USGS Home
Contact USGS
Search USGS

U.S. GEOLOGICAL SURVEY
Scientific Investigations Report 2007–5117

Back to Table of Contents

Information-Theoretic Approach to Multivariable Analysis of Water-Quality Conditions

Although the exploration of the univariate relations proposed by the hypotheses yielded no direct correlations between lake level and water-quality conditions, it was suspected that some combination of variables, including lake level, was affecting water-quality conditions. To examine these more complex interrelations, multivariable analyses were performed and then assessed using an information-theoretic approach.

The analyses were designed to examine the effect of multiple physical and biological variables on some measure of water-quality conditions. July and August were chosen for further exploration because poor water-quality conditions commonly occur during this time period and create stressful conditions in the lake for the suckers. Four lakewide water-quality measures were evaluated—the percentage of dissolved-oxygen concentrations less than 4 mg/L for July and August (table 7) and the percentage of pH values greater than 9.7 for July and August (table 9). The explanatory variables included in the analysis were peak chlorophyll‑a concentrations as a measure of the strength of the bloom, degree-days from April 1 to May 15 as a measure of how warm the spring was, 75^th-percentile water temperature, median October–May discharge in the Williamson River, median monthly wind speed measured at Klamath Falls airport, and median monthly lake level in Upper Klamath Lake.

Poor water-quality conditions related to low dissolved-oxygen concentrations or high pH values are expected to be a result of bloom dynamics. Therefore, the explanatory variables selected for further examination were chosen because of their observed or theorized effects related to the occurrence, strength, or duration of the algal bloom. Peak chlorophyll‑a concentration is an obvious water-quality variable to include when considering conditions related to the bloom, and was included in the analysis as a measure of the strength of the bloom. Kann and Smith (1999) have proposed that a reduction in chlorophyll‑a concentration in the lake from 200 µg/L to 100 µg/L would decrease the probability of potentially harmful pH values of greater than 9.5 by about half. Welch and Burke (2001) demonstrated that the growth and strength of the bloom are strongly affected by available light, air temperature, and indirectly lake level as it relates to these two variables. Unfortunately, long-term cloud-cover data were not available to assess the effects of available light in this analysis. Long-term air temperature data, however, were available, and cumulative degree-days were calculated from April 1 to May 15 as a measure of springtime warming. Kann (1998) proposed using cumulative degree-days for this same time period as an index of lake warming. To this extent, water temperature was included in this analysis as a direct measure of lake warming.

Nutrient inputs to the lake are recognized as an important factor affecting the growth and development of the bloom. The Williamson River accounts for about half of the inflow to Upper Klamath Lake and transports nutrients from the surrounding landscape to the lake with this inflow. Wood and others (2006) found a correlation between ammonia concentrations in the lake and the Williamson River discharge from the previous October–May, representing the period of nutrient loading to the lake before the onset of the bloom. Therefore, the mean October–May discharge from the Williamson River was included in this analysis to represent the effect of nutrient loading on bloom dynamics. Another source of nutrient loading to consider because of the potential effect on bloom strength is increased phosphorus loading in the lake itself due to bed-sediment resuspension. Low lake levels combined with summer wind conditions were shown by Laenen and LeTourneau (1996) to affect as much as 75 percent of the lake with bed-sediment resuspension, and consequently increased phosphorus loading.

As discussed earlier, wind speed is suspected to be an important factor affecting water-quality conditions in Upper Klamath Lake, but good-quality wind-speed data representing conditions on the lake itself is lacking. Kann and Welch (2005) noted that the severity of both low dissolved oxygen and high ammonia was related to water-column stability, which was dependent on wind speed. An inverse correlation between wind speed and the surface-to-bottom difference in dissolved oxygen averaged over July–August was reported for profile data for data collected from 1990 to 2000. Wood and others (2006), however, found insignificant or weak correlations between daily wind speed and daily minimum dissolved oxygen for the 2002–05 data collected by USGS. They reported that wind speed did appear to influence the degree of stratification in the water column, but was found to have has less influence over when the lowest dissolved-oxygen concentrations occurred. Therefore, even though the wind-speed data available for inclusion in this analysis was not ideal, it was considered as a factor for this analysis.

Lake level, being the only factor of those included in this analysis that can be directly controlled by Bureau of Reclamation, has already been discussed in this report as a factor related to water-quality conditions. The 2002 Biological Opinion for the Lost River and shortnose suckers (U.S. Fish and Wildlife Service, 2002), acknowledges “evidence that water levels directly or indirectly affect factors that affect water quality, and that water quality impacts suckers.” Barbiero and Kann (1994) have applied these lake level considerations to the recruitment of algal cells from the sediments, which has been shown to be an important contributor to water column biomass increases in AFA. It is proposed that greater light intensity at the sediment surface, as a result of low lake levels in early spring, may speed sediment recruitment.

Model sets were developed with the first five explanatory variables (peak chlorophyll-a concentration, degree-days, water temperature, Williamson River discharge, and wind speed) using different combinations of one variable, two variables, three variables, and so on. These model sets were regressed against each water-quality measure, and then each of these model combinations was repeated including lake level as an additional explanatory variable in an effort to assess the effect of lake level on water-quality conditions. These multiple regression models also were compared to each other as a means of explaining the variation observed between years in water-quality conditions in the lake.

The Akaike Information Criterion (AIC_c), corrected for small sample size, and the corresponding Akaike weights (w_i) were used to evaluate the relative likelihood of each multiple regression model (Burnham and Anderson, 2002). The AIC_c statistic provides a relative measure and is designed to compare “sets” of models to each other and is not designed to analyze a model independently. Models with a smaller AIC_c value fit the data better and do a better job of explaining the variance observed in the dataset. The AIC_c statistic also is designed to reward parsimony, such that the reduction in the residual sum of squares (RSS) that is obtained by adding an explanatory variable to a model will only result in a lower AIC_c statistic if the reduction is more than enough to compensate for the degree of freedom that has been added to the model. For all four water-quality measures tested and for all possible variable combinations, two possible models were generated—one including lake level as an explanatory variable and one without lake level. In every instance, the addition of lake level to the regression resulted in an increase in the AIC_c value of somewhere between 2 and 5, indicating that the model without lake level as a variable offered a better fit of the data. In other words, the addition of lake level does not explain enough additional variance in the observed water-quality condition to justify its inclusion as an additional explanatory variable, relative to the same model without lake level as an explanatory variable.

The generation of these multiple models with varying numbers of explanatory variables yielded no single overarching equation to explain the variation in water-quality conditions observed between years. For each water-quality measure tested, the models with the lowest AIC_c statistics, and therefore the best fit, were the models that considered each physical variable individually. For example, the model set considering water temperature alone was a better fit for explaining the variance observed in the percentage of June dissolved-oxygen concentrations of less than 4 mg/L than the models of water temperature and any other explanatory variable. These univariate models were examined as a set for each water-quality measure (percent DO less than 4 mg/L and percent pH greater than 9.7, July and August) and the Akaike weights were calculated to provide more insight into each variable’s role in the larger context of water-quality conditions in the lake (tables 14 and 15).

These models do not reveal the presence or absence of a statistically significant relation between the variables as determined by the acceptance or rejection of a null hypothesis of no relation, but rather provide insight into variables that help to explain the variance observed in the water-quality parameters. For instance, for the occurrence of low dissolved-oxygen concentrations in July, wind speed appears to be the best variable of those examined at explaining the variance observed in the data. The Akaike weight of 0.49 for the wind speed model, relative to the weights of all the other univariate models, which range from 0.02 to 0.39 (table 14), indicates that there is a 49‑percent chance that wind speed is the best model of the set. The next best model, with an Akaike weight of 0.39, reveals that water temperature also may help explain the observed variance. By removing the wind-speed model from the first set and reevaluating the remaining 5 models, the water-temperature model does rise to the top (with an Akaike weight of 0.76), as the next best model for explaining the variance in the July dissolved-oxygen data. Likewise for the occurrence of low dissolved-oxygen concentrations in August, water temperature and wind speed play important roles.

Water temperature emerges as an important variable in explaining the variance observed in the occurrence of high pH values in July and August as well (table 15), based on the Akaike weights of that model (0.48 in July and 0.35 in August) relative to the rest of the models in the set, which varied from 0.08 to 0.17 in July and from 0.06 to 0.20 in August. Removing the water-temperature model from the first set and reevaluating the remaining five models, reveals that the wind-speed model for July (with an Akaike weight of 0.32) and the peak chlorophyll‑a concentration model for August (with an Akaike weight of 0.31) were the next best models. The Akaike weights for the remaining variables were fairly evenly distributed, indicating that there was no clear hierarchy of importance among those variables.

These analyses were based on water-quality measures calculated from the lakewide dataset. To explore whether the explanatory variables could be defined more clearly for a localized measure of water quality, similar analyses were performed to examine the minimum dissolved-oxygen concentrations measured at Midnorth during the July-August time period (table 16). Midnorth was chosen because it was shown to be an important indicator site for water quality in the adult sucker habitat located in the northern part of the lake. Wood and others (2006) found that, at the Midnorth site, “die-off years could be successfully identified in the historical data by screening for water characterized by exceptionally low chlorophyll‑a concentration, exceptionally low dissolved-oxygen concentration throughout the water column (not just near the bottom), and exceptionally high ammonia concentration and water temperature, just prior to or coincident with the start of a fish die-off.” As with the lakewide analysis, the best models were those with just one explanatory variable, and water temperature provided the best fit when examining the individual parameter models. Removing water temperature and reevaluating revealed that lake level also played an important role. The relation with water temperature was explored further by examining a set of models that included water temperature along with each of the other variables (table 17). The best model was still the one with water temperature alone.

The implication that wind speed may be an important variable in predicting the occurrence of low dissolved-oxygen concentrations was noted in the lakewide analysis, but was not apparent for the Midnorth-only dataset. A possible explanation for this may be that wind speed does not affect all of the lake equally. For example, lower wind speeds increase residence time in the trench and can result in low dissolved-oxygen concentrations there, where oxygen consumption exceeds photosynthetic production, but Wood and Cheng (2006) showed that another consequence of low wind speeds is that they are less effective at pushing water from the trench into the northern part of the lake. This would complicate the relation between water quality and wind speed at the Midnorth site, particularly in comparison to the trench, where the lowest dissolved-oxygen concentrations in the lake often occur. These mechanisms, along with the fact that the wind-speed data used in this analysis was of questionable quality and perhaps not very representative of actual wind conditions on the lake, may make wind speed less effective as an explanatory variable.

Because this report has focused on the relation of water-quality conditions and lake level, and lake level was revealed as perhaps an important explanatory variable of the variance observed in the minimum dissolved-oxygen concentrations at Midnorth in July–August, models with lake level were explored further (table 18). Model sets were developed to compare lake level with each physical or biological variable used in this analysis, and Akaike weights were calculated. These sets revealed that the univariate lake-level model always was a better fit than when combined with some other variable, except for the univariate water-temperature model, which provided a better fit than lake level alone. So, water temperature and lake level do appear to be important variables in explaining the variance observed in minimum dissolved-oxygen concentrations at Midnorth. Wood and Cheng (2006) did not address the question of how lake level affects transport from the trench into the northern part of the lake, so the emergence of lake level as an alternative second-most-important explanatory variable remains as a question to be explored with further modeling work.

Although water temperature and wind speed appear to be important explanatory variables when examining the variance observed in different water-quality measures throughout the lake, no one overarching variable or combination of variables was revealed. It is suspected that these variables all play important roles in affecting water quality in Upper Klamath Lake to varying degrees and at varying times based on the overall interrelation between these influencing factors. This conclusion makes sense when considering that no clear pattern has emerged from multiple approaches to examining interrelations among water-quality conditions and lake level and climatic factors. No two years from the 17 years considered in this analysis were exactly the same for any of the considered variables, and poor water-quality conditions occur at the same time in every year. This analysis does reveal that many of these variables are important, and perhaps with better resolution of the datasets (wind-speed data for conditions affecting the lake itself and continuous water temperature rather than measurements every two weeks, for instance) and more examples of the interrelations to examine (more years of data), the relations can be further refined.

Back to Table of Contents