulated dye concentrations were similar to measured concentration distributions at the first four transects downstream from the dye-injection site that were considered vertically mixed. Farther downstream, however, simulated concentrations did not match the attenuation of maximum concentrations or cross-channel transport of dye that were measured. The difficulty of determining a consistent RMA4 Peclet was related to the two-dimension model assumption that velocity distributions are closely approximated by their depth-averaged values. Analysis of velocity data showed significant variations in velocity direction with depth in channel reaches with curvature. Channel irregularities (including curvatures, depth irregularities, and shoreline variations) apparently produce transverse currents that affect the distribution of constituents, but are not fully accounted for in a two-dimensional model. The two-dimensional flow model, using channel resistance to flow parameters of 0.0234 and 0.0275 for deep and shallow areas, respectively, and an RMA2 Peclet number of 38.3, and the RMA4 transport model with a Peclet number of 37.2, may have utility for emergency-planning purposes. Emergency-response efforts would be enhanced by continuous streamgaging records downstream from Meldahl Dam, real-time water-quality monitoring, and three-dimensional modeling. Decay coefficients are constituent specific.