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**Geoscientific Model Development**
An interactive open-access journal of the European Geosciences Union

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- Abstract
- Introduction
- Model description
- Model application: a case study
- Discussions
- Conclusions
- Code availability
- Data availability
- Sample availability
- Appendix A: The theoretical basis for core version 0.2
- Appendix B: A short user guide for the MARCS model
- Author contributions
- Competing interests
- Acknowledgements
- Financial support
- Review statement
- References

**Model description paper**
09 Jul 2019

**Model description paper** | 09 Jul 2019

The probabilistic hydrological MARCS^{HYDRO} (the MARkov Chain System) model: its structure and core version 0.2

^{1}Finnish Meteorological Institute, Helsinki, 00560, Finland^{2}National Research University Higher School of Economics, Nizhny Novgorod, 603155, Russia

^{1}Finnish Meteorological Institute, Helsinki, 00560, Finland^{2}National Research University Higher School of Economics, Nizhny Novgorod, 603155, Russia

**Correspondence**: Elena Shevnina (elena.shevnina@fmi.fi)

**Correspondence**: Elena Shevnina (elena.shevnina@fmi.fi)

Abstract

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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^{HYDRO} (the MARkov Chain System) model that can be applied to
predict future runoff extremes. The MARCS^{HYDRO} model simulates
statistical estimators of multi-year runoff in order to perform future
projections in a probabilistic form. Projected statistics of the
meteorological variables available in climate scenarios force the model.
This study introduces the new model's core version and provides a user guide
together with an example of the model set-up in a single case study. In this
case study, the model simulates the projected EPCs of annual runoff under
three climate scenarios. The scope of applicability and limitations of the
model's core version 0.2 are discussed.

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Shevnina, E. and Silaev, A.: The probabilistic hydrological MARCS^{HYDRO} (the MARkov Chain System) model: its structure and core version 0.2, Geosci. Model Dev., 12, 2767–2780, https://doi.org/10.5194/gmd-12-2767-2019, 2019.

1 Introduction

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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^{HYDRO} (the MARkov Chain System) were accepted for the new version, version 0.2 (see Sect. 1). This model core allows the prediction of a skewness parameter of a Pearson
type III distribution. An example of the model set-up, forcing and output
for a case study of the Iijoki River is given in Sect. 2. The main
features of the model and the limitations of the AFA method are formulated
in the Discussions section in order to better place the MARCS^{HYDRO}
model among other hydrological models.

2 Model description

Back to toptop
The probabilistic hydrological MARCS^{HYDRO} model consists of six blocks
(Shevnina, 2015). Figure 1 shows the tools for data analysis grouped into
blocks: two blocks for the analysis and screening of observed data (DPB and
DSB); a block with the model parametrization, cross-validation and hindcasts (“forecasts in the
past”; PHP); a block to visualize the model's results (VAB); and a block
with socio-economic applications (EAB). Shevnina and Gaidukova (2017)
provided details about the algorithms already implemented in each block in
the model. In this paper, only version 0.2, for the model's core, is
introduced. The formulas behind the model's core version 0.1 are published in
the Appendix of the work of Shevnina et al. (2017).

The MARCS^{HYDRO} model simulates three non-central statistical moments of
multi-year runoff based on the means of precipitation calculated over a
period of 20–30 years. Now, the model's application is only limited by a
prediction on the climate scale. The development of a socio-economic
infrastructure also needs the climate-scale prediction of river runoff
(Milly et al., 2008) because water extremes, such as floods and droughts,
lead to economical losses. The AFA approach has found practical applications
in building construction (Shevnina et al., 2017; Kovalenko, 2009). The
MARCS^{HYDRO} model allows a quick analysis of the runoff extremes
under different climate scenarios. The model needs fewer computational
resources because it simulates the parameters of the distribution, while
conceptual hydrological models simulate the runoff time series.

The MARCS^{HYDRO} model parametrization, cross-validation and hindcasts
need observations of the river water discharges of a hydrological network
for a period in the past (Kovalenko, 1993). For the cross-validation, the
yearly time series of river runoff are split into two sub-periods, namely
the training period and the control period (Shevnina et al., 2017). The splitting
year corresponds to the year when the statistically significant difference
in observations within two periods is detected by the Student and
Kolmogorov–Smirnov tests (Kovalenko, 1993; Kovalenko et al., 2006). The
description of the analysis and screening of the observed river runoff time
series, as well as the model cross-validation procedure, fell outside the
topics of this paper. We focused on the equations behind the model's core
version 0.2 and its limitations.

Two blocks of the MARCS^{HYDRO} model are needed to analyse and screen the
observations. The time series of river runoff and precipitation are required
for a period as long as possible. However, the length of yearly time
series on water discharges does not usually exceed 80–90 years.
Hydrological yearbooks or runoff data sets provide observations at sites of
national hydrological networks, and the river runoff is expressed as a
volumetric flow rate (water discharge, m^{3} s^{−1}). In the data
preparation block of the model, the volumetric flow rate (m^{3} s^{−1})
is converted to a specific water discharge (ARR, mm yr^{−1}):

$$\mathrm{ARR}=\mathrm{1000}QT/A,$$

where *Q* is the yearly average water discharge (m^{3} s^{−1}), *T* is the number
of seconds in a year and *A* is the catchment area (m^{2}). In the data
screening block of the model, the yearly time series of ARR are used in the
analysis of homogeneity and trends (Dahmen and Hall, 1990) and to define a
period for the model parametrization (called the “reference period” by
Shevnina et al., 2017). Then, the reference three non-central moments
${m}_{k}({m}_{k}=\mathrm{1}/n\sum _{i=\mathrm{1}}^{n}{\mathrm{ARR}}_{i}^{k}$ for *k*=1,
2, 3) are estimated from time series of ARR using the method of moments (van
Gelder et al., 2006).

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^{−1}) is calculated
from the observed yearly time series for the reference period. The mean
annual precipitation rate for the future period can be calculated from an
output of any global/regional climate model or even a set of models. In a
study on the catchment scale, the time series of water discharges can be
extracted from the Global Runoff Data Centre (GRDC), while the precipitation
rate can be estimated from gridded data sets (Willmott and Robeson, 1995).
These two data sets were used to perform an example of the model application
on the Iijoki River basin.

The MARCS^{HYDRO} model allows the simulation of the non-central moments
of runoff that can be used for the construction of probability distribution
(or an EPC); in other words, it provides a probabilistic form of prediction.
The end product of the model is the probability density function (PDF) (or
the EPC), and there are no simulated time series of runoff to compare with
the observations. Kovalenko (1993) suggested comparing the simulated PDF
with an empirical PDF by using known statistical tests such as the
Kolmogorov–Smirnov test (Smirnov, 1948). In the PHP block of the
MARCS^{HYDRO} model, a specific cross-validation procedure allows
conclusions to be drawn about the model's validation and the quality of
hindcasts. For the model's cross-validation, the observed time series of
river runoff is divided into two sub-periods, namely the training period and
the control period. The splitting year corresponds to the year when a
statistically significant difference in mean values is estimated over two
periods. In this study, we did not pay much attention to the
cross-validation procedure since it is described in
detail in Shevnina et al. (2019) for the model version 0.2.

In our study, the core version 0.2 of the probabilistic MARCS^{HYDRO} model
was suggested instead of version 0.1 (Shevnina et al., 2017). Version 0.2
allows the evaluation of the skewness parameter of the Pearson type III
distribution. In the new core, the non-central statistical moments of the
ARR were calculated as follows:

$$\begin{array}{}\text{(1)}& {\displaystyle}{m}_{\mathrm{1}}=a-{b}_{\mathrm{1}},\text{(2)}& {\displaystyle}{m}_{\mathrm{2}}=-{b}_{\mathrm{0}}-\mathrm{2}{m}_{\mathrm{1}}{b}_{\mathrm{1}}+{m}_{\mathrm{1}}a,\text{(3)}& {\displaystyle}{m}_{\mathrm{3}}=-\mathrm{2}{m}_{\mathrm{1}}{b}_{\mathrm{0}}-\mathrm{3}{m}_{\mathrm{2}}{b}_{\mathrm{1}}+{m}_{\mathrm{2}}a,\end{array}$$

where *m*_{1}, *m*_{2} and *m*_{3} are the moment estimates of the non-central
statistical moments of the ARR; and *a*, *b*_{0}, *b*_{1} and *b*_{2} are the
parameters of the distributions of the Pearson equation (Andreev et al., 2005).

To set up the MARCS^{HYDRO} model, observations of water discharges are
needed. For the reference period (denoted by low index *r*) the moments'
estimates for the non-central moments (*m*_{1r}, *m*_{2r}, *m*_{3r}) were first
calculated from observed times series of runoff (mm yr^{−1}); then the
non-central moments were used to evaluate the parameters of the Pearson
equation *a*, *b*_{0} and *b*_{1}:

$$\begin{array}{}\text{(4)}& {\displaystyle}a=\mathrm{0.5}(\mathrm{5}{m}_{\mathrm{1}r}{m}_{\mathrm{2}r}-\mathrm{4}{m}_{\mathrm{1}r}^{\mathrm{3}}-{m}_{\mathrm{3}r})/({m}_{\mathrm{2}r}-{m}_{\mathrm{1}r}^{\mathrm{2}}),\text{(5)}& {\displaystyle}{b}_{\mathrm{0}}=\mathrm{0.5}({m}_{\mathrm{1}r}^{\mathrm{2}}{m}_{\mathrm{2}r}-\mathrm{2}{m}_{\mathrm{2}r}^{\mathrm{2}}+{m}_{\mathrm{1}r}{m}_{\mathrm{3}r})/({m}_{\mathrm{2}r}-{m}_{\mathrm{1}r}^{\mathrm{2}}),\text{(6)}& {\displaystyle}{b}_{\mathrm{1}}=\mathrm{0.5}(\mathrm{3}{m}_{\mathrm{1}r}{m}_{\mathrm{2}r}-\mathrm{2}{m}_{\mathrm{1}r}^{\mathrm{3}}-{m}_{\mathrm{3}r})/({m}_{\mathrm{2}r}-{m}_{\mathrm{1}r}^{\mathrm{2}}).\end{array}$$

Then, the parameters of the linear filter model (see Appendix 1 for details),
$\stackrel{\mathrm{\u203e}}{c}$, ${G}_{\stackrel{\mathrm{\u0303}}{N}}$ and ${G}_{\stackrel{\mathrm{\u0303}}{c}\stackrel{\mathrm{\u0303}}{N}}$, denoted
with a low index *r*, were calculated:

$$\begin{array}{}\text{(7)}& {\displaystyle}{\stackrel{\mathrm{\u203e}}{c}}_{r}={\stackrel{\mathrm{\u203e}}{N}}_{\mathrm{r}}/\left(a-{b}_{\mathrm{1}}/\mathrm{2}\right),\text{(8)}& {\displaystyle}{G}_{\stackrel{\mathrm{\u0303}}{N}r}=-\mathrm{2}{b}_{\mathrm{0}}{\stackrel{\mathrm{\u203e}}{N}}_{\mathrm{r}}/\left(a-{b}_{\mathrm{1}}/\mathrm{2}\right),\text{(9)}& {\displaystyle}{G}_{\stackrel{\mathrm{\u0303}}{c}\stackrel{\mathrm{\u0303}}{N}r}={b}_{\mathrm{1}}{\stackrel{\mathrm{\u203e}}{N}}_{\mathrm{r}}/\left(a-{b}_{\mathrm{1}}/\mathrm{2}\right),\end{array}$$

where ${\stackrel{\mathrm{\u203e}}{N}}_{\mathrm{r}}$ is the mean annual precipitation rate (mm yr^{−1})
estimated from the observed time series as an average over any chosen reference
period.

To force the MARCS^{HYDRO} model, the outputs from global-/regional-scale
climate models are needed. The Coupled Model Intercomparison Project 5 (CMIP5;
Taylor et al., 2012) is one collection of data sets that is available for
climate-scale hydrological studies. At present, the model only needs to be
forced by mean precipitation (mm yr^{−1}), evaluated for a future
period of 20–30 years. A low index pr indicated that the values were estimated
for the future, and ${\stackrel{\mathrm{\u203e}}{N}}_{\mathrm{pr}}$ is estimated from climate scenarios.
Following the assumption that $\stackrel{\mathrm{\u203e}}{c}$, ${G}_{\stackrel{\mathrm{\u0303}}{N}}$ and ${G}_{\stackrel{\mathrm{\u0303}}{c}\stackrel{\mathrm{\u0303}}{N}}$
are constant for both periods,
${\stackrel{\mathrm{\u203e}}{c}}_{r}={\stackrel{\mathrm{\u203e}}{c}}_{\mathrm{pr}}$, ${\mathrm{G}}_{\stackrel{\mathrm{\u0303}}{N}r}={G}_{\stackrel{\mathrm{\u0303}}{N}\mathrm{pr}}$ and
${G}_{\stackrel{\mathrm{\u0303}}{c}\stackrel{\mathrm{\u0303}}{N}r}={G}_{\stackrel{\mathrm{\u0303}}{c}\stackrel{\mathrm{\u0303}}{N}\mathrm{pr}}$
(a “basic parametrization scheme” according to Kovalenko, 1993); new
parameters of the Pearson equation are calculated from ${\stackrel{\mathrm{\u203e}}{N}}_{\mathrm{pr}}$:

$$\begin{array}{}\text{(10)}& {\displaystyle}a=\left({G}_{\stackrel{\mathrm{\u0303}}{c}\stackrel{\mathrm{\u0303}}{N}\mathrm{pr}}+\mathrm{2}{\stackrel{\mathrm{\u203e}}{N}}_{\mathrm{pr}}\right)/\left(\mathrm{2}{\stackrel{\mathrm{\u203e}}{c}}_{\mathrm{pr}}\right),\text{(11)}& {\displaystyle}{b}_{\mathrm{0}}=-{G}_{\stackrel{\mathrm{\u0303}}{N}\mathrm{pr}}/\left(\mathrm{2}{\stackrel{\mathrm{\u203e}}{c}}_{\mathrm{pr}}\right),\text{(12)}& {\displaystyle}{b}_{\mathrm{1}}={G}_{\stackrel{\mathrm{\u0303}}{c}\stackrel{\mathrm{\u0303}}{N}\mathrm{pr}}/{\stackrel{\mathrm{\u203e}}{c}}_{\mathrm{pr}}.\end{array}$$

Finally, the non-central moments of runoff are calculated for the projected period (denoted by a low index pr):

$$\begin{array}{}\text{(13)}& {\displaystyle}{m}_{\mathrm{1}\mathrm{pr}}=a-{b}_{\mathrm{1}},\text{(14)}& {\displaystyle}{m}_{\mathrm{2}\mathrm{pr}}=-{b}_{\mathrm{0}}-\mathrm{2}{m}_{\mathrm{1}\mathrm{pr}}{b}_{\mathrm{1}}+a\phantom{\rule{0.25em}{0ex}}{m}_{\mathrm{1}\mathrm{pr}},\text{(15)}& {\displaystyle}{m}_{\mathrm{3}\mathrm{pr}}=-\mathrm{2}{m}_{\mathrm{1}\mathrm{pr}}{b}_{\mathrm{0}}-\mathrm{3}{m}_{\mathrm{2}\mathrm{pr}}{b}_{\mathrm{1}}+a\phantom{\rule{0.25em}{0ex}}{m}_{\mathrm{2}\mathrm{pr}}.\end{array}$$

It should be noted that in core version 0.2 the linear filter model includes
the multiplicative stochastic component (see Appendix 1 for details). It may
lead to unstable solutions for the Fokker–Planck–Kolmogorov equation
(*m*_{k}→∞) for statistical moments of high orders. Two
methods for getting stable solutions for the Fokker–Planck–Kolmogorov
equation are suggested by Kovalenko (2004), and one of them is already
implemented in core version 0.1 (Shevnina et al., 2017).

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):

$$\begin{array}{}\text{(16)}& {\displaystyle}\mathrm{CV}=\sqrt{\left({m}_{\mathrm{2}}-{m}_{\mathrm{1}}^{\mathrm{2}}\right)}/{m}_{\mathrm{1}},\text{(17)}& {\displaystyle}\mathrm{CS}=\left({m}_{\mathrm{3}}-\mathrm{3}{m}_{\mathrm{2}}{m}_{\mathrm{1}}+\mathrm{2}{m}_{\mathrm{1}}^{\mathrm{3}}\right)/{\mathrm{CV}}^{\mathrm{3}}{m}_{\mathrm{1}}^{\mathrm{3}}.\end{array}$$

The MARCS^{HYDRO} model output includes the estimates of the mean values of CV and CS, calculated for the reference period from observations, as well as
these estimates simulated from mean precipitation for the projected period.
The ordinates of the EPC available from lookup tables then allow the
calculation of the runoff values together with their exceedance probability.

3 Model application: a case study

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In our study, we chose the basin of the Iijoki River at the Raasakka gauge
(lat 25.4^{∘}, long. 65.3^{∘}) in order to give an
example of the application of the MARCS^{HYDRO} model on the catchment
scale. The Iijoki River is located in north-west Finland, and the Raasakka
gauge outlines a watershed area of over 14 191 km^{2}. The catchment has a
small population, and there are no hydropower plants of multi-year regulation
to affect the natural regime of the annual cycle. Thus, one can expect that
historical yearly time series of the annual runoff rate do not contain
trends connected to artificial regulation. This case study shows an example
of the set-up and output of the probabilistic MARCS^{HYDRO} model.

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 = 6 854 600) cover the period
1911–2014, and they do not contain gaps. This period was considered as the
reference period. The annual specific water discharge (ARR, mm yr^{−1})
was calculated from the average volumetric water discharge for each year in
the reference period. Then, the non-central moments were calculated from the
yearly time series of the ARR with the method of moments (see Table 1). The
reference climatology (the means of precipitation and air temperature) was
evaluated from the data set of NOAA (NOAA/OAR/ESRL PSD, Boulder, Colorado,
USA) at a grid node nearest to the watershed centroid (this technique will
be discussed in a separate paper, as will the methods of a forcing
pre-analysis).

The values of *m*_{1r}, *m*_{2r} and *m*_{3r}, the moments of runoff, and the mean of precipitation (${\stackrel{\mathrm{\u203e}}{N}}_{\mathrm{r}})$ were evaluated from observations. The mean air
temperature (${\stackrel{\mathrm{\u203e}}{T}}_{\mathrm{r}}{)}^{\ast}$ was not used in the model set-up in
the case of the Iijoki River; however this value allows advancement of the model
parametrization (Shevnina et al., 2017).

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^{HYDRO} model. For each RCP scenario,
the projections of annual precipitation rate were applied to test how the
MARCS^{HYDRO} model simulates the EPC under different forcing
trajectories. For the period of 2020–2050 (considered the projected
period), the mean values of the precipitation rate (mm yr^{−1}) were
calculated based on four world-leading global climate models. We used the
outputs from the global models CaESM2 (Chylek et al., 2011), HadGEM2-ES
(Collins et al., 2011), INM-CM4 (Volodin et al., 2010) and MPI-ESM-LR
(Giorgetta et al., 2013) (see Table 2). The mean values of the precipitation
rate varied by 2–5 % of the model's average over the RCP scenarios;
however, these values alter substantially between the climate models. Among
the outputs considered, the MPI-ESM-LR model projects the highest changes in
the mean values of the precipitation rate compared to the reference period
(see Tables 1 and 2). The HadGEM2-ES model gives the lowest values for the
mean values of the precipitation rate. The projected means of the
precipitation rate varied slightly between the scenarios. At the same time,
they exhibited a significant range of changes among the climate models (the
mean values of the precipitation rate ranges from 619 to 737 mm yr^{−1})
for the case of the Iijoki River at Raasakka.

Projected mean air temperature (${\stackrel{\mathrm{\u203e}}{T}}_{\mathrm{pr}}{)}^{\ast}$ is needed for a regional parametrization scheme (see details in Shevnina, 2011), and these values were not used in the model forcing in the case of the Iijoki River at Raasakka. ${\stackrel{\mathrm{\u203e}}{N}}_{\mathrm{pr}}$ is the projected mean annual precipitation amount.

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^{HYDRO} model. Table 3 shows the modelling results for
the HadGEM2-ES and MPI-ESM-LR global models, for which the water discharges with
10 % and 90 % exceedance probabilities are given. The ordinates of the
Pearson type III distribution were extracted from the lookup tables used in
hydrological engineering (Druzhinin and Sikan, 2001), and they allow
runoff to be expressed as water discharge (m^{3} s^{−1}). For the Iijoki River
at Raasakka, the mean values of ARR and CV vary under the RCP scenarios by
over 7 % and 5 % correspondingly. The maximum alteration in the
projected mean values of ARR was obtained under RCP8.5 (619 to 737 mm yr^{−1}). Under the projections of the MPI-ESM-LR model, the mean ARR
increases by over 17 %.

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^{HYDRO} model (Shevnina et al., 2019). In the further development of
the visualization block, it would be important to involve water managers and
decision makers in order to better outline practical applications for the
probabilistic hydrological model.

4 Discussions

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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^{HYDRO} model is
computationally cheaper when compared to lumped or semi-distributed
physically based hydrological models. It can easily be coupled with global
and regional climate models, and it can provide the express analysis of
water resources under a modern version of the future climate.

In this paper we described the structure for the probabilistic hydrological
MARCS^{HYDRO} model, together with the AFA method that lies behind
the new model's core version 0.2. The AFA method has a more than
25-year-long history; however, most of the studies are published in Russian
(Kovalenko, 1993, 2004, 2009; Kovalenko et al., 2010). The AFA method is
based on the statistical theory of automatic systems (Pugachev et al.,
1974), which is an outsider among the classical hydrological
disciplines. The AFA method is one simplification of the
Fokker–Planck–Kolmogorov equation approach that has been developed in the
Russian State Hydrometeorological University. It has been tested in many
case studies on river basins located in Russia, Colombia, Bolivia and Mali, etc.
There are also a number of publications in English (Rosmann and
Domínguez, 2017; Shevnina et al., 2017; Kovalenko, 2014; Domínguez
and Rivera, 2010; Viktorova and Gromova, 2008). In this paper we
formulated the theory logically in an attempt to provide the equations for
the new core 0.2 of the MARCS^{HYDRO} model; however, it also needs to
describe the AFA method that lies behind it.

The probabilistic hydrological MARCS^{HYDRO} model includes the core
versions 0.1 and 0.2. In both cores, only three non-central moments are
evaluated to construct the EPC within the theoretical distribution the
Pearson III type, which is among the traditional distributions of the
frequency and risk analysis in hydrology (Kite, 1977; Rogdestvenskiy and
Chebotarev, 1974; Sokolovskiy, 1968; Elderton, 1969; Benson, 1968). The
model simulates three estimates of non-central moments of runoff instead of
a runoff time series, and this circumstance makes the computations by the
MARCS^{HYDRO} model low-cost compared to conceptual hydrological
models (Arheimer and Lindström, 2015; Veijalainen et al., 2012). The
MARCS^{HYDRO} model allows the projections of runoff to be put in terms of
probability; that is, they appear as runoff values together with their
exceedance probability.

The MARCS^{HYDRO} model includes six modules, and each module allows
improvements by including new methods. In this paper, the new model – core
version 0.2, extended to simulate the third statistical estimator (skewness)
– is presented. The applicability of core version 0.2 is limited by the
assumptions behind the AFA approach. Among others, there is the
“quasi-stationarity” assumption for the expected climate change. In this
case, the climate is described by the statistical estimators (i.e. mean
value, variability, etc.) of precipitation, air temperature,
evapotranspiration and river runoff, etc., for the period of 20–30 years. It is
assumed to consider two time period periods with statistically different
climates, namely the reference period and the projected period. Another
limitation is connected to the linear filter stochastic model (for details,
see Appendix 1) used in core version 0.2. It should be noted that there is a
multiplicative component in the model core, and it may lead to unstable
solutions of the Fokker–Planck–Kolmogorov equation. Kovalenko (2004) suggests
two solutions that result in stable solutions of the Fokker–Planck–Kolmogorov
equation. One of the solutions was given by Kovalenko et al. (2010) and is
coded in the model core version 0.1 (Shevnina et al., 2017). However, a
checking procedure needs to be applied before using this core version. In
the checking procedure we plan to use the “beta criterion” method
suggested by Kovalenko (2004) to further develop the MARCS^{HYDRO} model.

Further improvements of the MARCS^{HYDRO} model are going to be
implemented in the block of parametrization and hindcasts. Recently, only
the basic parametrization scheme (Kovalenko, 1993) has been included. This
basic scheme gives over 70 %–80 % successful hindcasts (forecasts in the
past) in the model cross-validation (Shevnina et al., 2017), and the
implementation of a regionally oriented parametrization scheme (Shevnina,
2011) is the next step. It needs to include a mean value of the air
temperature of the parameter, connected to “noised” watershed physiography
in Eq. (A4), the inverse of the runoff coefficient in the work of
Kovalenko (1993). It is also important to study the role of the spatial resolution of
meteorological forcing in affecting the modelling uncertainties for the
simulated mean values of the CV and CS of runoff.

To place the probabilistic MARCS^{HYDRO} model among other hydrological
models, its practical applications need to be better outlined. The model
serves as a probabilistic form of long-term hydrological projections, and they
require adaptation to the needs of water engineers and water managers as a
tool for risk analysis under the expected climate change. The projected EPCs
of multi-year river runoff can be applied in designing bridges, pipes and dams,
etc., in order to minimize the future risks connected to extreme floods
(Shevnina et al., 2017; Kovalenko et al., 2014; Kovalenko, 2009) and to
water shortage due to droughts (Viktorova and Gromova, 2008). It is
important to define informative forms for the outputs of the MARCS^{HYDRO}
model that can be adapted to the needs of practice, and the development of
the block of economic application is among the others studies that are to be
continued in close cooperation with water managers and decision makers.

5 Conclusions

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The paper describes the theory and assumptions of the AFA approach, as well
as the probabilistic hydrological MARCS^{HYDRO} model's structure and core
version 0.2. The features of the model are the close connection to water
engineering due to providing the runoff projection in terms of probability,
cheapness in terms of computational cost and a wide range of techniques
allowing model improvement. In the new core, the third moment linked to the
location parameter of the Pearson type III distribution (or asymmetry) was
implemented for simulation. In the previous version of the model core, a
constant CS∕CV ratio is used to calculate the location parameter of the
distribution.

To give a practical example of how to set up the MARCS^{HYDRO} model, the
case of the Iijoki River at Raasakka located in Finland was considered. The
model simulated the tailed values of 10 % and 90 % of annual water
discharge from the outputs of global climate models. We showed two forms of
the probabilistic projections of runoff: an EPC and the runoff values with
their exceedance probability. This case study of the Iijoki River at
Raasakka shows that the MARCS^{HYDRO} model gives reasonable results for
the meteorological projections considered. The practical applications in
water management and decision-making should be clarified in further studies
in close co-operation with water engineers.

Code availability

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Code availability.

Currently, the MARCS^{HYDRO} model code is hosted at https://github.com/ElenaShe000/MARCS (last access: 2 July 2019; Shevnina, 2015), with details of its applications for
catchment-scale case studies. The model source code for core version 0.2 is
distributed under the Creative Commons Attribution 4.0 License and can be
downloaded from https://zenodo.org/record/1220096/\#.XSLG1qLRaUl (last access: 4 May 2019; Shevnina and Krasikov, 2018) and used freely in
scientific research with reference to this publication. We hope that this
type of licence provides the best way to create a community of motivated
people to further develop the model. Then, the source code will be
distributed under the terms of a user agreement.

Data availability

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Data availability.

The following data sets can be used to set up and force the MARCS^{HYDRO} model: the
GRDC (GRDC, 56068 Koblenz, Germany), the NOAA/OAR/ESRL PSD (Boulder,
Colorado, USA) and CMIP5 (Taylor et al., 2012).

Sample availability

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Sample availability.

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

Appendix A: The theoretical basis for core version 0.2

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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):

$$\begin{array}{}\text{(A1)}& \begin{array}{rl}& {a}_{n}\left(t\right){\displaystyle \frac{{d}^{n}Q}{\mathrm{d}{t}^{n}}}+{a}_{n-\mathrm{1}}\left(t\right){\displaystyle \frac{{d}^{n-\mathrm{1}}Q}{\mathrm{d}{t}^{n-\mathrm{1}}}}+\mathrm{\dots}+{a}_{\mathrm{1}}\left(t\right){\displaystyle \frac{\mathrm{d}Q}{\mathrm{d}t}}+{a}_{\mathrm{0}}\left(t\right)Q=\\ & {b}_{n}\left(t\right){\displaystyle \frac{{d}^{n}P}{\mathrm{d}{t}^{n}}}+{b}_{n-\mathrm{1}}\left(t\right){\displaystyle \frac{{d}^{n-\mathrm{1}}P}{\mathrm{d}{t}^{n-\mathrm{1}}}}+\mathrm{\dots}+{b}_{\mathrm{1}}\left(t\right){\displaystyle \frac{\mathrm{d}P}{\mathrm{d}t}}+{b}_{\mathrm{0}}\left(t\right)P,\end{array}\end{array}$$

where *Q* is the runoff in volumetric flow rate, *P* is the precipitation in
volumetric flow rate (rain, snowmelt), and the coefficients *a*_{i} and
*b*_{i} are the empirical parameters of a translating system. These
coefficients are the lumped parameters of the black box model. The solution
to Eq. (A1) for zero initial conditions gives

$$\begin{array}{}\text{(A2)}& Q\left(t\right)=\underset{\mathrm{0}}{\overset{t}{\int}}h(t,\mathit{\tau})P\left(\mathit{\tau}\right)\mathrm{d}\mathit{\tau},\end{array}$$

where the function *h*(*t*,*τ*) represents the response of a river
basin at time *t* to a single portion of precipitation *P*(*τ*) at time (*τ*). In the AFA approach, a river basin is considered as a linear system, transforming the
annual precipitation into the annual runoff:

$$\begin{array}{}\text{(A3)}& {a}_{\mathrm{1}}\left(t\right){\displaystyle \frac{\mathrm{d}Q}{\mathrm{d}t}}+{a}_{\mathrm{0}}\left(t\right)Q={b}_{\mathrm{0}}\left(t\right)P.\end{array}$$

On the other hand, a river basin can be considered as a linear system with stochastic components in the input function and the model parameter (Kovalenko, 1993):

$$\begin{array}{}\text{(A4)}& \mathrm{d}Q=[-(\stackrel{\mathrm{\u203e}}{c}+\stackrel{\mathrm{\u0303}}{c}\left(t\right))Q+(\stackrel{\mathrm{\u203e}}{N}+\stackrel{\mathrm{\u0303}}{N}\left(t\right)\left)\right]\mathrm{d}t,\end{array}$$

where ${a}_{\mathrm{0}}\left(t\right)=\stackrel{\mathrm{\u203e}}{c}+\stackrel{\mathrm{\u0303}}{c}\left(t\right)$ is the stochastic parameter
of the system (a noised watershed physiography, the inverse of runoff
coefficient), ${b}_{\mathrm{0}}\left(t\right)P=\stackrel{\mathrm{\u203e}}{N}+\stackrel{\mathrm{\u0303}}{N}\left(t\right)$ is the stochastic
input for the system (a noised precipitation) and *a*_{1}=1. The
stochastic components of $\stackrel{\mathrm{\u0303}}{c}\left(t\right)$ and $\stackrel{\mathrm{\u0303}}{N}\left(t\right)$ are the Gaussian
“white noise” with zero means, and their intensities are ${G}_{\stackrel{\mathrm{\u0303}}{c}}$ and ${G}_{\stackrel{\mathrm{\u0303}}{N}}$. The intensities are mutually correlated as
${K}_{\stackrel{\mathrm{\u0303}}{c}\stackrel{\mathrm{\u0303}}{N}}\left(\mathit{\tau}\right)=E\left(\stackrel{\mathrm{\u0303}}{c}\right(t\left)\stackrel{\mathrm{\u0303}}{N}\right(t+\mathit{\tau}\left)\right)={G}_{\stackrel{\mathrm{\u0303}}{c}\stackrel{\mathrm{\u0303}}{N}}\mathit{\delta}\left(\mathit{\tau}\right)$. It should be noted that the
multiplicative parameter $\stackrel{\mathrm{\u203e}}{c}+\stackrel{\mathrm{\u0303}}{c}\left(t\right)$ in Eq. (A4) is the
sum of the constant $\stackrel{\mathrm{\u203e}}{c}$ and Gaussian white noise $\stackrel{\mathrm{\u0303}}{c}\left(t\right)$,
and it may lead to unstable solutions of the Fokker–Planck–Kolmogorov
equation (i.e. it may lead to infinite statistical moments of high orders).
It limits the application of the AFA method (Kovalenko, 1993). Kovalenko (2004) suggests two solutions, and we will introduce them in a further paper.

The Fokker–Planck–Kolmogorov equation can be applied to simulate the
probability density function (PDF) for the stochastic *Q*(*t*) in Eq. (A4)
(Kovalenko, 1993; Pugachev et al., 1974):

$$\begin{array}{}\text{(A5)}& \begin{array}{rl}{\displaystyle \frac{\partial p(Q,t)}{\partial t}}& =-{\displaystyle \frac{\partial}{\partial Q}}\left(A\right(Q\left)p\right(Q,t\left)\right)+\mathrm{0.5}{\displaystyle \frac{{\partial}^{\mathrm{2}}}{\partial {Q}^{\mathrm{2}}}}\\ & \left(B\right(Q\left)p\right(Q,t\left)\right),\end{array}\end{array}$$

where *p*(*Q*,*t*) is the PDF of *Q* at time *t*; and the drift
coefficient (*A*(*Q*)) and diffusion coefficients (*B*(*Q*)) are calculated as
follows (Kovalenko, 1993; Pugachev, 1974):

$$\begin{array}{}\text{(A6)}& {\displaystyle}A\left(Q\right)=-(\stackrel{\mathrm{\u203e}}{c}-\mathrm{0.5}{G}_{\stackrel{\mathrm{\u0303}}{c}})Q-\mathrm{0.5}{G}_{\stackrel{\mathrm{\u0303}}{c}\stackrel{\mathrm{\u0303}}{N}}+\stackrel{\mathrm{\u203e}}{N},\text{(A7)}& {\displaystyle}B\left(Q\right)={G}_{\stackrel{\mathrm{\u0303}}{c}}{Q}^{\mathrm{2}}-\mathrm{2}Q{G}_{\stackrel{\mathrm{\u0303}}{c}\stackrel{\mathrm{\u0303}}{N}}+{G}_{\stackrel{\mathrm{\u0303}}{N}}.\end{array}$$

The analytical solution of Eq. (A5) is difficult and not always needed for
practical applications in water engineering since the PDFs of runoff are
modelled from a set of statistical estimators, and the moments are from,
among others, van Gelder et al. (2006). The PDFs are described with the set
of moments ${m}_{k}=\underset{-\mathrm{\infty}}{\overset{+\mathrm{\infty}}{\int}}{Q}^{k}p(Q,t)\mathrm{d}Q$ (where *k* is the number of the moment, *k*→∞). To obtain the equations for *m*_{k}, both sides of Eq. (A5) were multiplied by a differentiable function *ψ*(*Y*) and then
integrated within limits from −∞ to +∞ by
*Q* (however, it is supposed that *Q*>0):

$$\begin{array}{}\text{(A8)}& \begin{array}{rl}& {\displaystyle \frac{d\left(\underset{-\mathrm{\infty}}{\overset{+\mathrm{\infty}}{\int}}\mathit{\psi}\right(Q\left)p\right(Q,t\left)\mathrm{d}Q\right)}{\mathrm{d}t}}=\underset{-\mathrm{\infty}}{\overset{+\mathrm{\infty}}{\int}}p(Q,t)A\left(Q\right){\displaystyle \frac{\partial \mathit{\psi}\left(Q\right)}{\partial Q}}\mathrm{d}Q\\ & +\mathrm{0.5}\underset{-\mathrm{\infty}}{\overset{+\mathrm{\infty}}{\int}}p(Q,t)B\left(Q\right){\displaystyle \frac{{\partial}^{\mathrm{2}}\mathit{\psi}\left(Q\right)}{\partial {Q}^{\mathrm{2}}}}\mathrm{d}Q.\end{array}\end{array}$$

Then, *ψ*(*Q*) was replaced with *ψ*(*Q*)=*Q*^{k}, and Eq. (A8) was written as

$$\begin{array}{}\text{(A9)}& \begin{array}{rl}& {\displaystyle \frac{\mathrm{d}{m}_{k}\left(t\right)}{\mathrm{d}t}}=\underset{-\mathrm{\infty}}{\overset{+\mathrm{\infty}}{\int}}p(Q,t)A\left(Q\right){\displaystyle \frac{\partial \left({Q}^{k}\right)}{\partial Q}}\mathrm{d}Q+\mathrm{0.5}\\ & \underset{-\mathrm{\infty}}{\overset{+\mathrm{\infty}}{\int}}p(Q,t)B\left(Q\right){\displaystyle \frac{{\partial}^{\mathrm{2}}\left({Q}^{k}\right)}{\partial {Q}^{\mathrm{2}}}}\mathrm{d}Q.\end{array}\end{array}$$

For a stationary random process, $\mathrm{d}{m}_{k}\left(t\right)/\mathrm{d}t=\mathrm{0}$, and the drift and diffusion coefficients are constant. Thus, Eq. (A9) was simplified as follows:

For *k*=1,

$$\begin{array}{}\text{(A10)}& -(\stackrel{\mathrm{\u203e}}{c}-\mathrm{0.5}{G}_{\stackrel{\mathrm{\u0303}}{c}}){m}_{\mathrm{1}}-\mathrm{0.5}{G}_{\stackrel{\mathrm{\u0303}}{c}\stackrel{\mathrm{\u0303}}{N}}+\stackrel{\mathrm{\u203e}}{N}=\mathrm{0}\end{array}$$

For *k*≥2,

$$\begin{array}{}\text{(A11)}& \begin{array}{rl}& -k(\stackrel{\mathrm{\u203e}}{c}-\mathrm{0.5}{\mathrm{kG}}_{\stackrel{\mathrm{\u0303}}{c}}){m}_{k}+k\stackrel{\mathrm{\u203e}}{N}{m}_{k-\mathrm{1}}-k(k-\mathrm{0.5}){G}_{\stackrel{\mathrm{\u0303}}{c}\stackrel{\mathrm{\u0303}}{N}}{m}_{k-\mathrm{1}}\\ & +\mathrm{0.5}k(k-\mathrm{1}){G}_{\stackrel{\mathrm{\u0303}}{N}}{m}_{k-\mathrm{2}}=\mathrm{0}.\end{array}\end{array}$$

Further, the summands in Eqs. (A10)–(A11) were divided by $(\mathrm{2}\stackrel{\mathrm{\u203e}}{c}+{G}_{\stackrel{\mathrm{\u0303}}{c}})$, and new notation was introduced as suggested in the work of Kovalenko (1993) and Pugachev et al. (1974):

$$\begin{array}{rl}& a={\displaystyle \frac{{G}_{\stackrel{\mathrm{\u0303}}{c}\stackrel{\mathrm{\u0303}}{N}}+\mathrm{2}\stackrel{\mathrm{\u203e}}{N}}{\mathrm{2}\stackrel{\mathrm{\u203e}}{c}+{G}_{\stackrel{\mathrm{\u0303}}{c}}}};{b}_{\mathrm{0}}=-{\displaystyle \frac{{G}_{\stackrel{\mathrm{\u0303}}{N}}}{\mathrm{2}\stackrel{\mathrm{\u203e}}{c}+{G}_{\stackrel{\mathrm{\u0303}}{c}}}};{b}_{\mathrm{1}}={\displaystyle \frac{\mathrm{2}{G}_{\stackrel{\mathrm{\u0303}}{c}\stackrel{\mathrm{\u0303}}{N}}}{\mathrm{2}\stackrel{\mathrm{\u203e}}{c}+{G}_{\stackrel{\mathrm{\u0303}}{c}}}};\\ & {b}_{\mathrm{2}}=-{\displaystyle \frac{{G}_{\stackrel{\mathrm{\u0303}}{c}}}{\mathrm{2}\stackrel{\mathrm{\u203e}}{c}+{G}_{\stackrel{\mathrm{\u0303}}{c}}}}.\end{array}$$

Then, for *k*= 1, 2, 3, 4 the system of Eqs. (A10)–(A11) includes

$$\begin{array}{}\text{(A12)}& {\displaystyle}& {\displaystyle}{m}_{\mathrm{1}}(\mathrm{2}{b}_{\mathrm{2}}+\mathrm{1})-a+{b}_{\mathrm{1}}=\mathrm{0},\text{(A13)}& {\displaystyle}& {\displaystyle}(\mathrm{3}{b}_{\mathrm{2}}+\mathrm{1}){m}_{\mathrm{2}}+(\mathrm{2}{b}_{\mathrm{1}}-a){m}_{\mathrm{1}}+{b}_{\mathrm{0}}=\mathrm{0},\text{(A14)}& {\displaystyle}& {\displaystyle}(\mathrm{4}{b}_{\mathrm{2}}+\mathrm{1}){m}_{\mathrm{3}}+(\mathrm{3}{b}_{\mathrm{1}}-a){m}_{\mathrm{2}}+\mathrm{2}{b}_{\mathrm{0}}{m}_{\mathrm{1}}=\mathrm{0},\text{(A15)}& {\displaystyle}& {\displaystyle}(\mathrm{5}{b}_{\mathrm{2}}+\mathrm{1}){m}_{\mathrm{4}}+(\mathrm{4}{b}_{\mathrm{1}}-a){m}_{\mathrm{3}}+\mathrm{3}{b}_{\mathrm{0}}{m}_{\mathrm{2}}=\mathrm{0}.\end{array}$$

The set of four moments (*m*_{1}, *m*_{2}, *m*_{3}, *m*_{4}) is sufficient to
model distributions from the Pearson equation (Andreev et al., 2005;
Elderton and Johnson, 1969). However, in water engineering we usually only
use three-parameter probability distributions fitted to observations
(Guidelines, 2004, 1984; Bulletin 17-B, 1982). In this case,
${G}_{\stackrel{\mathrm{\u0303}}{c}}\ll \stackrel{\mathrm{\u203e}}{c}$ is assumed; thus it leads to
${b}_{\mathrm{2}}=-{G}_{\stackrel{\mathrm{\u0303}}{c}}(\mathrm{2}\stackrel{\mathrm{\u203e}}{c}+{G}_{\stackrel{\mathrm{\u0303}}{c}})\approx \mathrm{0}$ and
$(\mathrm{4}{b}_{\mathrm{2}}+\mathrm{1})\approx \mathrm{1}$, $(\mathrm{3}{b}_{\mathrm{2}}+\mathrm{1})\approx \mathrm{1}$,
$(\mathrm{2}{b}_{\mathrm{2}}+\mathrm{1})\approx \mathrm{1}$. To model the PDFs (or EPCs) of annual river
runoff within the Pearson type III distribution, the system of Eqs. (A12)–(A15) is simplified as follows:

$$\begin{array}{}\text{(A16)}& {\displaystyle}& {\displaystyle}-a+{b}_{\mathrm{1}}=-{m}_{\mathrm{1}},\text{(A17)}& {\displaystyle}& {\displaystyle}{b}_{\mathrm{0}}+\mathrm{2}{m}_{\mathrm{1}}{b}_{\mathrm{1}}-a{m}_{\mathrm{1}}=-{m}_{\mathrm{2}},\text{(A18)}& {\displaystyle}& {\displaystyle}\mathrm{2}{m}_{\mathrm{1}}{b}_{\mathrm{0}}+\mathrm{3}{m}_{\mathrm{2}}{b}_{\mathrm{1}}-a{m}_{\mathrm{2}}=-{m}_{\mathrm{3}}.\end{array}$$

Denoting $\mathit{l}\mathit{k}=\left(\begin{array}{l}-{m}_{\mathrm{1}}\\ -{m}_{\mathrm{2}}\\ -{m}_{\mathrm{3}}\end{array}\right)$, $\mathit{x}=\left(\begin{array}{l}{b}_{\mathrm{1}}\\ {b}_{\mathrm{0}}\\ a\end{array}\right)$ and $\mathbf{A}=\left(\begin{array}{ccc}\mathrm{1}& \mathrm{0}& -\mathrm{1}\\ \mathrm{2}{m}_{\mathrm{1}}& \mathrm{1}& -{m}_{\mathrm{1}}\\ \mathrm{3}{m}_{\mathrm{2}}& \mathrm{2}{m}_{\mathrm{1}}& -{m}_{\mathrm{2}}\end{array}\right)$, the parameters *a*, *b*_{0} and *b*_{1} are calculated as
${\mathit{x}}_{i}={D}_{i}/D$, where *D* is the determinant of matrix
**A**, and *D*_{i} is the determinant of the matrix obtained by replacing of the
column *i* (1, 2, 3) in matrix **A** by the vector *l*** k**. Finally, the parameters

$$\begin{array}{}\text{(A19)}& {\displaystyle}{b}_{\mathrm{1}}& {\displaystyle}=\mathrm{0.5}(\mathrm{3}{m}_{\mathrm{1}}{m}_{\mathrm{2}}-\mathrm{2}{m}_{\mathrm{1}}^{\mathrm{3}}-{m}_{\mathrm{3}})/({m}_{\mathrm{2}}-{m}_{\mathrm{1}}^{\mathrm{2}}),\text{(A20)}& {\displaystyle}{b}_{\mathrm{0}}& {\displaystyle}=\mathrm{0.5}({m}_{\mathrm{1}}^{\mathrm{2}}{m}_{\mathrm{2}}-\mathrm{2}{m}_{\mathrm{2}}^{\mathrm{2}}+{m}_{\mathrm{1}}{m}_{\mathrm{3}})/({m}_{\mathrm{2}}-{m}_{\mathrm{1}}^{\mathrm{2}}),\text{(A21)}& {\displaystyle}a& {\displaystyle}=\mathrm{0.5}(\mathrm{5}{m}_{\mathrm{1}}{m}_{\mathrm{2}}-\mathrm{4}{m}_{\mathrm{1}}^{\mathrm{3}}-{m}_{\mathrm{3}})/({m}_{\mathrm{2}}-{m}_{\mathrm{1}}^{\mathrm{2}}).\end{array}$$

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, *a*_{0}, *a*_{1} and *b*_{0}, and two of them are assumed
to be noised. These noised parameters include a constant component
(indicated with a bar) and a Gaussian white noise component (indicated with
a tilde) with their own intensities.

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.

Appendix B: A short user guide for the MARCS model

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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 https://zenodo.org/record/1220096\#.WyTXxxxRVhw, last access: 4 May 2019)
as should the ID number of the catchment (line 51 of model_core.py). To force the model, the projected mean value of the annual
precipitation rate should be evaluated from an output of a climate model,
and then the model_core.py can be run in the Unix command
line: ./model_core.py XXX (where XXX is the mean of the
annual precipitation rate for the projected period). The output of
model_core.py is stored in the output file
model_GPSCH.txt and included in line with the following
format: the ID of the catchment, the first non-central moment estimate of the annual
runoff rate (mm yr^{−1}) for a reference period, the mean value of
the annual precipitation rate (mm yr^{−1}) for a reference period, the
coefficient of variation for a reference period, the coefficient of skewness
for a reference period, the model parameters
$\stackrel{\mathrm{\u203e}}{c}$, ${G}_{\stackrel{\mathrm{\u0303}}{N}}$ and ${G}_{\stackrel{\mathrm{\u0303}}{c}\stackrel{\mathrm{\u0303}}{N}}$, the first non-central
moment estimate of annual runoff rate (mm yr^{−1}) for a projected
period, the mean value of the annual precipitation rate (mm yr^{−1}) for a
projected period, the coefficient of variation for a projected period and the
coefficient of skewness for a projected period.

Author contributions

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Author contributions.

ES contributed to the MARCS^{HYDRO} model coding and to the
Russian–English translation of some parts from Kovalenko (1993),
Kovalenko (2004) and Pugachev et al. (1974). She is responsible for writing the text. AS supported the theoretical part of the AFA method (Pugachev et al., 1974; Kovalenko, 1993) and formulated the equations step by step. Both authors took care of the terms used in the paper, which we tried to write in “a language in common” between water engineering and
radiophysics.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

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.

Financial support

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Financial support.

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

Review statement

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Review statement.

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

References

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Short summary

The paper provides a theory and assumptions behind an advance of frequency analysis (AFA) approach in long-term hydrological forecasting. In this paper, a new core of the probabilistic hydrological model MARkov Chain System (MARCS^{HYDRO}) was introduced, together with the code and an example of a climate-scale prediction of an exceedance probability curve of river runoff with low computational costs.

The paper provides a theory and assumptions behind an advance of frequency analysis (AFA)...

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