ESTIMATING JOINT FLOW PROBABILITIES AT STREAM CONFLUENCES USING COPULAS

Transcription

1 ESTIMATING JOINT FLOW PROBABILITIES AT STREAM CONFLUENCES USING COPULAS Roger T. Kilgore, P.E., D. WRE* Principal Kilgore Consulting and Management 2963 Ash Street Denver, CO David B. Thompson, Ph.D., P.E., D.WRE R.O. Anderson Engineering 1603 Esmeralda Minden, NV Submitted July 29, 2010 *Corresponding Author

2 ABSTRACT Highway drainage structures are often located near the confluence of two streams where they may be subject to inundation by high flows from either stream. These structures must be designed to meet specified performance objectives for floods. Because the flooding of structures on one stream can be affected by high flows on the other stream, it is important to know the relationship between the joint exceedance probabilities on the confluent stream pair, that is, the joint probability of the coincident flows. The objective of the research summarized in this paper was to develop practical procedures for estimating joint probabilities of design coincident flows at stream confluences and guidelines for applying the procedures. The scope was limited to riverine areas and did not include coastal areas. Two practical strategies emerged from the analyses. One is a strategy for determining a set of exceedance probability combinations associated with the desired joint probability for design based on copulas. The second is a series of conditional probability matrices for use in the total probability method. Only the copula strategy is addressed in this paper. The Gumbel-Hougaard copula method performed better than the other three methods. Therefore, it is the recommended bivariate method. However, the designer is not required to have a background in statistics because the guidance has been transformed to a series of tables. An example application of the copula-based procedure is included in the paper. Kilgore and Thompson i

3 INTRODUCTION The joint probability question arises when a structure is located where the hydraulic behavior of some combination of the main stem and tributary may result in critical hydraulic design conditions. Specifically, there is a portion of the tributary stream that is influenced by the discharge of the tributary stream and the backwater caused by the main stem stream. This is referred to as the influence reach. The location of the structure within the influence reach, as well as the joint hydrologic behavior of the confluent streams, will determine the importance of the confluent streams on the appropriate design conditions for the structure. The definition of influence reach and three variations of relative importance of the design location - A) structure very close to the confluence; B) structure some distance, x, from the confluence; and C) structure beyond the influence reach - are illustrated in Figure 1. The influence of the main stem and tributary varies depending on the design location and the watershed characteristics. For a given distance from the confluence, x, there is a unique maximum for stage, y, for a given probability of exceedance at the design location. Defining the following variables: x = distance from confluence X max = maximum range of the zone of influence y = stage Q M = discharge on the main stem Q T = discharge on the tributary Influence reach MAIN x Xmax TRIB A B C 41 FIGURE 1 Problem Definition Schematic. Kilgore and Thompson 1

4 For the three cases, the stage may be stated as a function of the following: Case A: x/x max is close to zero. Stage is a function of the sum of the two flows. y A = f(q M +Q T ) (1) Case B: 0 < x/x max < 1. Stage is a function of the sum and the tributary flow. y B = f(q M +Q T, Q T ) (2) Case C: x/x max >1. Stage is only a function of the tributary flow. y C = f(q T ) (3) For Case A, the structure is sufficiently close to the confluence so that the stage is determined by the sum of the flows from the main stem and tributary. This case essentially reduces to an analysis of the flow just downstream of the confluence; therefore, this case is not of great interest for this paper. Case B represents the joint probability problem and the primary focus of this paper. The objective in this case is to find the stage, y B, which corresponds to the exceedance probability appropriate for the design. The stage at location B at any time is a function of the flow at the confluence, which establishes the downstream control elevation, and the flow in the tributary, which determines the water surface profile from the downstream control to the design location. For Case C, the structure is outside of the influence reach and is not representative of the joint probability problem so there will be no further discussion of this case. This research addresses the joint probability of Q T and Q M as shown in Figure 2. This is described as: where, ' QM Q M and ' QT Q T (4) ' Q M = Threshold discharge for defining an event on the main stream ' Q T = Threshold discharge for defining an event on the tributary Characteristic of the exceedance probability isoline is that all points on the line have the same joint exceedance probability. However, a designer will apply estimated discharges to determine an appropriate water surface elevation or velocity to consider in the design. Although several combinations of discharge pairs may have the same exceedance probability, the one that should be used for design will depend on the site location within the influence reach. Finding that critical combination for a given site may involve selecting two or more potential combinations from the appropriate isoline and selecting the one that exhibits the most severe consequences at the site. It is assumed in the characterization of the problem that the design location and, therefore, the influence reach are on the tributary (smaller) watershed. However, the analysis is analogous if the design location is on the main stem. Earlier assessments of joint probability focused on characterization of the issues (1). The theoretical foundation for joint probability and a practical application using copulas are briefly described in this paper. Kilgore and Thompson 2

5 Tributary Peak Discharge, QT Exceedance probability isoline Main Stem Peak Discharge, Q M Potential critical combinations FIGURE 2 Isoline for Annual Bivariate Exceedance Probability. Several general types of strategies were explored for their suitability in addressing the objective. They were: 1. Bivariate probability distributions 2. Univariate probability distributions with linking copulas 3. Total probability theorem 4. Regression analyses 5. Marginal analysis 6. Synthetic storm cell/runoff modeling 7. Tabular summaries Preliminary evaluations eliminated all but the first three strategies. The NCHRP research report provides discussion of all of the strategies (2). This paper is focused on univariate probability distributions with copulas. DATABASE Three databases were developed and used for the research. The first was a collection of confluent gage pairs from the coterminous United States. This was the primary database for developing and testing the strategies and included 85 gage pairs located throughout the coterminous United States. The second database included watershed and meteorological data associated with the confluent gage pairs. A third database is comprised of instantaneous flow records from the USGS Instantaneous Data Archive. The foundation of the analyses was the daily data, primarily because the instantaneous data records were not available across the country and the record lengths were too short, in most Kilgore and Thompson 3

6 cases, to be useful. However, there was significant concern that use of daily data would obscure statistical relationships, particularly correlative relations between confluent watersheds. To illustrate the concern, two different flood events are displayed in Figures 3 and 4. In Figure 3, the instantaneous data trace (solid lines) and the corresponding daily flow values (symbols) for a flood event in October of 2005 are plotted for both gages from gage pair 06. Gage pair 06 is located in New Jersey. The drainage areas for gages and are 36.7 and 106 mi 2, respectively. Agreement between the instantaneous and daily data is good. By contrast, the analogous data for a flood event in October 1998 for gage pair 43 are plotted in Figure 4. Gage pair 43 is located in Washington. The drainage areas for gages and are 70.3 and 133 mi 2, respectively. In this case the daily averaging greatly affects the magnitude, though not necessarily the timing. It is expected, especially for smaller watersheds, that daily data will smooth the flood hydrographs. However, it is of greater importance to preserve the correlation relations for estimating the probabilities of confluent flooding. Several approaches were taken to evaluate the validity of using daily data. One approach was to correlate the annual peak series data from instantaneous data with the annual peak series data from mean daily data. The instantaneous annual peak series data is the traditional series prepared by the USGS. The annual peak series from mean daily data were derived from the daily flow record as part of this study. The correlation is assessed through two quantitative measures, Pearson s ρ and Kendall s τ. This evaluation was completed for five gage pairs that included the smallest watershed in the study database. The results of these analyses, plotted against drainage area, are summarized in Figure 5. For Pearson s ρ, the correlation is 0.8 or greater for all of the gages except for the two smallest watersheds. Similarly, Kendall s τ is 0.7 or greater for all but the two smallest watersheds. Although climate and other factors will influence these observations, the data suggest that the annual series based on daily data is highly correlated with annual series based on instantaneous data for watersheds greater than 6.5 square miles. Lower correlations, though not appreciably lower in some cases, are observed for smaller watersheds. Kilgore and Thompson 4

7 Flow (cfs) /8/05 10/10/05 10/12/05 10/14/05 10/16/05 10/18/05 10/20/ Daily Daily 130 FIGURE 3 Gage Pair 06 October 8-20, Flow (cfs) /15/88 10/16/88 10/17/88 10/18/88 10/19/88 10/20/88 10/21/88 10/22/ Daily Daily FIGURE 4. Gage Pair 43 October 15-22, Kilgore and Thompson 5

8 Correlation Drainage Area (mi 2 ) Pearson's Rho Kendall's Tau FIGURE 5 Daily Instantaneous Correlation versus Drainage Area. COPULA TECHNOLOGY Copulas are a relatively new approach to bivariate problems in hydrology. The term copula refers to a function, called the dependence function, used to link two univariate distributions in such a way as to represent the bivariate dependence between the two random variables. The potential of a copula is realized in that the copula is independent from the form of the univariate marginal distributions. Therefore, the marginal distributions can be chosen such that they provide a best fit of the univariate random variables, with the copula used to model the dependence behavior. Many copulas are available for application to bivariate random variables. Those of interest to hydrologists typically fall into the Archimedean family of copulas. For two random variables, X and Y with cumulative distribution functions of F X (x) and F Y (y), respectively, define U = F X (X) and V = F Y (Y). Then, U and V are uniformly distributed random variables and u will denote a specific value of U and v will denote a specific value of V. The Gumbel-Hougaard and Frank copulas were examined in detail for addressing the joint probability problem. As is described later, the Gumbel-Hougaard was ultimately selected because it fit the gage pair data best. The Gumbel-Hougaard copula is described as follows (3): C 1 θ θ { [ ] } θ [ 1 ] θ ( u,v ) = exp ( ln(u) ) + ( ln(v) ) θ, where, C θ (u,v) = bivariate distribution function for the copula θ = the dependence parameter u,v = univariate nonexceedance probabilities The relation between Kendall s τ and θ is given by 1 τ = 1 θ (6) Copulas (and bivariate distributions) provide a relation between the marginal distributions in the form of exceedance probability isolines. An example for gage pair 06 and the 10-percent probability of exceedance is shown in Figure 6. The 10-percent exceedance probability isolines are shown for the bivariate Gumbel (green), bivariate log-normal (blue), Gumbel-Hougaard copula (black), and Frank copula (red). Each point on the isoline represents a (5) Kilgore and Thompson 6

9 percent exceedance probability. The values on the abscissa and ordinate are the marginal distribution nonexceedance probabilities for the main and tributary and tributary streams FIGURE 6 10-percent Nonexceedance Probability for Gage Pair 06. The Gumbel-Hougaard copula was recommended for use based on evaluation of the fit to observed data in the 85 gage pair database. The root mean squared error (RMSE) was adopted to compare the four strategies. Because the focus on this study is on the extreme events, only the events with a probability of exceedance less than 0.5 were considered for this exercise. The RMSE results are summarized in Table 1 for each of the four strategies for both the peak on main and peak on tributary datasets. Except for the bivariate Gumbel strategy, all 85 gage pairs are represented in the analysis. (The bivariate Gumbel is limited to paired data where the Pearson s ρ is less the two-thirds, meaning this distribution was not defined for those gage pairs with a higher correlation.) Two separate datasets were evaluated. The Peak on Main (POM) dataset was developed for each gage pair taking the annual peak series (daily data) on the main stem gage and pairing it with the coincident flow (same day) on the tributary. Similarly, the Peak on Tributary (POT) dataset was developed taking the annual peak series on the tributary gage and pairing it with the coincident flow on the main stem. For both the POM and POT datasets, the Gumbel-Hougaard copula results in the lowest average error of the four strategies, indicating the best fit. The same result is observed for the median and minimum errors. For the maximum error, the table reports that the bivariate Gumbel is the lowest for both the POM and POT datasets followed by the Gumbel-Hougaard. However, by inspection of the data it is revealed that the one RMSE error (gage pair 48) is the largest for the Gumbel-Hougaard method, but is undefined for the bivariate Gumbel method because the Pearson s ρ is greater than two-thirds. Therefore, the RMSE analyses show that the Gumbel- Hougaard copula is the best-fit alternative. Kilgore and Thompson 7

10 TABLE 1 Summary of RMSE Computations. Bivariate Gumbel (POM) Bivariate Normal (POM) Gumbel- Hougaard (POM) Bivariate Gumbel (POT) Bivariate Normal (POT) Gumbel- Hougaard (POT) Frank Frank Parameter (POM) (POT) Average Median Minimum Maximum n Once the Gumbel-Hougaard copula was selected, the next task was to develop a method for estimating Kendall s τ for an ungaged watershed pair. Kendall s τ is needed to estimate the dependence parameter, θ, in Equation (6), which in turn is used in the copula relation, Equation (5). Several strategies for developing relations between Kendall s τ and various watershed and meteorological characteristics were implemented using regression techniques. The characteristics evaluated included, but were not limited to, drainage area ratio, total drainage area, distance between watershed centroids, mean annual precipitation, and 2-yr/24-h precipitation. Another strategy for developing relations between watershed and meteorological characteristics and Kendall s τ was to group data with common characteristics. After evaluating the performance of the regression and the grouping strategies, the latter approach was adopted. The recommended grouping approach for estimating Kendall s τ is summarized in Table 2. Watershed pairs were divided into one of four groups based on the drainage area ratio, R A, and the total area of the combined watersheds, A TOT. The dividing line for R A is 7 and for A TOT is 350 mi 2. To estimate Kendall s τ, the designer determines the ratio of the main stem watershed area to the tributary watershed area, R A, and the total watershed area, A TOT. The designer also chooses to use either the Best-fit or Envelope method. The best-fit Kendall s τ in a group is the mean value of the group. The designer would choose this value in most cases. Alternatively, the envelope Kendall s τ is the 90 th percentile value from the group. The designer would choose this value when a more conservative selection is appropriate, perhaps because of the greater downside of failure in a particular situation. TABLE 2 Kendall s τ Based on Watershed Characteristics. Method Best-fit Envelope Drainage Area Ratio A TOT < 350 mi 2 A TOT > 350 mi 2 R A < R A > R A < R A > The research database did not have sufficient data in the group where R A is greater than 7 and A TOT is less than 350 mi 2. To be conservative, watershed pairs with these characteristics are treated as watersheds with R A < 7 and A TOT less than 350 mi 2. Kendall s τ, combined with a desired joint probability, establishes the marginal probabilities for each stream. An example of this is shown in Table 3 for a 50-yr joint return period. (Other joint return periods are found in the research report (2).) Kilgore and Thompson 8

11 TABLE 3 50-yr Joint Return Period for Coincident Events for the Paired Stream. Best Fit Envelope Return Period on One Stream Kendall s τ Equal R A < 7 A TOT < 350 mi R A < 7 A TOT > 350 mi R A > 7 A TOT < 350 mi 2 R A > 7 A TOT > 350 mi R A < 7 A TOT < 350 mi R A < 7 A TOT > 350 mi R A > 7 A TOT < 350 mi 2 R A > 7 A TOT > 350 mi For example, if a designer determined that the best-fit equation is appropriate and the joint return period is 50-years, then the designer would consult Table 3 for the potential combinations of flow on the main stem and tributary. Assuming the site has a drainage area ratio of 5 and a total area of 50 mi 2, the potential combinations are (10,50), (25,48), and (38,38). In total, this represents five potential combinations; the three previously listed with the main stem first and the tributary second and two additional combinations where the pairs are with the tributary first and the main stem second. APPLICATION An illustrative example of copulas is presented based on the Nisqually River and Mineral Creek watershed pair in Washington State. The design situation calls for use of the best-fit Kendall s τ and a 50-yr joint return period. Table 4 summarizes the relevant site data. Although a gaged pair from the project database is used in this example, none of the site-specific flow data for the pair is used. Therefore, these watersheds illustrate the application on an ungaged watershed pair. TABLE 4 Example Watershed Pair Data. Nisqually River Mineral Creek Gage ID A (mi 2 ) P M (in) The United States Geological Survey (USGS) regression equations are applied (4) to estimate the flows for each watershed considered independently. The watershed pair is located in USGS region 2 for which the following regression equation applies: where, b b 1 Q = aa P 2 (7) T M Q T = design flow for return period, T, ft 3 /s A = drainage area, mi 2 P M = mean annual precipitation, in a,b 1,b 2 = regression constants (See Table 5) Kilgore and Thompson 9

12 The typical return period estimates of discharge are summarized in Table 5. Discharges for return periods not shown can be estimated algebraically from the computed values by interpolation. TABLE 5 Example Regression Discharge Results. Return Period a b 1 b 2 Nisqually River (ft 3 /s) Mineral Creek (ft 3 /s) Q ,260 3,810 Q ,300 6,920 Q ,900 8,550 Q ,400 10,100 Q ,400 11,400 A summary of the flow combinations based on Table 3, as previously discussed, along with the corresponding flows estimated for each return period is provided in Table 6. Each combination represents a joint return period of 50-yrs. TABLE 6 Potential Combination Discharges. Combination Nisqually River Return Period (yrs) Mineral Creek Return Period (yrs) Nisqually River Discharge (ft 3 /s) Mineral Creek Discharge (ft 3 /s) ,300 10, ,900 10, ,300 9, ,200 8, ,400 6,920 After development of the discharge pairs for the potential combinations, a hydraulic model such as HEC-RAS is applied to site conditions using those discharge combinations. The initial data collection, setup, and calibration of HEC-RAS are completed. Then, in the case of this example, the five combinations in Table 6 are executed changing only the main stem and tributary discharges. Although HEC-RAS was not executed for this example, hypothetical results in Table 7 illustrate the process. TABLE 7 Hypothetical Stage and Velocity at the Example Project Site. Combination Stage (ft) Velocity (ft/s) Depending on the design objective (stage or velocity) and location within the influence reach, one of the combinations will yield the extreme condition for that design objective. In this example, combination 4 results in the highest stage such that 30.4 ft would be taken as the 50-yr water surface elevation at the site. Similarly, combination 1 yields the highest velocity such that 14.2 ft/s would be taken as the 50-yr velocity. Kilgore and Thompson 10

13 Results from different combinations are not combined. If the design objective is most concerned about stage, the maximum stage is used. If the design objective is most concerned about velocity, the maximum velocity is used. For scour analyses, which depend on both stage and velocity, scour computations using each combination are completed, and the maximum scour is used. The extreme condition for the appropriate design objective is considered the design condition corresponding to that return period. As in this example, a different combination will likely be the design condition for stage and velocity. The proposed methodology based on copulas is a straightforward method to estimate the joint probability of confluent flows. It is based on a nationwide dataset of flow data. ACKNOWLEDGEMENTS The authors would like to acknowledge the Nationally Coordinated Highway Research Program (NCHRP) for funding the original research effort on which this paper is based and David Reynaud, the NCHRP Manager. REFERENCES 1. Koltun, G.F. and J.M. Sherwood. Factors Related to Probability of Joint Flooding on Paired Streams in Ohio. In Transportation Research Record: Journal of the Transportation Research Board, No. 1690, Transportation Research Board of the National Academies, Wahshington, C.C., 1999, pp Kilgore, Roger T., David B. Thompson, and David Ford. Estimating Joint Probabilities of Design Coincident Flows at Stream Confluences, NCHRP Project 15-36, Zhang, L. and V.P. Singh. Bivariate Flood Frequency Analysis Using the Copula Method, Journal of Hydrologic Engineering 11(2), , Sumioka, S.S., D.L. Kresch, and K.D. Kasnick. Magnitude and Frequency of Floods in Washington, Water-resources Investigations Report , USGS, Kilgore and Thompson 11