## 1. Introduction

[2] Recent decades have seen a surge in analyses of hydrological and climatological data to seek evidence of trends brought about by anthropogenic influences. In terms of trends in river flows, the international literature reports many recent studies of trends in annual and seasonal flows [e.g., *Hannaford and Buys*, 2012; *Burn et al*., 2010; *Wilson et al*., 2010; *Novotny and Stefan*, 2007; *Hodgkins and Dudley*, 2006] some of which have included analyses of trends in high flows [*Marsh and Harvey*, 2012; *Petrow and Merz*, 2009; *Hannaford and Marsh*, 2008; *Svensson et al*., 2006]. The present paper addresses some issues relating to the analysis of trends in high flows, in a region where land-use change from native forest to arable cropping is likely to be at least as influential on extreme river flows as any existing or potential climate change over the last 80 years. In particular, the paper is concerned with the apparently simple issue of how to estimate trends in annual maximum 1 day river flows, although the methods discussed also have relevance to the estimation of trends in annual maximum rainfalls of any given duration, and trends in annual maxima of climatological variables.

[3] There is a very extensive literature on the use of extreme value distributions for describing the variability amongst “block maxima” (such as the series obtained by abstracting maximum values during periods or blocks, typically years) with the generalized extreme value (GEV) distribution given by

where *μ*, *σ*, and *ξ* are parameters of location, scale, and shape, respectively; *q* denotes the annual maximum 1 day flow. The methods of *Hosking and Wallis* [1997] based on estimation of the GEV parameters (*μ*, *σ*, *ξ*) by *L*-moments have proved simple to use, and software is also widely available for fitting GEV distributions by maximum likelihood (ML). *Coles* [2001, chap. 6] has described how, in the presence of trend in the series of block maxima, the GEV distribution can be adapted to estimate trends in any of the parameters, typically and most commonly by fitting the modified distribution GEV(*μ*(*t*), *σ*, *ξ*) where *μ*(*t*) = *α* + *βt*; extensions to GEV(*μ*(*t*), *σ*(*t*), *ξ*(*t*)) are also possible. In all such approaches, it is assumed that annual maxima are statistically independent. This paper considers only the simpler model GEV(*α* + *βt*, *σ*, *ξ*), with particular emphasis on the estimation of *β*, and on the uncertainty in this estimate as measured by its 95% confidence interval. The linear trend parameter *β* is estimated by ML using the statistical package GenStat [*VSN International*, 2012]; other packages (e.g., *ismev*, www.ral.ucar.edu/~ericg/softextreme.php [*Heffernan and Stephenson*, 2013], and related packages *evd*, *evdbayes*, *lmom*, *POT*) allow more general GEV models to be fitted with link functions relating parameters to predictors [*Coles*, 2001, section 6.1], which may include other predictors as well as time. Also, since the GEV distribution reduces to the widely used Gumbel distribution when the GEV shape parameter *ξ* is zero, the paper includes this too, using a form in which the Gumbel parameter *μ* is *μ*(*t*) = *α* + *βt*.

[4] A GEV distribution with time-variant location parameter is not the only way to estimate trend parameters when annual maxima are statistically independent. The many alternatives include bootstrap estimation [e.g., *Venables and Ripley*, 1999] and Theil-Sen distribution-free estimation [e.g., *Hannaford and Buys*, 2012], two methods that are used in this paper. Ordinary least squares also provides a valid estimate of the trend parameter *β*, but the fact that extreme flows are commonly heteroscedastic, with larger extremes having greater variances, means that confidence limits for *β* calculated on the assumption that the residual variance *σ _{e}*

^{2}is constant will no longer be valid. Another estimate of

*β*explored in this paper is given by using least squares to estimate the trend coefficient

*β*, then calculating the residuals {

*ε*} to which a GEV (or Gumbel) is fitted, the location parameter then being no longer time-dependent. An advantage of such a procedure is that ML estimation requires a search over the 3-D (

_{t}*μ*,

*σ*,

*ξ*) space instead of the 4-D (

*μ*,

*σ*,

*ξ*,

*β*): a considerable advantage if many sequences are to be analyzed for trend. Other methods of trend estimation (M-estimators, least median of squares (LMS), least trimmed squares, S-estimation, MM-estimation) might have been included, but most have drawbacks [

*Venables and Ripley*, 1999] and in any case, are not widely used by hydrologists and climatologists. Nor does the paper include the estimation of trends in “peaks-over-threshold” (PoT) models, which can also be modeled, in very general forms, by the

*ismev*software referred to above.

[5] Thus, the purpose of the paper is to compare estimates of the trend coefficient *β* obtained from a number of possible estimation procedures and to compare their uncertainties as measured by these estimates' approximate confidence intervals. The following sections describe the data used, the analytical procedures by which they were analyzed, and the results. A discussion follows, and conclusions are stated.