## 1. Introduction

[2] Rating curves are used for transforming stage data recorded at gauging stations into discharge so that streamflow time series can be obtained and applied in hydrological, hydraulic, and geomorphologic analysis as well as for water resources management and in climatic studies. Rating curves are, in most cases, obtained by fitting a predetermined stage-discharge model to measurements of discharge and concurrent values of stage.

[3] It is well known in hydrometry that rating curve assessment in natural rivers can be plagued with several complicating factors [*Di Baldassarre and Montanari*, 2008], of which unstable channel conditions are the most difficult to handle and also the most widespread. Temporal changes in the elements that govern the stage-discharge relationship at the gauge, such as channel geometry, roughness characteristics, and approach conditions, cause the stage-discharge relationship to vary with time. Shifting control behavior is a result of complex river processes that are both diverse and difficult to interpret. There is a large amount of literature on the causes to changes in natural watersheds, e.g., the flood regime and climatic changes. The mechanisms behind river adjustments are investigated extensively by researchers. Channel instability induced by human activity in the catchments (e.g., river regulation, land use change, mining, etc.) is also well covered in the literature. It is beyond the scope of this paper to give a satisfactory review on the geomorphologic literature. Works such as those by *Schumm* [1977] and *Knighton* [1998] provide a thorough treatment of the fluvial processes that cause channel changes through time together with a long list of references.

[4] The procedure of setting up rating curves for unstable channel sections has not been adequately solved in the literature. Virtually all standard hydrometric literature [e.g., *Rantz et al.,* 1982; *Herschy,* 1995; *International Standards Organization,* 1998] tackles the problem by segmenting the available stage-discharge measurements into time periods for which the hydraulic control is assumed to be stable. Rating curves with unique sets of parameters for each of the time periods are then set up. Some measurements are sometimes used across two or more time periods when it is believed that instability is affecting only a limited range, typically the lower one, of the stage-discharge relationship. The rating curves are usually derived from one of two methods. The first way is to fit the rating curves to the measurements using statistical or manual methods. Another approach is to start with the last valid rating curve, or a base rating curve, and then adjust some, often one, of its parameters until the resulting curve fits adequately to the measurements taken after the date when the shift is deemed to have taken place. *Burkham and Dawdy* [1970] illustrate this approach with a practical example and an error analysis.

[5] There are several problems with the common practice. First and foremost, defining all changes of the rating curve with measurement is, in most cases, not possible because of economic and personnel constraints. This fact introduces considerable uncertainty about the segmentation dates and the validity of the current and past rating curves. Also, the information about the stage-discharge relationship contained in the measurements will vary, sometimes greatly, between consecutive time periods. As a result, the computed streamflow time series characteristics will more or less jump from one time period to the next. The traditional approach does not attempt to rectify the problem appropriately, especially in cases where channel changes vary continuously with time. It is similar to the procedure of approximating a complex nonlinear function with linear segments, where the number of segments available is constrained by the number of observations of the nonlinear function. Also, a general and clear definition of when to introduce a shift is lacking. A common method is to assess the deviation between the current rating curve and recent consecutive validation measurements. If the deviations appear to be systematic and larger than one would expect from measurement imprecision exclusively, a shift is indicated. This approach can be implemented with experience and knowledge of observations of the changing control. Clearly, the success of this technique for detecting rating shifts is highly variable and depends on factors such as measurement frequency, knowledge about the site-specific conditions dictating the discharge measurement accuracy, and information about the characteristics of the changing control as well as the skill and experience of the analyst assessing the problem. Even if the hydraulic control is stable within the chosen time periods, there normally would be few measurements available to fit or adjust the rating curves. Consequently, modelers using the streamflow data are faced with a large and varying degree of rating curve uncertainty. It should be mentioned that some studies propose methods where a theoretical flow equation (e.g., a uniform flow formula or a critical flow equation) is applied to leveled channel geometry and then fitted to measurements. *Leonard et al.* [2000] utilized this approach to model a stage-fall-discharge relationship in an unstable river using two stage gauges. To overcome the problems of varying geometry, they assumed that changes in cross-sectional geometry were manifested in the slope measurement characteristics. Although they found evidence for this assumption in results from several geometrical surveys of the channel reach, it can hardly be recognized as a general rule. Nonetheless, this method might be a viable but costly way for setting up rating curves in some unstable rivers.

[6] The aforementioned criticism implies that a better approach would be to model the channel changes continuously in the time domain. Explicitly, this means that the rating curve parameters should be viewed as appropriate functions of time *t*. The simplest way is to formulate some of the parameters as deterministic functions of time, e.g., polynomials of *t* with unknown coefficients that can be obtained through standard regression methods. This naïve approach provides an incomplete approach to the problem because it assumes a too simplistic course of changes and also requires an accurate prior knowledge about the progress of the changing control. A better approach is to view the rating curve parameters as continuous time stochastic processes. This paper proposes a method where the rating curve parameters are assumed to follow Ornstein-Uhlenbeck processes, which will be described in section 3. Analysis is performed using a Bayesian framework and Markov chain Monte Carlo simulation techniques. The appropriateness and computational feasibility of the method is examined by three Norwegian case studies.