## Introduction

Perhaps the most fundamental and important information to track the status and trends of river water quality are records of discharge and suspended-sediment concentration that are collected by government resource agencies, academic institutions and private citizens (Milliman and Farnsworth, 2011). In combination, these records of discharge and suspended-sediment concentration can provide insights to the patterns and variability of river sediment discharge in time and space (Meade *et al*., 1990; Meade, 1994; Asselman, 1999; Hicks *et al*., 2000; Walling and Fang, 2003) and the effects of disturbances such as wildfire, earthquakes, torrential rainfall, landslides, land use change, and dam construction and removal (e.g. Dadson *et al*., 2004; Syvitski *et al*., 2005; Hicks and Basher, 2008; Kao and Milliman, 2008; Wang *et al*., 2008; Horowitz, 2009; Meade and Moody, 2010; Huang and Montgomery, 2013; Curran *et al*., 2014). Thus, multiyear to decadal sets of river suspended-sediment concentrations are valuable because they can reveal trends related to the effects of natural and human-caused changes to watersheds (Figure 1).

To characterize patterns and trends in river sediment concentrations, data are often fit with regression techniques, such as the commonly used linear regression between log-transformed discharge (*Q*) and suspended-sediment concentration (*C*) data:

where *a* and *b* are the rating parameters found from regression (Syvitski *et al*., 2000). In this technique, *b* is the slope between the log-transformed *Q* and *C* data, and log(*a*) is the ‘*y*-intercept’ or ‘offset’ value of log(*C*) defined where log(*Q*) is equal to zero (i.e. where Q is equivalent to 1 in the units of discharge, which are most commonly m^{3}/s). Note that Equation (1) is often reported in the power-law formulation:

A global analysis suggested that *a* and *b* are related to river basin characteristics such as topographic relief and runoff (Syvitski *et al*., 2000), although they can also vary significantly in time owing to sediment availability in the watershed and other factors (Asselman, 1999; Dadson *et al*., 2004; Yang *et al*., 2007; Huang and Montgomery, 2013; Figure 1).

Equations (1) and (2) are often referred to as ‘sediment rating curves’ owing to their potential usefulness for predicting suspended-sediment concentrations for intervals of time without samples (Asselman, 2000). Predictions of continuous suspended-sediment concentrations are valuable because these records allow for calculation of river sediment discharge from the product of continuous sediment concentration, water discharge and log-transform bias correction, if appropriate (Ferguson, 1986; Cohn *et al*., 1992). The utility of suspended-sediment rating curves in these calculations varies from river to river and is largely a product of the patterns of supply and transport of suspended-sediment over multiple timescales (Asselman, 1999; Horowitz, 2003; Aulenbach and Hooper, 2006). As such, there have been disparate conclusions about the utility of sediment rating curves for calculating river sediment loads (e.g. Walling, 1977; Horowitz, 2003; Wright *et al*., 2010).

Changes in sediment rating curve parameters, *a* and *b*, over time have been noted for many rivers systems (e.g. Syvitski *et al*., 2000; Warrick and Rubin, 2007; Yang *et al*., 2007; Huang and Montgomery, 2013), and there is a general assumption that these changes reflect alteration of the erodibility and/or supply of sediment in the watershed, the power of the river to erode and transport sediment or the spatial scale of the basin (Asselman, 2000). Owing to the widespread alteration and development of the global land surface (Goldewijk *et al*., 2011; Lambin and Meyfroidt, 2011), hydrologic changes that may result from human-caused alterations of the global climate system (Vörösmarty and Sahagian, 2000; Milly *et al*., 2008; Elsner *et al*., 2010), the need to restore river sediment supplies as part of ecosystem resilience and recovery plans (Kirwan *et al*., 2010; Duda *et al*., 2011), and the inherent variability of river sediment transport even without these human pressures, it is likely that river sediment concentrations and the associated sediment rating parameters will continue to change in the future. It is, therefore, important to have adequate techniques to quantify and characterize changes to these river systems.

Many types of changes in sediment rating curves are possible, including the alteration of the vertical offset, the slope or both variables (Figure 2). Although the examples in Figure 1 show changes dominated by the vertical offset, there are numerous examples of time-dependent changes in rating curve slope (e.g. Warrick and Rubin, 2007; Wang *et al*., 2008) a few of which will be highlighted later in this paper. Although trends in sediment rating curves are intriguing, it is arguably more important to determine the causes of change. For example, changes in the vertical offset of a rating curve (Figure 2d and f) may be associated with changes in watershed sediment production (e.g. Wang *et al*., 2008; Warrick *et al*., 2012), or – as shown by Warrick and Rubin (2007) for the Santa Ana River of California (*cf*. Figure 1c) – associated with increases in river discharge that, in effect, dilute the relatively steady suspended-sediment supplies with time. These contrasting results show the need for a better synthesis of the effects of sediment and water supply on sediment rating curve parameters, so that future investigations can better assess the causes of sediment rating curve changes.

The goal of this paper is to evaluate the utility of sediment rating curves for measuring and determining water quality trends in rivers. Two questions about the utility of sediment rating curves are assessed: (i) how well to the parameters, *a* and *b*, characterize trends in the data, and (ii) are trends in rating curves diagnostic of changes to river water or sediment discharge? These questions are addressed by exploring the mathematical implications of Equations (1) and (2), investigating empirical data from a range of river systems and simulating the effects of simple river water and sediment discharge changes using Monte Carlo techniques. As such, this paper does not attempt to supersede the synthesis of statistical techniques and trend analyses provided by Helsel and Hirsch (2002); rather, the work here provides narrow focus upon the use and utility of sediment rating curves for trend analyses. In fact, as noted in the Discussion and Conclusion Sections, there are an additional techniques suggested by Helsel and Hirsch (2002), Aulenbach and Hooper (2006), Horowitz (2009), Hirsch *et al*. (2010) and several others that can provide better sediment discharge and trend analyses than those provided by simple rating curve analyses. Furthermore, it is concluded that all sediment rating curve analyses require additional assessments of river water and sediment discharge trends. Yet, sediment rating curves are popular, easy to understand and – if used properly – useful. Thus, it is important to better describe their utility.