# Understanding Flow Ecology Statistics: The Median of Minimums

Simple and Complex Flow Statistics
The practice of flow characterization can be thought of as having two general types of statistical calculations: simple and complex.  An example of a simple calculation, i.e., one with a single function, is the mean daily flow, where we are literally asking the question: “what is the average of all of the daily flows that ever occur in a stream?”  This can be contrasted with the calculation of something such as a 7Q10 (a common water-quality metric), where we ask “what is 7-day low-flow that we would expect to occur no more frequently than once in every 10 years?”  The 7Q10 can be considered a complex calculation because we first calculate a statistical function, the annual 7 day low flow, and then apply another function on top of that, the calculation of the frequency distribution of those 7 day low flows.  The calculation of flow-ecology metrics is also what we might consider a “complex” calculation.  These flow metrics are similar to the 7Q10 in that we are interested in the frequency distribution that describes some annually recurring flow event, but by introducing the element of timing into the analysis (month, season, etc.) we attempt to describe the coincidence of these flows with some critical life-stage of stream biota.  Ultimately, we wish to use these metrics to quantify the characteristics of the “natural flow regime” and the extent to which it has been altered (see Poff et al, 1997 for more).

Example Flow Statistic: the Median of Minimums
The basic process for calculating a complex flow statistic is to identify a statistic of interest, for example, the minimum flows occurring in each month during every year of the record.  Then, those numbers are analyzed to determine the shape of the distribution of those individual statistics, for example, the mean, median, or some quantile of interest.  In this way the flow conditions are characterized in terms of their magnitude (the monthly distribution) and the frequency of annual occurrence (see Richter, 1996 for a discussion of the Indicators of Hydrologic Alteration set of metrics).

For example, it is believed that decreases in late summer flows might negatively impact species diversity.  Table 1 shows the minimum daily flow during each August for a sample stream with a simulated  record of 10 years (2001-2010).  By applying a frequency analysis to this data set we will get a sense of how often certain flows would be expected to occur.  For example, the median of this data set is 115 cfs.  In other words, at least 1 of every 2 years we would expect to see a single  August daily flow of less than or equal to 115 cfs.  If we were to ask the question “how many times during a given organisms life span would we expect it to experience a daily flow in August of less than or equal to 115 cfs?”, we could use this to produce an estimate.  In other words, an organism with a 10 year life span could expect to see August flows below 115 cfs approximately 5 times in its life.

Table 1:  Simulated August minimum flow for a stream in the Piedmont region of Virginia.

 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Pre-Dam 134 3 486 456 227 51 106 13 122 108

Figure 1: A plot of # of taxa of benthic insectivores as a function of alteration in August minimum flow (median of min) in Virginia streams draining to the Atlantic Ocean, but outside of the coastal plain. Stream flow alteration output by VAHydro model of altered and unaltered hydrology, # of benthic insectivores calculated with EDAS database during EPA sponsored Healthy Watersheds Initiative study 2011.

Identifying Ecologically Significant Flows
In order to precisely evaluate flow-ecology relationships we would need a pre-alteration hydrologic record, a pre-alteration ecological assessment, a post-alteration flow and and a post-alteration ecological assessment.  Because of the vastness of our stream network, the cost of funding extensive ecological assessments, and the relatively short period of time during which we have been aware of the need for this type of assessment, there are actually very few streams in which this full set of data is available.  As a result, the best that we can do in the short term is to produce models of pre and post-alteration hydrology, and then compare those models to the historical record of biological observations and look for trends in the biological monitoring that might reflect the impact of flow alterations.  Figure 1 is an example of one such comparison.  Modeled alteration in August minimum low flow (the change in median of minimum August flow) is compared to a measurement of the Number of Taxa of Benthic Insectivores (we will refer to this as the NT score).  In this analysis a statistically significant decrease in the NT score was observed with both an increasing (p=0.034, R2=0.09) and decreasing (p=0.004, R2=0.34) August minimum flow.

While both of these relationships are statistically significant (as evidenced by the p-values < 0.05), there are a few reasons why we might be more likely to interpret the relationship between decreasing diversity and decreases in flow (decrease:decrease) as representing a more plausible flow-ecology connection.  Quantitatively speaking, the relationship between decreasing flows and the diversity measure (shown on the left hand side of the graph in Figure 3) had a much steeper slope, indicating that there is a more abrupt relationship between loss of water and diversity.  This decrease:decrease relationship also had a much larger R2, with a value of 0.34 suggesting that approximately 34% of the variation in diversity can be explained by changes in the August low-flow.  While the relationship between increasing August low flow and decreasing diversity (increase:decrease) is statistically significant , it’s slope is much more gradual, with the greatest decreases in diversity showing up after 60-70% increases in flow, and it’s low R2 value suggesting that even if this is a genuine relationship, its explanatory power is small (i.e., 91% of the variation in diversity is not explained by this correlation).  Perhaps most importantly, the decrease:decrease relationship is explained by a flow-ecology hypothesis, namely that decreases in late summer flows would negatively impact species diversity.  Furthermore, the increase:decrease relationship actually has several non flow-ecology hypotheses to explain it, namely, increases in low flow can be attributed to point source discharges and/or urbanization and loss of forest cover, both of which can lead to decreases in aquatic community health for water quality reasons.

Application: Assessing Impacts of Reservoir Release Rules
From a a management perspective, the above relationship suggests that in streams having greater than a 15-20% decrease in August low-flow there is substantially lower aquatic diversity, as measured by the number of taxa of benthic insectivores.  These flow ecology metrics and suspected relationships can be used to guide our choices when evaluating the operational rules for surface water management permits.  By using both operational models and flow-ecology models we can look forward in time to predict the areas where we might 1) observe some ecological degradation as a result of necessary operational policies, and/or 2) where we might seek to avoid degradation if we can possibly do so.  Table 2 shows the simulated August minimum flow for a stream in the Piedmont region of Virginia both prior to the construction of a water supply and hydro-power dam, and after the dams construction.

Table 2:  Simulated August minimum flow for a stream in the Piedmont region of, prior to and after the construction of a dam and large water withdrawal.

 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Pre-Dam 134 3 486 456 227 51 106 13 122 108 Post-Dam 107 11 366 367 170 52 99 13 98 97

Figure 2 shows a plot, and table 3 shows the data for the median-minimum monthly flow for the months of May-August for 3 management scenarios for this stream.  The blue bar represents that “baseline” or “pre-dam” scenario.  The green and red bars represent two variations on future management, the first (in green) where the rules were set to allow a 53 MGD sustained withdrawal (or “safe yield”) and the second (in red) where the rules permitted a 63 MGD sustained withdrawal with operational goals of maintaining lake levels for recreational purposes and in-stream flows for migratory species.  While both of these scenarios demonstrated significant hydrologic alteration, the change in median-minimum August flow was 15% for the 53 MGD scenario, while it was 28% for the 63 MGD scenario.  Based on the apparent threshold at 15-20% loss of August low flow and decreased downstream diversity, we can surmise that the 53 MGD scenario is preferable to the 63 MGD scenario from the standpoint of the downstream ecological community.

Table 3:

 Month Q(Pre-Dam) Q(WD=53) % of Pre-Dam Q(WD=63) % of Pre-Dam 5 399 311 78% 311 78% 6 243 159 65% 151 62% 7 167 132 79% 117 70% 8 115 98 85% 83 72%

Figure 2: Comparison of the median of the minimum monthly flows for 3 management scenarios over a 10 year period in the Appomattox River basin below Lake Chesdin. Scenario 20 (blue) is the baseline, 28 (green) is a tiered release scheme with a 53 MGD yield, and scenario 29 (red) is a tiered release schema with a 68 MGD yield.