The question of the environmental risks of social and
economic infrastructure has recently become apparent due to an increase in
the number of extreme weather events. Extreme runoff events include floods
and droughts. In water engineering, extreme runoff is described in terms of
probability and uses methods of frequency analysis to evaluate an exceedance
probability curve (EPC) for runoff. It is assumed that historical
observations of runoff are representative of the future; however, trends in
the observed time series show doubt in this assumption. The paper describes a
probabilistic hydrological MARCS

Streamflow runoff serves as a water resource for humans, food production and energy generation, while the risks of water-sensitive economics are usually connected to runoff extremes. In fact, runoff extremes are always connected to human activity since they do not exist in a natural water cycle. Engineering science considers runoff extremes as critical values of runoff that lead to the damage of infrastructure or water shortages, and it introduces the extremes in terms of probability. In particular, in water engineering runoff extremes are evaluated from the tails of exceedance probability curves (EPCs) that are used in risk assessment for water infrastructure and decision-making in cost–loss situations (Mylne, 2002; Murphy, 1977, 1976). The EPC of multi-year runoff allows the estimation of the runoff extremes and supports the designing of building construction, bridges, dams and withdrawal systems, etc.

Modern hydrology uses two approaches to evaluate runoff extremes with their exceedance probability: conceptual modelling (Lamb, 2006) and frequency analysis (Kite, 1977; Benson, 1968; Kritsky and Menkel, 1946). In the conceptual modelling approach, synthetic runoff series are simulated from meteorological series in order to calculate the runoff values of a chosen exceedance probability (Arheimer and Lindström, 2015; Veijalainen et al., 2012; Seibert, 1999). In the frequency-analysis approach, historical yearly time series of runoff are used to evaluate statistical estimators, that is, the mean value, the coefficient of variation (CV) and the coefficient of skewness (CS) (van Gelder, 2006). These estimators are applied to calculate runoff values with their exceedance probability (Guidelines SP 33-101-2003, 2004; Guidelines, 1984; Bulletin 17–B, 1982) needed to support the designing of roads, dams, bridges or water-withdrawal stations. The basic assumption of this approach is that the future risks during an infrastructure's operational period are equal to the risks estimated from the past observations. The runoff extremes are simply extrapolated for the next 20–30 years on the assumption that past observations are representative of the future: the “stationarity” assumption (Madsen et al., 2013).

The number of weather extremes – including hurricanes, wind, rain- and snowstorms, floods and droughts – has increased (Vihma, 2014; Wang and Zhou, 2005; Manton et al., 2001). Historical time series of many climate variables show evident trends which are statistically significant, among which are the series of streamflow runoff (Wagner et al., 2011; Dai et al., 2009; Milly et al., 2005). Rosmann et al. (2016) applied the Mann–Kendall test to analyse a time series of daily, monthly and yearly river discharges for the last four decades. The highest number of trends was detected for the yearly time series of annual runoff. The statistically significant trends are founded on historical time series; thus the water engineers and managers are motivated to revise the basic stationarity assumption that lies behind infrastructures' risk assessment since past observations are not representative of the future (Madsen et al., 2013; Kovalenko, 2009; Milly at al., 2008).

In this paper, we described a method that combines conceptual modelling and frequency analysis in order to estimate runoff extremes in a changing climate. The method adapts the theory of stochastic systems to water engineering practice, and it was further named as advance of frequency analysis (AFA). It was introduced by Kovalenko (1993) and relied on the theory of stochastic systems (Pugachev et al., 1974). The basic idea behind the method is to simulate the statistical estimators of multi-year runoff (annual, minimal and maximal runoff) from the statistical estimators of precipitation and air temperature on a climate scale (Budyko and Izrael, 1991). The simulated statistical estimators of runoff are used to construct EPCs with distributions from the Pearson system (Pearson, 1895). Kovalenko (1993) suggested modelling the EPCs within a Pearson type III distribution based on traditional practice in water engineering (Rogdestvenskiy and Chebotarev, 1974; Matalas and Wallis, 1973; Sokolovskiy, 1968). However, the distribution can also be chosen by fitting (Laio et al., 2009), defined in accordance with local hydrological guidelines (Bulletin 17-B, 1982), or somehow more advanced (Andreev et al., 2005).

A linear “black box” (or a “linear filter model”) with stochastic components is suggested as a catchment-scale hydrological model (Kovalenko, 1993). For this linear model, the theory of stochastic systems provides methods to direct the simulation of probability distributions for a random process (Pugachev et al., 1974). The theory of stochastic systems is applied to analyse and predict runoff extremes on various timescales, ranging from days (Rosmann and Domínguez, 2017) to seasons (Domínguez and Rivera, 2010; Shevnina, 2001), and on various climate scales (Shevnina et al., 2017; Kovalenko, 2014; Viktorova and Gromova, 2008). The AFA approach is a simplification of the theory of stochastic systems on a climate scale. Kovalenko et al. (2010) gave guidelines for water engineers to estimate runoff extremes in a changing climate.

AFA was suggested about 30 years ago; however, a full description of this
approach has still not been published in English. Moreover, previous
publications in Russian contain many typewriting mistakes in the formulas
(Kovalenko, 1993; Kovalenko et al., 2006), and this makes understanding them
troublesome, even for native Russians. In this paper, the theory and
assumptions of the AFA approach were formulated step by step (see Appendix 1), and the formulas behind the core of the probabilistic hydrological model
MARCS

The probabilistic hydrological MARCS

The MARCS

The MARCS

The MARCS

Two blocks of the MARCS

The observations on precipitation are collected from meteorological sites,
and they may be interpolated into grids in order to better estimate a
precipitation rate over a river basin area. In the data preparation block of
the model, the mean annual precipitation rate (mm yr

The MARCS

In our study, the core version 0.2 of the probabilistic MARCS

To set up the MARCS

To force the MARCS

In our study, the EPC of runoff was modelled within a Pearson type III
distribution. This distribution is commonly used by water engineers to
estimate water extremes (Koutrouvelis and Canavos, 1999; Rogdestvenskiy and
Chebotarev, 1974; Matalas and Wallis, 1973). The water engineering
guidelines provide the ordinates of EPCs from lookup tables (Guidelines,
1984) depending on the CV and CS. These coefficients are calculated from
non-central moments' estimates (Rogdestvenskiy and Chebotarev, 1974):

In our study, we chose the basin of the Iijoki River at the Raasakka gauge
(lat 25.4

The yearly time series of volumetric water discharge of the Iijoki River
were extracted from a data set of the GRDC (GRDC 56068 Koblenz, Germany). The
observations at the Raasakka gauge (ID

The MARCS

The values of

Climate scenarios provide a range of projections for temperature and
moisture regimes in the future. This range is produced by different
assumptions about climate scenarios as well as specific climate models.
However, the climate projections include precipitation and air temperature,
and they give a forcing to hydrological models in order to simulate
projections of runoff. In the case study of the Iijoki River, the data from
CMIP5 (Taylor et al., 2012) for three representative concentration pathways
(RCPs) were used to force the MARCS

The forcing of the MARCS

Projected mean air temperature (

The projected non-central moments' estimates were simulated for the
scenarios and models listed in Table 2. These estimates were used to calculate
the mean values of CV and CS (see Eqs. 16–17) that were included in the
output of the MARCS

The projected climatology and statistics of annual runoff: the case of the Iijoki River.

In the case of the Iijoki River at Raasakka gauge, the 10 % water
discharge exceedance probability will increase in the future under the
scenarios and models considered (see Table 3). It may lead to risks of energy
spills at hydropower stations located within the catchment of the Iijoki
River in the period 2020–2050. At the same time, risks connected with water
shortages may be fewer since they are connected to a 90 % water discharge
exceedance probability, which is predicted to increase. Figure 2 shows
another way in which the model performs the EPC of the annual runoff rate
for the Kyrönjoki River at Skatila gauge (GRDC ID: 6854900). The set of
EPCs was simulated under three RCP scenarios using a similar set-up to the
MARCS

The variability of the tails of the EPCs for annual runoff (DR, mm yr

Nowadays, the future vision of the climate is changing continuously. Climate
projections are updated almost every 5–6 years, and many climate models
generate meteorological projections for variables such as precipitation and
air temperature. Hydrological models are needed to perform an express
analysis of future changes in water resources and water extremes
(floods and droughts) on a climate scale. The climate scale means that the
express analysis is provided for a period of 20–30 years. Lumped or
semi-distributed physically based hydrological models are traditionally used
on a short-term or seasonal scale to simulate a runoff time series from a
time series of meteorological variables (Seibert, 1999). In many
catchment-scale hydrological studies, these models are driven by the outputs
of climate models or their ensembles in order to evaluate water resources
and extremes in the near future (Arheimer and Lindström, 2015;
Veijalainen et al., 2012; Yip et al., 2012). The simulation of the runoff
time series from a time series of meteorological variables (see Fig. 2 in
Veijalainen et al., 2012) leads to high computational costs for such
estimations that need to be provided in terms of probability in economical
applications (Murphy, 1976). The probabilistic MARCS

In this paper we described the structure for the probabilistic hydrological
MARCS

The probabilistic hydrological MARCS

The MARCS

Further improvements of the MARCS

To place the probabilistic MARCS

The paper describes the theory and assumptions of the AFA approach, as well
as the probabilistic hydrological MARCS

To give a practical example of how to set up the MARCS

Currently, the MARCS

The following data sets can be used to set up and force the MARCS

The sample data set for the case study of the Iijoki River at Raasakka is given by Shevnina and Krasikov (2018).

Advance of frequency analysis (AFA) is based on the theory of stochastic systems, specifically, the Fokker–Planck–Kolmogorov equation, which is simplified into a system for three non-central statistical moments (Pugachev et al., 1974). The time series of annual runoff is considered as a realization of a random-process Markov chain type that is assumed to be “stationary”. It means that the statistical estimators (mean, variance and skewness) do not change over the period considered. The statistical estimators are used to model an exceedance probability curve (EPC) of the annual runoff with a Pearson type III distribution. The AFA approach is developed with an assumption of quasi-stationarity (Kovalenko et al., 2010; Kovalenko, 1993). The quasi-stationarity assumption suggests that the statistical estimators of multi-year runoff are different for two periods (the reference period and the projected period). For the reference period, the statistical estimators are evaluated from historical yearly time series of runoff. For the projected period, the statistical estimators of runoff are simulated based on the outputs of global- or regional-scale climate models under any climate scenario.

In this context, models replace a complicated hydrological system using maths
abstractions and aim to reveal the spatial and temporal river runoff
features which are important depending on the goals of study. Among other
models, black box hydrological models consider a river basin as a
dynamic system with lumped parameters. These models are “based on analysis
of concurrent inputs and temporal output series” (WMO-No. 168, 2009) and
transform series of meteorological variables (precipitation, air
temperature) into series of river runoff. Both input and output series are
functions of time (WMO-No. 168, 2009):

The Fokker–Planck–Kolmogorov equation can be applied to simulate the
probability density function (PDF) for the stochastic

For

Further, the summands in Eqs. (A10)–(A11) were divided by

The set of four moments (

Denoting

There is too much notation used to describe the model's core version 0.2;
thus the secondary parameters of equations were grouped by the model behind
it. Table A1 shows the notation and description of the secondary parameters
for the linear filter stochastic model. Equation (A3) is a simplification of Eq. (A1) that limits the first-order ordinal differential equation. It includes
three parameters,

The notation and description of the parameters for a linear filter stochastic model.

Table A2 gives a description of the parameters of the Fokker–Planck–Kolmogorov equation and the Pearson system. It should be noted that we do not solve the Fokker–Planck–Kolmogorov equation, and only its simplification for the system of three non-central moments is applied. These non-central moments are estimated from runoff observations for the reference period. For the projected period the moments are calculated from the mean of precipitation.

The notation of the Fokker–Planck–Kolmogorov equation and the Pearson equation.

To set up the model for a single river catchment, the non-central moments
should be calculated from historical time series of the annual river runoff
rate as well as from a mean value of annual precipitation rate. These values
should be placed manually (lines 45–48 in model_core.py,
located at

ES contributed to the MARCS

The authors declare that they have no conflict of interest.

The authors would like to thank two anonymous referees, the editor and Anatoly Frolov for their comments. Our special thanks are given to Alexander Krasikov, who supported the model coding.

This research has been supported by the Academy of Finland (contract no. 283101).

This paper was edited by Wolfgang Kurtz and reviewed by two anonymous referees.