Skill Transfer from Meteorological to Runoff Forecasts in Glacierized Catchments

Abstract

Runoff predictions are affected by several uncertainties. Among the most important ones is the uncertainty in meteorological forcing. We investigated the skill propagation of meteorological to runoff forecasts in an idealized experiment using synthetic data. Meteorological forecasts with different skill were produced with a weather generator and fed into two different hydrological models. The experiments were repeated for two glacierized catchments of different sizes and morphological characteristics, and for scenarios of different glacier coverage. The results show that for catchments with high glacierization (>50%), the runoff forecast skill is more dependent on the skill of the temperature forecasts than the one for precipitation. This is because snow and ice melt are strongly controlled by temperature. The influence of the temperature forecast skill diminishes with decreasing glacierization, while the opposite is true for precipitation. Precipitation starts to dominate the runoff skill when the catchment’s glacierization drops below 30%, or when the total contribution of ice and snow melt is less than about 60%. The skill difference between meteorological forecasts and runoff predictions proved to be independent from the lead time, and all results were similar for both the considered hydrological models. Our results indicate that long-range meteorological forecasts, which are typically more skillful in predicting temperature than precipitation, hold particular promise for applications in snow- and glacier-dominated catchments.

1. Introduction

Numerical runoff predictions are crucial for hydropower and drinking water management e.g., [1,2] as well as for drought preparedness e.g., [3,4]. They are usually produced by forcing a hydrological model with meteorological forecasts or climate projections e.g., [5,6,7]. Runoff predictions are affected by various sources of uncertainty that include, among others, the hydrological model input forcing (such as temperature, precipitation, soil moisture, or snow extent) and the hydrological model structure e.g., [8,9].

This study focuses on how the uncertainty of meteorological forecasts transfer to runoff forecasts. In particular, it investigates how the skill (i.e., quality) of temperature and precipitation forecasts impact on the skill of the resulting runoff forecasts. While the skill of the runoff forecasts in precipitation-driven basins is largely independent from the temperature forecasts e.g., [10], the influence of precipitation and temperature in snow-dominated basins is more nuanced. At lead times up to 10 days, Hay et al. [11] and Clark and Hay [12] found that variations in runoff for snow-dominated basins are more influenced by variations in temperature than by variations in precipitation. Using seasonal forecasts (lead times up to one year) and still referring to snow-dominated basins, Olsson and Lindström [13] highlighted that although temperature has a strong impact in spring, precipitation remains the main forcing with respect to runoff generation. Other studies also showed that the large influence of temperature on the runoff skill in early summer is due to the presence of ice and snow at high altitudes [8,14]. Forecasts covering longer lead times, such as decadal forecasts which aim at lead times up to several years e.g., [15,16], were to our knowledge never used to assess the transfer of skill between meteorological and runoff forecasts.

Future climate projections anticipate virtually unanimously an increase in temperature, and, in general, also an increase in precipitation e.g., [17]. Shrinking snow-covered areas e.g., [1,18,19] and reduced glacier extents e.g., [20,21] are therefore expected. This in turn suggests an increasing importance of precipitation in runoff generation also for snow- and ice-dominated catchments e.g., [22,23,24]. In fact, several studies e.g., [25,26,27,28] showed that with decreasing glacierization, the ice melt contribution to runoff will decrease, while snowmelt and precipitation will increase. In the future, precipitation might thus have a larger impact on runoff than today, and the importance of temperature for runoff formation is therefore expected to decrease accordingly. Although the question of how the forecasts skill of precipitation and temperature reflect into runoff forecasts in snow-dominated basins has been partially investigated e.g., [8,11,12,13,14], it has to our knowledge never assessed with respect to glacierized catchments. In particular, quantitative statements about the relation between the skill of meteorological forecasts and the corresponding runoff predictions are missing.

The objective of this study was threefold: First, to investigate how the forecasts skill of the input meteorological time series is transferred to the runoff predictions over different lead times in glacierized catchments; second, to analyze how the forecasts skill is influenced by the degree of glacierization in a catchment; and third, to evaluate the impact of the hydrological model structure on the results.

2. Methods

2.1. General Methodological Workflow

The general methodological workflow of the study is shown in Figure 1. The main elements of the workflow are (i) the meteorological input time series required to drive a hydrological model (Section 2.2); (ii) the hydrological model itself (Section 2.3); and (iii) the resulting runoff time series. Conceptually, we performed all analyses in a virtual world in which the hydrological model is assumed to be prefect. A perfect meteorological input time series thus results in a perfect runoff time series. With this in mind, we used a weather generator to produce a synthetic reference meteorological time series (assumed to be perfect; Section 2.2.1), and superimposed various signals on it to simulate sets of forecasts with varying skill (Section 2.2.2). These synthetic meteorological forecasts were then fed into the same hydrological model, producing a corresponding set of runoff forecasts. The skill of the individual runoff forecasts (Section 2.4) was then computed with respect to the reference runoff time series (the latter is obtained by feeding the reference meteorological time series through the model). The relation by which variations in the skill of the meteorological input propagates to the skill of the hydrological output is the object of our study.

All simulations were performed for two virtual catchments, “A” and “B” (Figure 2), characterized by a glacial hydrological regime. Both catchments have a similar mean air temperature but have a different size, aspect and mean annual precipitation. Individual simulations were performed over several years (decadal time scale) and the degree of glacierization of the two catchments was artificially varied in a set of “Scenarios” (Figure 2) to investigate the glaciers’ role in skill propagation.

2.2. Generation of Synthetic Meteorological Time Series

2.2.1. Reference Meteorological Time Series

The reference meteorological time series (Figure 1b₁), comprising temperature ($T_{ref}$) and precipitation ($P_{ref}$), were generated with the weather generator described in Farinotti [29]. In a nutshell, the weather generator decomposes a given meteorological time series (Figure 1a) in a long term trend, long term monthly and daily anomalies, and three stochastic variability components capturing (i) year-to-year; (ii) month-to-month; and (iii) day-to-day fluctuations (Figure 3b). The three variability components are characterized by a corresponding set of variability parameters, $\mathit{\sigma }=[{\sigma }_{day},{\sigma }_{mon},{\sigma }_{yr}]$. For generating synthetic time series, the weather generator uses a sequence of Auto-Regressive Moving-Average (ARMA) models. In the case of precipitation, a Markov chain model is additionally employed to produce realistic sequences of wet and dry days.

To estimate the variability parameters required to produce the synthetic reference time series for Catchments A and B, the weather generator was trained on real-world meteorological observations used in Farinotti et al. [28] (example for temperature, Figure 3a). The weather generator was then used to produce a reference time series with an arbitrary length of 18 years. The time series had daily resolution, which is the resolution at which all our simulations are performed. For additional details on the weather generator, refer to Farinotti [29].

2.2.2. Synthetic Meteorological Forecasts

The synthetic meteorological forecasts ($T_{fcst}$ and $P_{fcst}$; Figure 1b₂) were created by (a) using the weather generator described above to produce sets of additional meteorological time series with different variability, and (b) superimposing several components – namely a trend, a bias, a long-term oscillation (i.e., a signal varying over several decades) and random noise – to them (Figure 3).

More specifically: Let $T_{\mathrm{WG}}(t,{\mathit{\sigma }}_T)$ and $P_{\mathrm{WG}}(t,{\mathit{\sigma }}_P)$ be the temperature and precipitation generated with the weather generator (WG) for a given point in time t and a given set of variability parameter ${\mathit{\sigma }}_i$. Let, moreover, $[t_0,t_1]$ be the time period over which the synthetic forecasts are generated. $T_{fcst}\left(t\right)$ and $P_{fcst}\left(t\right)$ were then given by:

$$T_{fcst}\left(t\right)=\left(T_{\mathrm{WG}}(t,{\mathit{\sigma }}_T)+\frac{a}{t_1-t_0}·(t-t_0)+b+\omega (t,c)\right)+\epsilon (t,{\mu }_T,l_0,l_1)$$

(1)

$$P_{fcst}\left(t\right)=\left(P_{\mathrm{WG}}(t,{\mathit{\sigma }}_P)·(1+\frac{a}{t_1-t_0}·(t-t_0))·(1+b)·(1+\omega (t,c))\right)·\epsilon (t,{\mu }_P,l_0,l_1)$$

(2)

where a is the trend magnitude at $t_1$, b is a bias, $\omega (t,c)$ is a long-term oscillation of the form

$$\omega (t,c)=c·sin\left(\frac{\pi }{2·(t_1-t_0)}·(t-t_0)\right)$$

(3)

and $\epsilon (t,\mu ,l_0,l_1)$ are noise terms (see below). In the long-term oscillation, c is the amplitude. Note also that the periodicity is much longer than our forecast period, which aims at representing long-term natural variability. For the noise term, we chose the arbitrary form

$$\epsilon (t,{\mu }_i,l_0,l_1)=\mathcal{N}\left({\mu }_i,l_0+(l_0-l_1)·\frac{t-t_0}{t_1-t_0}\right)$$

(4)

where $\mathcal{N}({\mu }_i,\sigma )$ is a random realization of a normal distribution with with mean ${\mu }_i$ (${\mu }_T=0$ and ${\mu }_P=1$ in our case) and standard deviation $\sigma$. In the equation, parameters $l_0$ and $l_1$ can be interpreted as the noise level at the beginning ($t=t_0$) and the end ($t=t_1$) of the forecast, respectively. We choose $l_0=0$, which means that there is no noise at the beginning of the forecast, and $l_1>l_0$, which means that the noise intensity increases with forecasting lead time. The form of the various components added to the time series obtained from the weather generator are shown in Figure 3c.

It is important to note that the exact form of the elements in Equations (1)–(4) is not of further importance, since precisely reproducing the structure of deviations in real-world forecasts was outside the scope of the study. Rather, we aimed at obtaining a set of individual forecasts with a wide range of skills: From very good ones close to the reference time series, to very poor ones close to arbitrary noise.

For a given set of parameters ($\sigma ,a,b,c,{\mu }_i,l_0,l_1$), we created forecasts that start every months during 9 years. This makes a total of $9·12=108$ forecasts per parameter set (Figure 3d,e). We call such a set of 108 forecasts a “forecast block”. The length of the synthetic forecasts was also 9 years (in the above notation: $t_1-t_0=9$ yr). With this, every forecasts could be validated against the reference meteorological time series over a period of 9 years.

To imitate a variety of possible forecast, we varied parameters ${\sigma }_{day},{\sigma }_{mon},{\sigma }_{yr},a,b,c$ and $l_1$ over three levels (

Table 1. Parameters used to create the synthetic temperature and precipitation forecasts (see Equations (1) and (2) for details). Note that the values for $\sigma$ are given relative to the ones used to produce the synthetic reference time series (a value of, e.g., “2” means that ${\sigma }_{day}$, ${\sigma }_{mon}$, ${\sigma }_{yr}$ were doubled).

Parameter		Forecast
		Temperature	Precipitation
$\mathit{\sigma }$	variability (relative)	$0.5,1.0,2.0$	$0.5,1.0,2.0$
a	trend	$-1.5,0,+1.5$ (${}^{\circ }$C)	$-20,0,+20$ (%)
b	bias	$-2,0,+2$ (${}^{\circ }$C)	$-10,0,+10$ (%)
c	oscillation amplitude	$-3.0,+0.5,+2.0$ (${}^{\circ }$C)	$-30,5,20$ (%)
${\mu }_i$	mean of the noise term	0	1
$l_0$	noise level at $t_0$	0	0
$l_1$	noise level at $t_1$	$0.4,0.5,0.6$	$0.4,0.5,0.6$

). The parameter values were chosen arbitrarily. Out of the $3^7=2187$ possible combinations, and due to computational limitations, we then selected a random subset of 150 realizations for further analyses.

2.3. Runoff Forecasts Generation

The reference meteorological time series (Section 2.2.1) and the synthetic forecasts (Section 2.2.2) were fed into two hydrological models (see below) to obtain corresponding reference runoff time series $Q_{ref}$ (Figure 1d₁) and runoff forecasts $Q_{fcst}$ (Figure 1d₂). For every simulation, the hydrological models were initialized with a one-year spin-up phase driven by the reference meteorological time series. To isolate the impact of glacierization on the skill transfer, both models were run with five different scenarios for glacier extents and ice volumes (

Table 2. Glacierization (glac.) and glacier ice volume ($V_{gl}$) in each scenario. Scenario 1 has the largest glacier volume and Scenario 5 has no glacier left. The spatial extent of the glaciers is shown in Figure 2.

Scenario	Catchment A		Catchment B
	glac. (%)	${\mathit{V}}_{\mathit{gl}}$ (km${}^3$)	glac. (%)	${\mathit{V}}_{\mathit{gl}}$ (km${}^3$)
Scenario 1	61.02	0.43	53.56	1.29
Scenario 2	49.70	0.29	49.56	0.99
Scenario 3	32.38	0.17	36.03	0.58
Scenario 4	15.63	0.06	24.19	0.36
Scenario 5	0.00	0.00	0.00	0.00

, Figure 2).

The two hydrological models were the Glacier Evolution Runoff Model (GERM) and the Hydrologiska Byråns Vattenbalansavdelning-light (HBV-light). GERM [26,28] is a fully distributed, deterministic model simulating catchment runoff at daily time scales. The model includes different modules for snow accumulation, snow and ice ablation, glacier evolution, evapotranspiration, and runoff routing. It requires continuous temperature and precipitation time series as input, a digital surface model of the catchment, as well as the glacier ice thickness distribution. To simulate glacier evolution, the model implements the so-called $\Delta h$-parametrization [30], which is a simple mass conserving approach based on the observation that the largest changes in ice thickness occur at the glacier tongue. HBV-light [31] is a semi-distributed conceptual model constituted of four different routines, namely a snow and glacier routine, a soil moisture routine, a response routine, and a runoff routine. For representing the glaciers, the new “Glacier Area Change Routine” presented in Seibert et al. [32] was used. The latter is based on the $\Delta h$-parametrization as well. The required input data for HBV-light, including the glacier routine, are continuous daily temperature and precipitation, monthly mean temperatures and mean potential evapotranspiration, as well as information about the glacier thickness per elevation band. For more details on the HBV-light model, refer to Seibert [33] and Seibert and Vis [31].

2.4. Skill Scores

The performance of the various forecasts ($T_{fcst}$, $P_{fcst}$, $Q_{fcst}$) was quantified with two different and complementary skill scores: The Root Mean Square Error (RMSE, Equation (5)) and the Nash-Sutcliffe efficiency coefficient (NSE, Equation (6)). Let $X_{ref,i}$ and $X_{fcst,i}$ be the reference and forecasted values of a given variable X and lead time i, respectively. RMSE and NSE are then given by:

$$RMS{E}_i=\sqrt{\frac{1}{N_i}\sum {(X_{fcst,i}-X_{ref,i})}^2}$$

(5)

$$NS{E}_i=1-\frac{\frac{1}{N_i}\sum {(X_{fcst,i}-X_{ref,i})}^2}{var{〈X_{ref,i}〉}^2}$$

(6)

where N is the number of data pairs ($N=108$ in our case, see below), and $var〈X_{ref}〉$ is the variance of $X_{ref}$. In the above equations, X can either be temperature (T), precipitation (P) or runoff (Q). In the following, we denote the skill scores (RMSE or NSE) for the temperature, precipitation and runoff forecasts with $T_{skill}$, $P_{skill}$ and $Q_{skill}$ respectively.

RMSE measures the average magnitude of the deviation between reference and forecast time series. NSE, instead, measures the deviation relative to the variability in the reference time series. RMSE values range between 0 (best score) and ∞, and the ones of NSE between $-\infty$ and 1 (best score).

In our analyses, all skill scores were evaluated by aggregating the individual time series to monthly values. Since all forecasts were 9 years in length, skill scores were computed for $9·12=108$ lead times. Similarly, since the forecasts were started every months over 9 years, the score of each of the 108 lead times was computed from $N=108$ value pairs (Figure 3e).

3. Results and Discussion

The results generated by forcing the two different hydrological models (GERM and HBV) with the 150 forecast blocks are presented hereafter. We first address the influence of lead time on the results (Section 3.1) and assess the skill transfer between meteorological and runoff forecasts afterwards (Section 3.2).

3.1. Dependence on Lead Time

The skill scores for runoff ($Q_{skill}$), temperature ($T_{skill}$), and precipitation ($P_{skill}$) were transformed as to have different units. To allow for intercomparison between them, we first normalized the values obtained for every lead time separately (this means that, for every lead time, the three variables have values ranging between 0 and 1). Figure 4 presents the difference between the skill in the meteorological forecasts (average over $T_{skill}$ and $P_{skill}$) and the skill in the resulting runoff predictions ($Q_{skill}$) for different lead times. This difference remains approximately constant over the lead times, despite the fact that the skill of both precipitation and temperature decreases with increasing lead time. This indicates that all further observations—and the inter-relations between $T_{skill}$, $P_{skill}$, and $Q_{skill}$ in particular—are independent from the lead time. Loosely speaking, this means that the combination of a good temperature forecast and a good precipitation forecast result in a good runoff forecast, while the combination of a poor temperature forecast and a poor precipitation forecast result in a poor runoff forecast, rather than in a disastrous one. This result also indicates that the skill in temperature and precipitation forecasts is interchangeable at every lead time, i.e., that a poor skill in precipitation forecasts can be compensated by a good skill in the temperature forecasts, and vice-versa.

3.2. Skill Transfer between Meteorological and Runoff Forecasts

The relationship between $T_{skill}$, $P_{skill}$ and $Q_{skill}$ from the 150 forecast blocks is shown in Figure 5. Since the dependence on lead time was assessed to be of minor relevance (Section 3.1), a mean value over all lead times is calculated for each forecast block. We observe that for Scenario 1 (highest glacierization), the relationship between $T_{skill}$ and $Q_{skill}$ is stronger (more pronounced horizontal color variations in Figure 5) than $P_{skill}$ and $Q_{skill}$. The observation holds true in both catchments, and is more visible in Catchment B than in Catchment A. With decreasing glacierization (Scenarios 2 to 5) the horizontal color gradient shifts towards a vertical gradient, suggesting an increasing influence of $P_{skill}$ (and decreasing influence of $T_{skill}$) on $Q_{skill}$. This shift is more pronounced in Catchment A than Catchment B.

In order to quantify the relationship between the meteorological and the runoff forecasts (i.e., the transfer of forecast skill), we express $Q_{skill}$ as a linear function of $T_{skill}$ and $P_{skill}$:

$$Q_{skill}=m·T_{skill}+n·P_{skill}+{ϵ}^{\prime }.$$

(7)

In the above equation, parameters m and n are determined by ordinary least-square regression, and ${ϵ}^{\prime }$ is the residual of the relation. The values of m and n can be interpreted as the strength by which the runoff forecasts are influenced by the temperature and precipitation forecasts, respectively. The mean values of m and n (mean over all lead times), along with the mean coefficient of determination $r^2$, are given in

Table 3. Parameters m and n and corresponding coefficient of determination ($r^2$) of Equation (7) for the two analyzed catchments and both skill scores. Results refer to the GERM model.

	Catchment A						Catchment B
	RMSE			NSE			RMSE			NSE
	$\mathit{m}$	$\mathit{n}$	${\mathit{r}}^{\mathbf{2}}$	$\mathit{m}$	$\mathit{n}$	${\mathit{r}}^{\mathbf{2}}$	$\mathit{m}$	$\mathit{n}$	${\mathit{r}}^{\mathbf{2}}$	$\mathit{m}$	$\mathit{n}$	${\mathit{r}}^{\mathbf{2}}$
Scenario 1	0.69	0.71	0.75	0.26	0.28	0.73	0.75	0.72	0.73	0.14	0.15	0.68
Scenario 2	0.60	0.62	0.69	0.33	0.34	0.67	0.75	0.70	0.74	0.10	0.10	0.68
Scenario 3	0.50	0.52	0.64	0.40	0.42	0.62	0.66	0.59	0.72	0.17	0.18	0.65
Scenario 4	0.38	0.39	0.58	0.46	0.48	0.57	0.56	0.50	0.68	0.19	0.20	0.60
Scenario 5	0.26	0.25	0.58	0.54	0.55	0.55	0.43	0.36	0.58	0.25	0.23	0.50

.

For Scenario 1 and RMSE, we found $m=0.69\pm 0.04$ and $n=0.26\pm 0.05$ for Catchment A, and $m=0.75\pm 0.04$ and $n=0.14\pm 0.04$ for Catchment B. This indicates that $T_{skill}$ has a larger impact on the resulting $Q_{skill}$ than has $P_{skill}$ (an equivalent result is found when considering NSE; cf. Table 1). This result matches our hypothesis that, in glacierized catchments, temperature has a larger influence on runoff compared to precipitation, which we interpret as a result of the direct impact of temperature on snow and ice melt, and thus runoff generation. In general, we observe that $T_{skill}$ has a larger influence on $Q_{skill}$ than $P_{skill}$ has (values of m four times the values of n, on average), and that the difference between m and n is larger in Catchment B than it is in Catchment A. This indicates that $Q_{skill}$ is more strongly influenced by the skill of the temperature forecasts than it is by the precipitation forecasts, especially in Catchment B. We suggest this to be related to the larger amount of runoff that originates from snow and ice melt in Catchment B compared to A (Figure 6). Note that due to the higher fraction of catchment area lying below the equilibrium line in the case of Catchment B, the runoff share from snow and ice melt is larger than in Catchment A despite the lower glacierization (Figure 2). With decreasing glacierization (from Scenario 1 to Scenario 5), the values of parameter m decrease, while the ones of n increase. This is true for both the considered skill scores (Table 3) and means that the impact of $T_{skill}$ on $Q_{skill}$ diminished with shrinking ice extent, while $P_{skill}$ increases its influence. Again, the diminishing amount of ice and snow melt, and the increasing one of precipitation, can explain this trend (Figure 6).

For Catchment A, the influence of $P_{skill}$ is stronger than the one of $T_{skill}$ for glacierizations below 30% (solid lines in Figure 7a). In Catchment B, however, $T_{skill}$ continues to have a stronger influence on $Q_{skill}$ even if there is no glacier left. We suggest this to be linked to the fact that in Catchment B the runoff share originating from ice and snow melt (sum of both components) is higher than the one originating from liquid precipitation, even in a scenario without glacier (Figure 6b, Scenario 5). This makes the catchment’s runoff more sensitive to temperature than for Catchment A, in which the proportion of ice and snow melt is lower (Figure 6a). Figure 7b suggests that temperature will have the larger impact on runoff compared to precipitation if the sum of the ice and snow melt contributions is higher than about 60%.

The above results are similar for both hydrological models (Figure 7c,d). The small differences could be related to the different runoff composition predicted by two models (Figure 6). For the same meteorological input time series, for example, the amount of runoff produced by HBV is lower than the one produced by the GERM model. We interpret this as an indication for the HBV model storing more water in one or several storage compartments (such as the snowpack or the soil) than GERM.

4. Conclusions

This study investigated the transfer of skill between meteorological and runoff forecasts. Synthetic meteorological forecasts for which the skill was manually controlled were generated, and fed into two separate hydrological models. Five scenarios representing different glacier ice volumes and ice extents were tested to assess the influence of glacierization on the results. The relation between the skill of the meteorological forecasts and runoff predictions was quantified through linear regression, and through a series of dedicated analyses.

For high degrees of glacierization (>50%), temperature showed an impact on runoff up to four times larger than precipitation, which we interpret as an expression of the temperature’s control on both ice and snow melt. With shrinking glaciers and smaller glacierization, the influence of temperature decreased steadily, while the opposite was true for precipitation. For the two investigated catchments, temperature maintained a larger control on runoff compared to precipitations if the glacierizations was above 30%, or if the ice- and snow-melt contribution to total runoff was more than ca. 60%. The two hydrological models gave similar results, and the relations were shown to hold true independently of the lead time considered.

Our results are of relevance for the application of new generations of long- and extended-range meteorological forecasts—including decadal predictions—as those have been shown to be more skillful in predicting temperature than precipitation e.g., [34,35,36]. In particular, we suggest that using such type of predictions to produce runoff forecasts in snow- and glacier-dominated catchments holds more promise than doing so in catchments dominated by liquid precipitation.