Stochastic modeling of flood peaks using the generalized extreme value distribution

Authors


Abstract

[1] The generalized extreme value (GEV) distribution is a standard tool for modeling flood peaks, both in annual maximum series (AMS) and in partial duration series (PDS). In this paper, combined maximum likelihood estimation (MLE) and L moment (LMOM) procedures are developed for estimating location, shape, and scale parameters of the GEV distribution. Particular attention is given to estimation of the shape parameter, which determines the “thickness” of the upper tail of the flood frequency distribution. Mixed MLE–LMOM methods avoid problems with both MLE (estimator variance) and LMOM (estimator bias) estimators of the shape parameter. The mixed MLE–LMOM procedure is extended to use the two largest flood peaks in a year. This extension is developed in a PDS framework. Estimation procedures are applied to flood peak observations from 104 central Appalachian basins. The estimated values of the shape parameter for the central Appalachian basins are more negative than has been considered physically reasonable, independent of the estimation procedure that is used. Twenty-eight percent of mixed method estimators of the shape parameter have values less than −0.5, implying that the moments of order 2 and above are infinite. The estimated shape parameters for the central Appalachian basins do not depend on basin morphological parameters (such as drainage area) or land cover properties (such as percent urban, forest, or agricultural land use). Estimated values of the location and scale parameters for the central Appalachian watersheds correspond well with GEV-based simple scaling theory. Estimated values of the shape parameter for central Appalachian watersheds are shown to differ markedly from those of southern Appalachian watersheds and the difference is shown to be linked to contrasting properties of extreme floods. To conclude the paper, analyses of mixture distribution models are presented to address the question of whether flood peaks really have extreme “heavy tail” behavior or whether the GEV distribution is not the appropriate model for flood peaks.

1. Introduction

[2] The development of stochastic methods for the characterization of flood peaks in drainage basins has both motivated and benefited from the treatment of classical problems in extreme value statistics. The generalized extreme value (GEV) distribution has been widely used for modeling the distribution of flood peaks in at-site and regional settings [Hosking et al., 1985a; Smith, 1987; Stedinger and Lu, 1995; Rosbjerg and Madsen, 1995; Hosking and Wallis, 1996]. In addition to flood modeling, the GEV distribution is commonly used to model many other natural extreme events [Smith, 1986; Bauer, 1996; Kuchenhoff and Themerus, 1996; Bruun and Tawn, 1998; Parret, 1998]. “Extreme events” are often defined to be the maximum value of a quantity over a given period of time, such as the maximum annual discharge in a river. Extreme value theory, in particular the extremal types theorem [Leadbetter et al., 1983], suggests that the distribution of these maxima should be close to one of the extreme value types. The GEV distribution, introduced by Jenkinson [1955], is a three-parameter distribution that combines all three extreme value types into a single form (see section 2).

[3] Parameter estimation procedures for the three-parameter GEV distribution have been extensively studied (see the work of Martins and Stedinger [2000] for a literature review). The most commonly used methods are the maximum likelihood estimation (MLE) [Prescott and Walden, 1980], the method of L moments (LMOM) [Hosking, 1990], and the method of moments (MOM). It has been noted that estimates of the shape parameter k of the GEV distribution for flood peak data are usually negative [Smith, 1987; Madsen et al., 1997; Martins and Stedinger, 2000], implying heavy tails in the distribution. It has been shown that MLE parameter estimators have a very large variance for negative values of k, and result in large errors in quantile estimation. Although both LMOM and MOM estimators tend to produce biased estimates, they are still considered preferable to MLE because of smaller variances in their quantile estimates [Hosking et al., 1985b; Madsen et al., 1997]. MLE-based methods, however, can easily incorporate additional information, such as censored data [Prescott and Walden, 1983] or a known prior distribution for k [Martins and Stedinger, 2000, 2001].

[4] In this paper (section 3), we show how MLE and LMOM methods can be combined to produce improved GEV parameter estimators based on annual maximum series (AMS) of flood peaks. The resulting “mixed” method estimators of the shape parameter of the GEV distribution have reduced variance compared to the MLE estimator and reduced bias compared to the LMOM estimator. The root mean square errors (RMSE) of mixed method estimators of the GEV location, scale, and shape parameters are superior to those of LMOM estimators for flood-size samples. The RMSE of LMOM estimators of extreme flood quantiles (100 and 1000 year flood magnitude analyses are presented) are slightly smaller than those for mixed method estimators. The contrasting properties of quantile and parameter estimators are examined and provide interesting insights to both LMOM and mixed method estimators.

[5] Partial duration series (PDS) [Shane and Lynn, 1964; Todorovich and Zelenhasic, 1970] models assume that the arrival times of peaks above a specified threshold form a Poisson process in time, and that the distribution of the peak magnitudes has a particular form. The attraction of these procedures is that additional information can be used, relative to AMS-based techniques. There are, for example, many flood records in which the second largest flood peak during a year is larger than the majority of other flood peaks. The generalized Pareto (GP) distribution is a common choice for the peak magnitude distribution both because it corresponds to a limiting distribution for excesses over a threshold as that threshold is increased [Leadbetter et al., 1983], and the resulting distribution of annual flood peaks is GEV [Smith, 1984; Madsen and Rosbjerg, 1997]. Madsen et al. [1997] showed that errors in parameter estimation under the GP/PDS approach under certain conditions are smaller than those of the GEV/AMS approach.

[6] In section 4 we introduce a MLE method that uses the values of the two largest observations for a given year (MLE2) and extend the method to a mixed method estimator. The MLE2 method is developed in the GP/PDS framework and tested via Monte Carlo simulations. Analyses illustrate the flexibility of the mixed method framework and the potential for improving parameter and quantile estimators through incorporation of flood observations from PDS records.

[7] The GEV distribution has played an important role in regional flood frequency analyses [Hosking et al., 1985a; Lettenmaier et al., 1987; Chowdhury et al., 1991; Stedinger and Lu, 1995; Hosking and Wallis, 1996]. A commonly used foundation for regional flood frequency analyses is the simple scaling theory, which assumes that appropriately scaled annual flood peaks have the same distribution in a hydrologically homogeneous geographical region. In the GEV approach, this means that the shape parameter k of the GEV distribution and the ratio of scale and location parameters are constant for all basins in the region.

[8] In the work of Smith [1992], a sample of 104 basin from the central Appalachian region was studied, and the hypothesis that simple scaling theory holds for this sample was rejected. In section 6 we further investigate the applicability of the index-flood theory to the central Appalachian region. In particular, we estimate the parameters of the GEV distribution for the same sample of basins used by Smith [1992] and Hosking and Wallis [1996] and study the dependence of parameter estimates on morphological and land cover characteristics of the basins.

2. MLE and LMOM Methods

[9] The GEV distribution combines into a single form all three Extreme Value (EV) distributions: Gumbel (EVI, k = 0), Frechet (EVII, k < 0), and Weibull (EVIII, k > 0). The GEV distribution has the following cumulative distribution function:

equation image

It has three parameters: scale a > 0, location b, and shape k. Here, −∞ < xb + a/k for k > 0, −∞ < x < ∞ for k = 0, and b + a/kx < ∞ for k < 0. To simplify our notation, we will write θ for the vector (b, a, k)T, and to refer to this distribution form with a particular set of parameters, we will write Gθ(x). The corresponding probability density function will be denoted as gθ(x). In this paper, we will focus on parameter estimation procedures for negative values of the shape parameter k. Because the GEV distribution does not have a third moment when k < −1/3, MOM estimators will not be considered here.

[10] The log likelihood function of a random sample {x1, x2,…, xn} from the GEV distribution is:

equation image

and the corresponding MLE estimator equation image = (equation image, equation image, equation image)T is the point at which log L(θ∣x) attains its maximum. It can also be expressed as the solution to the following optimization problem:

equation image

The constraints in the problem correspond to the condition that the probability density function of the GEV distribution must be positive at {x1, x2,…, xn}.

[11] Traditionally, the problem (3) is solved by setting the partial derivatives of the log likelihood function (2) to zero and then using Newton–Raphson iterations to solve for the parameters [Prescott and Walden, 1980; Hosking, 1985; MacLeod, 1989]. These methods have only local convergence and experience difficulty when the objective function is nonconvex. In this study, the method of steepest descent was used without second-order information about the objective function. It was found that convergence problems associated with nonconvexity of the objective function could be largely avoided in this manner. Although these methods require a larger number of function evaluations, the overall increase in computational time was nonetheless insignificant for the cases we tested.

[12] The LMOM estimator equation image for the shape parameter is the solution of the following equation:

equation image

The corresponding LMOM estimators for a and b are:

equation image
equation image

where equation image, equation image, and equation image are the estimators of the first two LMOMs and the L skewness obtained from the sample [Hosking, 1990]. Equation (4) is usually solved using Newton's method or by an approximate solution [see Hosking et al., 1985b].

[13] The bias of LMOM estimates of k increases with decreasing k, and is larger than 0.07 when k = −0.4. The MLE method produces almost unbiased estimates of k, but the variance of these estimates is large in comparison with those of LMOM. In addition, MLE frequently produces absurd estimates of k (< −1), which lead to very large errors in the quantile estimates [Martins and Stedinger, 2000]. In the next section, we present a combination of LMOM and MLE methods and show that it provides improved estimates of the shape parameter k.

[14] The quantile function of the GEV distribution is given by:

equation image

For a given value of p, quantile estimates are obtained by substituting estimated values of the parameters to the formula above. Of particular interest are large quantile values, for example, the 100 year return interval flood magnitude Q(0.99). In subsequent sections, we examine properties of quantile estimators equation image(p), in addition to properties of parameter estimators equation image, equation image, and equation image.

3. Mixed LMOMs: MLE Methods

[15] One of the ways to improve MLE estimates of k is to impose additional constraints on the optimization problem in (3). We would like these constraints to be based on our sample rather than on additional assumptions about the process that we have observed. One such constraint could be posed by LMOMs, for example, we can require the first LMOM of the estimated GEV distribution to be the same as determined from the sample. The addition of this constraint to the MLE problem (3) produces the first Mixed (MIX1) method.

[16] In the MIX1 method, we maximize the log likelihood function L(θ∣x), as a function of a and k after substituting b from the LMOM equation (6). The MIX1 estimator equation image of the parameters of the GEV distribution, then, is the solution to the following optimization problem:

equation image

[17] In the second mixed method (MIX2), we maximize the likelihood function L(θ∣x) as a function of k after taking both b and a from the LMOM (equation (5) and (6)). The optimization problem for this method is

equation image

[18] We can also consider the method MIX2 to be an LMOM method where, instead of using (4) to obtain estimates of k, we maximize the likelihood function to obtain k. In this case, we avoid using the estimator for τ3, which has a large bias if the true value of k is less than −0.2 and the sample size is small (less than 50).

[19] Both MIXed methods are based on the solution of a nonlinear optimization problem with nonlinear constraints involving 1 or 2 variables. Modern optimization solvers can solve similar problems with hundreds of variables in a matter of seconds, so, from the computational point of view, these problems are very tractable. In our implementation of the MIXed methods, we used the steepest descent method [Bazaraa et al., 1993] for solving these problems. The initial point was taken to be the LMOM estimate, and if it was infeasible (relative to the constraint in (9)), the value of k was adjusted to allow it to be feasible. At each iteration, the bounds on the step size were set such that the current solution remained feasible at all times. Although we coded the methods ourselves, standard optimization solvers (such as LOQO [Vanderbei, 1999] or MINOS [Mutagh and Saunders, 1998] can be readily used.

[20] In order to evaluate the performance of the MIXed methods, we conducted a series of simulation experiments. We simulated random samples of different sizes n from the GEV distribution corresponding to different values of k in the range −0.5 to 0.0. For each sample, we estimated the parameters of the GEV distribution using MLE, LMOM, MIX1, and MIX2 methods. We are most interested in the estimates of the shape parameter, but we also present the results for estimators of location, scale, and flood quantiles. Each simulation experiment was performed 10,000 times.

[21] The bias and RMSE (defined as (equation image[(equation imagek)2])1/2) were computed for MLE, LMOM, MIX1, and MIX2 estimators of k for sample size n = 30 (Figure 1). For k < −0.1, the MIX1 and MIX2 estimators have smaller biases than the LMOM methods and smaller variances than MLE, resulting in smaller RMSE in the estimation of k compared to LMOM and MLE. Neither MIXed method produced absurd estimates for k. The differences in performance decrease with increasing sample size (Table 1). An attractive feature of the MIX1 method is that the RMSE of the estimator equation image is insensitive to the value of k. This is a very desirable property for the estimator if we would like to examine the dependence of k on drainage basin and climatological properties. In section 5, we examine the regional distribution of estimators of k for central Appalachian drainage basins and examine the dependence of estimators of k on basin properties, such as basin area, land use and land cover (LULC), and drainage density.

Figure 1.

Bias (a) and RMSE (b) of the estimator equation image for a sample size n = 30 and four different estimation methods: MLE, LMOM, MIX1, and MIX2.

Table 1. RMSE of Parameter Estimates for MLE, LMOM, MIX1, and MIX2 Methods for Selected Sample Sizes and Negative Values of k
nkequation image RMSEequation image RMSEequation image RMSE
MLELMOMMIX1MIX2MLELMOMMIX1MIX2MLELMOMMIX1MIX2
30−0.50.210.260.210.210.220.230.220.220.220.210.180.17
30−0.40.190.230.200.190.220.220.220.210.210.190.170.17
30−0.30.180.210.190.190.220.220.220.220.200.180.170.16
30−0.20.170.190.180.170.210.210.210.210.190.170.170.16
30−0.10.170.170.170.170.210.210.220.210.180.160.170.17
3000.160.160.160.170.210.210.220.210.170.150.170.17
50−0.50.160.20.170.160.160.170.170.170.160.170.140.14
50−0.40.150.180.150.150.170.170.170.170.150.160.130.13
50−0.30.140.160.140.140.170.170.170.160.140.140.130.12
50−0.20.130.140.130.130.160.160.170.160.130.130.120.12
50−0.10.130.130.130.130.170.160.170.160.130.120.130.12
5000.120.120.120.120.160.160.160.160.120.110.120.12
100−0.50.110.140.120.110.120.120.120.120.100.130.100.10
100−0.40.100.130.110.100.120.120.120.120.100.120.090.10
100−0.30.100.110.100.100.120.120.120.110.090.100.090.09
100−0.20.0910.100.090.090.110.120.120.110.090.090.080.08
100−0.10.090.100.090.090.120.120.120.110.080.080.080.08
10000.080.080.080.080.110.110.110.110.080.070.080.08

[22] The RMSE of LMOM, MLE, MIX1, and MIX2 estimators of the quantile function Q(p) ((equation image[(equation image(p) − Q(p))2])1/2) were computed for p = 0.99 (100 year flood) and p = 0.999 (1000 year flood) for different values of n (Table 2). The ratio RMSEQ − MIX1/RMSEQ − LMOM takes values between 0.98 and 1.02, implying that, in terms of quantiles, the two methods perform almost equally well. This result seems somewhat unusual in comparison to the estimation results in parameter space, where similar ratios range from 0.84, when k = −0.5 to 0.97 when k = −0.1.

Table 2. RMSE of Quantile Estimates for MLE, LMOM, MIX1, and MIX2 Methods for Selected Sample Sizes and Values of k
nkequation image(0.99) RMSE/Q(0.99)equation image(0.999) RMSE/Q(0.999)
MLELMOMMIX1MIX2MLELMOMMIX1MIX2
30−0.51.530.560.580.569.121.101.191.14
30−0.41.030.530.5520.543.631.121.161.14
30−0.30.960.500.510.505.441.081.091.07
30−0.20.630.440.450.442.040.900.910.88
30−0.10.530.370.400.391.700.670.770.73
3000.370.300.330.330.770.520.610.60
50−0.50.740.480.540.501.870.971.141.01
50−0.40.620.450.460.461.860.960.950.96
50−0.30.480.390.390.391.050.790.730.74
50−0.20.390.340.340.340.770.650.620.62
50−0.10.320.290.300.290.620.500.530.51
5000.260.240.250.250.440.380.410.41
100−0.50.400.370.370.370.790.720.710.72
100−0.40.350.340.330.340.660.680.600.64
100−0.30.30.290.280.280.540.560.490.51
100−0.20.250.240.240.230.440.420.400.39
100−0.10.200.190.200.190.330.320.320.31
10000.170.160.170.170.260.250.260.26

[23] The differences between performance of MIXed method and LMOM estimators in parameter and quantile spaces lead to a more detailed analysis of the distribution of the quantile estimators. A surprising conclusion was that the bias of the LMOM estimator of k plays an important role in producing good LMOM quantile estimators. We demonstrate this feature with an example using n = 30 and parameter values of the GEV distribution of b = 0, a = 1, and k = −0.3. The covariance matrix of MIX2 estimators (Table 3, based on 10,000 simulation runs) is smaller than that for LMOM estimators for each element. Only the bias of equation image is larger for MIX2 than for LMOM.

Table 3. Comparison of Performance of LMOM and MIX2 Methods in Parameter and Quantile Spaces (See Text for More Details)a
 LMOMMIX2
  • a

    The true values of the parameters are (0, 1, −0.3)T and the true value for Q(0.99) is 9.9169.

Expected values of the estimators, equation image[(equation image, equation image, equation image)T]equation imageequation image
Covariance matrix of the estimators, Bequation imageequation image
det B3.2212 × 1052.0724 × 105
Theoretically computed values for Q(0.99), assuming unbiased parameter estimators
Expected value11.181711.0946
Variance36.245631.9329
RMSE6.15195.7723
Theoretically computed values for Q(0.99), accounting for biased parameter estimators
Expected value9.798610.1838
Variance26.309326.7793
RMSE5.13055.1817
Simulation results
Expected value9.775610.1427
Variance24.462624.6872
RMSE4.94754.9733

[24] Under the assumption that the estimators have a joint Gaussian distribution, we can compute the moments of equation image(99) by numerically integrating expression (7) with the appropriate pdf of the estimators. If we neglect the bias of the estimators and assume that they are centered at (0, 1, −0.3)T, we will find that the MIX2 method performs better (Table 3, center). Note that the bias of the quantile estimator that we obtain under this assumption is large, nearly 12%. Taking into account the biases of the parameter estimators in the computation, we obtain better results for quantile estimators: biases and variances of the quantile estimators are decreased and the RMSEQ obtained from LMOM is slightly smaller than that from MIX2. This agrees with the estimated RMSEQ obtained from the simulation experiments, where equation (7) is calculated for every simulation experiment and appropriate statistics are computed (Table 3, bottom).

[25] The difference between theoretically computed and simulated variances and RMSEs can be attributed to non-Gaussian properties of one or more estimators. Further analyses suggested that RMSEQ is most sensitive to equation image[equation image]. This is not surprising, because k contributes to (7) exponentially and RMSEQ decreases with increasing equation image[equation image], provided that the covariance between equation image and equation image is positive (see Table 3). This result agrees well with results of Lu and Stedinger [1992], who show that, for certain pairs of k and n, smaller RMSEQ can be achieved by setting equation image = 0. In addition, it is clear that decreasing covariance between equation image and equation image will result in better estimators of the quantiles. Although accurate estimation of the shape parameter k is important for proper characterization of the tail of the flood peak distribution, slight overestimation of k for the LMOM procedure results in smaller values of RMSEQ. Improvement of the parameter estimators, therefore, does not necessarily mean improvement of the quantile estimators.

[26] We also compared the performance of the four methods for positive values of k. We found that the MIXed methods and MLE performed roughly the same, and LMOM performed slightly better than the other methods.

[27] It is important to note here that, if we have prior knowledge of the underlying physical process, we can add associated constraints to the MLE optimization problem accordingly, and there is a good chance that the estimates of the model parameters will be improved. One example of such prior knowledge is including an estimate of the lower bound of the peak magnitude in the GEV distribution (b + a/k). Indeed, absurd estimates of k usually occur in situations where the smallest value in the sample is very close to the estimated lower bound. If, in addition to our sample {x1, x2,…, xn}, we know that a value x0 < min{x1,…, xn} is a possible flood value, then we must have b + a/k < x0. Adding this lower bound condition to the MLE problem (3) will reduce the chance of obtaining absurd estimates, if not eliminate it completely (see the work of Stedinger and Cohn [1986] for additional development of this idea). Martins and Stedinger [2000] discuss a sample of size 15, generated from the GEV distribution with parameters a = 1, b = 0, and k = −0.2 for which the MLE estimate of k is less than −2.4. This resulted in an estimate of the 0.999 quantile on the order of 6 × 106, while the value of the real quantile was only 14.9. The true distribution has a lower bound of −5, while the lowest value in the sample was −0.39. Suppose that we had the additional information that the value x0 = −1 is feasible, i.e., there is a strictly positive probability of obtaining −1 from the underlying process. If we add this information into the problem (3), our estimate of k will be −0.74, resulting in an estimate of the 0.999 quantile of 208. Although the error of this estimate is still quite large, it is almost 30,000 times smaller than without the condition. Simulation experiments show that adding this condition reduces the overall error in quantile estimates by a factor of 8 in comparison with the standard MLE method. In applications involving natural events (floods, winds, etc.), it is possible that some useful additional information of this kind is available. For instance, the data might say that there was no flood peak above a certain threshold during a certain year, but the value of the maximum flood peak for that year was not recorded. While it is very hard to incorporate such information into the LMOM method, it is very easy to insert it into MLE. We will discuss this approach further in later sections.

[28] To summarize this section, we conclude that for the negative values of k: (1) MIXed methods produce better estimates of the parameters of the GEV distribution than MLE and LMOM; (2) Quantile estimates produced by MIXed methods have RMSE close to that of LMOM; (3) Difference between relative performances of the MIXed and LMOM methods in the quantile and parameter spaces can be explained by the correlation between parameter estimates, nonlinear quantile function, and the favorable bias of the LMOM estimates; and (4) MIXed methods preserve the attractive large sample properties of MLE estimators (see the work of Morrison [2001] for derivation of large sample properties of MIXed method estimators; it is shown that estimators are strongly consistent, under certain regularity conditions, and have a limiting multivariate Gaussian distribution).

4. Extension to PDS Methods

[29] PDS models of flood peaks (sometimes referred to as the peaks-over-threshold approach) [Shane and Lynn, 1964; Todorovich and Zelenhasic, 1970] assume that the arrival times of peaks greater than a specified threshold form a Poisson process in time, and that the distribution of peak magnitudes has a particular form. If we assume that the distribution is a GP distribution (as in the works of Davison and Smith [1990] and Madsen and Rosbjerg [1997]), the annual flood peaks derived from this model have a GEV distribution (with an atom at zero) [see Smith, 1984; Hosking and Wallis, 1987].

[30] Assume that flood peaks above the threshold δ arrive according to a (stationary) Poisson process with rate λ, and that the peaks' magnitudes V1, V2,… are i.i.d. random variables independent of the arrival process, each having a GP distribution with location parameter δ, scale parameter α, and shape parameter κ. The cumulative distribution function of Vj is then

equation image

[31] To simplify our notation, we will write η to represent the vector (α, δ, κ)T, and to refer to the distribution function with this particular set of parameters η, we will write Fη(x). The corresponding probability density function will be denoted as fη(x). Under our assumptions, the distribution of annual flood peaks (for values greater than δ) is the same as the GEV distribution with parameters

equation image

After estimating the parameters for GP/PDS model, then, we can subsequently deduce the appropriate parameters for the GEV model. Madsen and Rosbjerg [1997] showed that the PDS approach can improve MLE and LMOM estimates if both the arrival rate of flood peaks above threshold is greater than 2 peaks per year and κ < 0.

[32] We developed a MLE method based on the magnitudes of the two largest floods each year (MLE2). Methods based on more than one peak per year, have been studied before in the application to sea level heights [Smith, 1986; Dupuis, 1997].

[33] Suppose that the flood peaks for a basin follow the GP/PDS process described above. Let X and Y be random variables representing the annual maximum flood peak and the second largest flood peak for a given year, respectively.

[34] Under the GP/PDS model, we can write the probability distribution for X:

equation image

The joint distribution for two largest peaks during a given year is then:

equation image

[35] Under the GEV/AMS approach, we approximate the expression in (12) by Gθ(x) with the parameters θ related to those of GP/PDS η through relationships (11). That means that (13) will be approximated by:

equation image

and the joint probability density function for X and Y is then just gθ(y)gθ(x)/Gθ(x) for δ < y < x. Using this argument, we can construct a likelihood function of the observations of the two largest peaks per year. Also note that

equation image

[36] Consider a basin with m years of PDS record with threshold δ. Among these m years of PDS record, there are m0 with no peaks above δ, m1 years with only one peak above δ (let z1, z2,…, zm1 denote the magnitudes of these peaks), and m2 years with 2 or more peaks (the largest peaks per year will be denoted by x1, x2, …, xm2 and the second largest by y1, y2,…, ym2). The log likelihood function of these observations is, then:

equation image

where the parameters θ, η, and λ are connected through the relationships in (11). Here, the first three terms on the right hand side correspond to the probability of having 0, 1, or “2 or more” peaks in a given year, respectively. The term on the second line is the log likelihood of obtaining the particular values z1, z2,…, zm1 of the single flood peaks over threshold that occurred during the m1 years, and the term on the last line corresponds to the log likelihood of obtaining the particular pairs of two largest peaks for the m2 years that we observed two or more peaks. Substituting into (16) the derivative of (10) for fη, and Gθ and gθ from (1), and using the relationships (11) we obtain the following expression for the likelihood function of the two maxima:

equation image

The time required to evaluate this expression computationally is not much more than the time necessary to evaluate the standard likelihood function for the MLE method.

[37] The maximum likelihood estimator based on the two largest maxima per year (MLE2) is the (local) maximum of the likelihood function in (17). This can be equivalently written as the solution to the following optimization problem:

equation image

The first three conditions correspond to the restriction that the measured peak magnitudes must be feasible values for the varying GP distribution. The last condition is the restriction that the scale parameter is positive. The second to last constraint (“the δ constraint”) is the condition that the threshold level itself must be a feasible value for the GEV distribution. This constraint is necessary because it ensures that the likelihood function can be evaluated. When k >0, since all zi, xj, and yj are greater than δ, the δ constraint never becomes binding on the problem (the upper bound of the GEV distribution is obviously greater than δ); when k = 0 the δ constraint becomes a ≥ 0, which is less restrictive than the last constraint. For k < 0, the δ constraint has an effect, as it requires that the lower bound for annual flood peaks be less than δ. This corresponds to the positive probability of having no peaks above the threshold during a given year.

[38] As with the regular MLE method, we investigated different possible constraints that we can add to the problem (18) in order to improve the estimates of the quantiles. The equivalent of the MIX1 method in this case involves adding the following constraint to the problem (18):

equation image

where equation image is the mean value (estimate of the first LMOM) of all annual maxima above the threshold δ, that is, equation image. This condition helps to eliminate all absurd estimates of k, and improves the estimation. We will refer to this method as MLE2–MIX1.

[39] The procedure that we used for Monte Carlo simulations in order to test our methods is the same as described by Madsen et al. [1997]. It is designed so that we can compare the performance of the AMS methods to that of PDS-based methods. It exploits the fact that in GP/PDS peaks above higher threshold levels from the same process also have GP distribution with the same value of k. Specifically, if peaks over the threshold level δ0 arrive according to a Poisson process with rate λ0, and the peaks' magnitudes are i.i.d., independent of the Poisson process, and have a GP distribution with parameters η0 = (α0, δ0, κ), then peaks above the higher threshold δ1 for λ1 < λ0

equation image

arrive according to a Poisson process with rate λ1, are i.i.d., and have a GP distribution with scale parameter α1 = α0 + κ (δ1 − δ0), location parameter δ1, and the same value of shape parameter κ [see Madsen et al., 1997].

[40] The simulation procedure can be described as follows:

  1. Pick parameters to simulate data from the GP/PDS model: Choose a large arrival rate λ0, so that the probability of obtaining zero peaks during any given year is very small. Choose a threshold level δ0, scale parameter α0, and the number of years in the record m. Also, choose the value of the shape parameter κ and the arrival rate λ1 for which we would like to test the procedure. Compute δ1 from (20) and α1 = α0 + κ (δ1 − δ0).
  2. Generate arrival times from the Poisson process with rate λ0, and generate a flood peak magnitude from the GP distribution with parameters δ0, α0, and κ for each arrival time.
  3. Extract the PDS sample corresponding to all peaks higher than δ1 and their arrival times: compute the number of years with no peaks, m0, the number of years with 1 peak, m1, and the number of years with more than one peak, m2 = mm0m1. For years with only one peak, record the magnitude of the peak zi, i = 1,…, m1, and for years with two or more peaks record the magnitudes of the largest and second largest peaks, xj and yj, j = 1,…, m2.
  4. Solve the optimization problem (18) and record the MLE2 estimates.
  5. Extract the AMS sample from the original process by recording the largest peak for each year.
  6. Use LMOM and MIX1 method to estimate the parameters of the GEV distribution based on the AMS sample.

[41] Monte Carlo simulations were performed according to the procedure above for records 20–100 years long, values of λ1 from 2 to 7, shape parameter k between −0.5 and 0.0. Results (Figure 2) show that incorporating information about second maxima leads to a decrease in the estimation error of the shape parameter for all sample sizes. The results obtained for all other sets of the model parameters were similar to this one (Table 4). This was especially significant for very negative values of k. Although the MLE2 method sometimes produces absurd results, it does so less frequently than MLE, in part due to the “δ condition” in problem (18) (see the example above). The MLE2–MIX1 method did not produce any absurd estimates of k, and its estimates of k have the smallest RMSE. In terms of quantiles, though incorporating second maximum decreased the RMSE in the quantiles by a factor of 8 in comparison with the standard MLE method, the LMOM/AMS and MIX1/AMS methods still produce better quantile estimates than MLE2, primarily due to the absurd estimates of k. MLE2–MIX1 has the smallest RMSEQ (see Table 5) for k ≤ −0.2.

Figure 2.

Bias (a) and RMSE (b) of the estimator equation image for a sample size n = 30 and four different estimation methods: MLE2, LMOM, MIX1, and MLE2–MIX1.

Table 4. RMSE of the Estimates of k LMOM, MLE2, and MLE2–MIX1 Methods for Selected Sample Sizes and Negative Values of k
nkequation image RMSE
LMOMMLE2MLE2–MIX1
35−0.50.190.140.13
35−0.40.170.140.12
35−0.30.170.130.12
35−0.20.150.120.11
35−0.10.140.110.11
3500.140.110.11
50−0.50.180.120.12
50−0.40.150.110.10
50−0.30.150.110.10
50−0.20.130.100.10
50−0.10.110.090.09
5000.110.090.09
100−0.50.130.080.08
100−0.40.120.080.08
100−0.30.110.070.07
100−0.20.090.070.07
100−0.10.080.070.07
10000.080.060.06
Table 5. RMSE of Quantile Estimates for LMOM, MLE2, and MLE2–MIX1 Methods for Selected Sample Sizes and Values of k
nkequation image(0.99) RMSE/Q(0.99)equation image(0.999) RMSE/Q(0.999)
LMOMMLE2MLE2–MIX1LMOMMLE2MLE2–MIX1
35−0.50.480.660.50x21.021.671.02
35−0.40.420.550.410.881.230.80
35−0.30.420.450.401.011.030.83
35−0.20.300.330.300.640.660.55
35−0.10.250.240.240.540.460.45
50−0.50.460.540.470.961.160.92
50−0.40.380.410.370.810.810.72
50−0.30.360.340.330.810.680.64
50−0.20.260.260.250.560.500.47
50−0.10.200.190.190.400.320.33
100−0.50.360.330.380.800.610.74
100−0.40.3210.270.310.720.490.60
100−0.30.260.220.230.600.390.43
100−0.20.190.170.170.370.300.30
100−0.10.150.140.150.270.250.25

5. Analysis of Flood Peak Data From the Central Appalachian Region

[42] GEV parameter estimation procedures were applied to flood peak observations from a sample of 104 USGS stream gauging stations in the central Appalachian region (see the works of Smith [1992] and Hosking and Wallis [1996] for previous analyses of this data set). These basins have at least 30 years of data and are not regulated by dams [see Smith, 1992]. The questions that we would like to address in our analysis are (1) How variable are at-site estimates of the shape parameter k within the region; (2) How can this variability be explained; and (3) How do the estimates of the three GEV parameters depend on morphological and land cover properties of the drainage basins? To address the third question, basin morphological and land cover information was computed for each of the basins from digital elevation data (DEM) and LULC data (based on Landsat Thematic Mapper images from 1990 to 1992). From these data sets we computed drainage area, measures of basin slope and shape (including basin relief, relief ratio and elongation ratio) [Rodriguez-Iturbe and Rinaldo, 1997], and percent cover for various LULC categories (including urban, forest and agricultural classifications) for each of the 104 basins.

[43] Based on analyses of previous sections and for ease of comparison with prior studies, the MLE2–MIX1 method and LMOM method were used to estimate the parameters of the GEV distribution for each of the basins. The range of LMOM estimates of k for the central Appalachian basins was between −0.74 and 0.02, with a median value of −0.37. For the MLE2–MIX1 method, estimates of k ranged from −0.82 to 0.01 with a median value of −0.40. As given by Hosking and Wallis [1996] the same sample of basins is divided into 5 groups, and the regional estimates of k for each group were determined. The values of these estimates ranged from −0.45 to −0.24, which agrees with our distribution of at-site estimates. These values are more negative than what has conventionally been considered physically reasonable [Martins and Stedinger, 2000]. When k is less than −1/3, the flood peak distribution has an infinite third moment and when k is less than −1/2, the distribution has infinite variance. Very negative values of k suggest that the distribution of flood peaks has very heavy tails.

[44] Figure 3 shows the values of at-site estimates of k obtained using the LMOM and MLE2–MIX1 methods plotted against the corresponding empirical quantiles of the standard Gaussian distribution. The lines on the plot correspond to a Gaussian approximation of the distribution of the respective estimators of k for the two methods, if the true value of k is −0.42 and the period of record is 46 (the average number of years of record available for our sample). From this plot, we conclude that the distribution of the estimates of k for our sample is approximately Gaussian, and the variability of the estimates can be explained by the variability of the estimators used.

Figure 3.

A normal QQ plot of at-site estimates of k for the central Appalachians basins. The lines on the plot correspond to a Gaussian approximation of the distribution of the LMOM and MIX1 estimator of k.

[45] The estimates of the shape parameter k for the central Appalachian basins do not exhibit systematic dependence on basin morphometric properties or land cover properties. The relationship between estimates of k and basin area (Figure 4) is representative of those for other basin descriptors. Regression analysis between the estimates of k and the basin drainage area produced an R2 value of 0.007. R2 values for regression analyses of estimates of k versus basin morphological and land cover variables were less than 0.06 for all variables.

Figure 4.

Dependence of MLE2–MIX1 estimate of k on basin's drainage area.

[46] According to simple scaling theory, values of the scaling parameter a and the location parameter b should exhibit a log–log relationship with drainage area A. For the central Appalachian basins, this property generally holds (Figures 5a and 5b). A significant contribution to the variability in this scaling relationship for estimates of the GEV location and scale parameters is related to land cover properties. It was found that basins with a higher percentage of urbanized land have higher values of the scaling parameter a and b (Figures 5a and 5b). These basins respond as though they have larger effective areas [see Leopold, 1968; Smith et al., 2002]. Notably, the urbanization effects on GEV parameter estimates are quite important for location and scale, but not for the shape parameter k (see the work of Iacobellis and Fiorentino [2000] for additional discussion).

Figure 5.

(a) Dependence of MLE2–MIX1 estimate of a on basin's drainage area. Solid circles denote urban basins and empty circles denote rural basins. (b) Dependence of MLE2–MIX1 estimate of b on basin's drainage area. Solid circles denote urban basins and empty circles denote rural basins.

Figure 5.

(continued)

[47] GEV flood estimation analyses were also carried out for a sample of 34 drainage basins from the southern Appalachians yielding estimates of k ranging from −0.53 to 0.24 with a median value of −0.11. Selection of the 34 basins (all of which are in North Carolina) was based on identical constraints to those used for the 104 central Appalachian basins. A systematic difference in flood peak distributions between the two regions is that the southern Appalachian basins exhibit markedly lower variability, as represented by the coefficient of variation of annual flood peaks, than central Appalachian basins. This contrast in flood peak distributions is reflected in contrasting magnitudes of extreme flood peaks, as illustrated in Figure 6 by envelope curves of flood peaks for the two samples. The “extreme” estimates of the shape parameter k for the central Appalachian region are linked to the hydrology and hydrometeorology of extreme floods in the region. The central Appalachian region has experienced some of the largest unit discharge flood peaks in the United States east of the Rocky Mountains [Smith et al., 1996; Eisenlohr, 1952; Hack and Goodlett, 1960] The Three Floods paradigm of Miller [1990] interprets the flood hydrology of the central Appalachian region in terms of the contributions of (1) organized systems of thunderstorms, (2) tropical storms, and (3) extratropical cyclones. The relative importance of these three flood-producing storm systems is scale dependent, with thunderstorm systems of most importance at the smallest basin scales (less that 100 km2) and extratropical cyclones of greatest importance at the largest basin scales (greater than 10,000 km2).

Figure 6.

Envelope curves of flood peaks for central and southern Appalachians region.

[48] A key question is whether extreme estimates of the GEV tail parameter k necessarily mean that flood peak distributions indeed have heavy tails or whether alternative stochastic models can explain the estimated values of k. One possible scenario for negative estimates of k from thin-tailed flood distributions is based on the GEV distribution not being the correct distribution for annual flood peaks. If, for example, we estimate the three parameters of the GEV distribution from samples of size n = 50 from the exponential distribution, then the estimated values of k will be centered at −0.38 for MIX1 estimators and at −0.2 for the LMOM estimators. The exponential distribution is from the Gumbel domain of attraction (k = 0), so it does not have thick tails. The exponential distribution, however, can be shown to be a poor choice for modeling flood peak distributions. Are there good alternative models for annual flood peak distributions with thin tails, but large negative estimates of k?

[49] The heuristic explanation for the contrasts in flood distributions between the central and southern Appalachians given above, rests on the influence of particular types of flood events. This notion points to a class of alternative models that involve a mixture of different distributions. Let V1 and V2 be independent random variables having distributions from the same family, with equation image[V2] > equation image[V1]. Again, let X be a random variable representing an annual flood, and suppose that X = V1 with probability p and X = V2 with probability (1 − p). V1 and V2 can be thought of as flood peaks occurring from different classes of storms. For example, V1 might represent flood peaks produced by tropical storms [Sturdevant-Rees et al., 2001] and V2 might then represent flood peaks due to summer thunderstorms [Smith et al., 1996]. If V1 and V2 have a GEV distribution with k = 0, i.e., both flood populations have a Gumbel distribution, is it possible to obtain estimates of k centered at −0.4 for flood-length samples? The answer is yes for the following formulation: (1) V1 ∼ GEV with k1 = 0, a1 = 1, b1 = 0, (2) V2 ∼ GEV with k2 = 0, a2 = 3.3, b2 = 6.6, and (3) p = 0.83.

[50] If one estimates parameters of the GEV distribution from samples of size 50 from the Gumbel mixture model above using the GEV LMOM estimators, the estimator of the shape parameter k is distributed about −0.4 (Figure 7). For samples of size 50 from a GEV distribution with parameters a = 1.4, b = 0.2, and k = −0.42 (a “best fit” GEV parameter set for the Gumbel mixture model) the LMOM estimator of k is centered about −0.4. The GEV distribution with a = 1.4, b = 0.2, and k = −0.42 is close to the Gumbel mixture distribution for the 10 year event, taking a value of 5.3 for the GEV distribution and 5.6 for the Gumbel mixture. The distributions are quite different in the upper tails with a 0.99 quantile of 18.7 for the GEV distribution versus 8.2 for the Gumbel mixture model and a 0.999 quantile of 52.2 for the GEV and 10.6 for the Gumbel mixture. The preceding analyses demonstrate that if we estimate parameters of the GEV distribution from a flood sample whose true distribution is a Gumbel mixture, we can overestimate the upper tail thickness. Furthermore, the differences in assessment of upper tail thickness can have a marked impact on estimates of extreme flood quantiles.

Figure 7.

Distribution of the estimates of k for GEV distribution and the Gumbel mixture model (with parameters described in text) with sample size n = 50.

6. Conclusions

[51] New Mixed Method parameter estimators for the GEV distribution are introduced based on a combination of the MLE and LMOM methods. These procedures can be viewed as LMOM-constrained MLE methods. The new estimation procedures were motivated by problems in estimating the GEV shape parameter using MLE and LMOM procedures. The performance of GEV parameter and quantile estimators is studied via simulation. MIX1 and MIX2 estimators do not produce absurd estimates of k (unlike MLE estimators), and the RMSE of these estimators are smaller than that of LMOM and MLE for k < −0.1. The Mixed Method estimators are based on the MLE principle and are readily extended to incorporate additional information. These estimators also possess the attractive large sample properties of MLE estimators.

[52] Despite the fact that Mixed Method estimators provide better estimators of the GEV location, shape, and scale parameters (for negative values of k), the resulting Mixed Method quantile estimators are not superior to LMOM quantile estimators. Analyses of the distribution of the quantile estimators demonstrate that bias of the LMOM estimator of the GEV shape parameter is an important element of the performance of LMOM quantile estimators.

[53] The Mixed Method estimators are extended to use the largest two flood peaks in a year. This extension is developed in a PDS framework. Performance of the MLE2 and MLE2–MIX1 estimators are also studied via simulation. It is shown that for negative k, incorporation of the additional information on flood peaks from the PDS record into Mixed Method estimators can result in quantile estimators with smaller values of RMSE than for LMOM quantile estimators.

[54] The GEV estimation techniques were applied to flood samples from 104 basins in the central Appalachians. Estimates of the GEV shape parameter, which are centered at −0.4, are more negative than what has conventionally been considered physically reasonable. Comparison of estimators of the shape parameter with basin descriptors uncovered no significant dependences. The dependence of estimated location and scale parameters on drainage area was consistent with simple scaling theory. It was also concluded that basins with a higher percentage of urban area have larger values of the scale and location parameters. The estimates of k for central Appalachian watersheds are shown to differ from those of southern Appalachian watersheds and the difference is linked to contrasting properties of extreme floods. Gumbel mixture distribution models were examined to determine whether the negative values of k might be explained by a stochastic model with tails thinner than those implied by the GEV distribution.

[55] The methods described in this paper were implemented by the authors in the EVANESCE (“Extreme Value Analysis Employing Statistical Copula Estimation”) package for S-Plus [Venables and Ripley, 1997]. The package is available from the authors free of charge.

Acknowledgments

[56] This research was funded in part by the U.S. Army Research Office (grant DAAD19-99-1-1063), NASA (grant NAG5-7544), and National Science Foundation (grant EAR-9706259). This support is gratefully acknowledged. The authors are thankful to Jery Stedinger for his valuable comments and suggestions.

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