Timely projections of seasonal streamflow extremes can be useful for the early implementation of annual flood risk adaptation strategies. However, predicting seasonal extremes is challenging, particularly under nonstationary conditions and if extremes are correlated in space. The goal of this study is to implement a space–time model for the projection of seasonal streamflow extremes that considers the nonstationarity (interannual variability) and spatiotemporal dependence of high flows. We develop a space–time model to project seasonal streamflow extremes for several lead times up to 2 months, using a Bayesian hierarchical modeling (BHM) framework. This model is based on the assumption that streamflow extremes (3 d maxima) at a set of gauge locations are realizations of a Gaussian elliptical copula and generalized extreme value (GEV) margins with nonstationary parameters. These parameters are modeled as a linear function of suitable covariates describing the previous season selected using the deviance information criterion (DIC). Finally, the copula is used to generate streamflow ensembles, which capture spatiotemporal variability and uncertainty. We apply this modeling framework to predict 3 d maximum streamflow in spring (May–June) at seven gauges in the Upper Colorado River basin (UCRB) with 0- to 2-month lead time. In this basin, almost all extremes that cause severe flooding occur in spring as a result of snowmelt and precipitation. Therefore, we use regional mean snow water equivalent and temperature from the preceding winter season as well as indices of large-scale climate teleconnections – El Niño–Southern Oscillation, Atlantic Multidecadal Oscillation, and Pacific Decadal Oscillation – as potential covariates for 3 d spring maximum streamflow. Our model evaluation, which is based on the comparison of different model versions and the energy skill score, indicates that the model can capture the space–time variability in extreme streamflow well and that model skill increases with decreasing lead time. We also find that the use of climate variables slightly enhances skill relative to using only snow information. Median projections and their uncertainties are consistent with observations, thanks to the representation of spatial dependencies through covariates in the margins and a Gaussian copula. This spatiotemporal modeling framework helps in the planning of seasonal adaptation and preparedness measures as predictions of extreme spring streamflows become available 2 months before actual flood occurrence.

Floods are a concern in mountainous regions such as the Upper Colorado River basin (UCRB), where streamflow extremes happen in spring, due to snowmelt in combination with precipitation

Operational streamflow forecasts are generally implemented using physically based models that use forecasts of hydrometeorological variables, such as rainfall, as their forcing

Only a few studies have tried to implement seasonal peak flow forecasts; e.g.,

Seasonal and subseasonal streamflow forecasting models rely on the skill of hydroclimatic variables from the previous season, such as snow cover

In the case of mountainous regions, projection models for spring flow extremes can take advantage of the fact that snow water equivalents (SWEs) accumulated until the beginning of spring are a skillful predictor of spring streamflow

How does the representation of nonstationarity (interannual variability) through suitable covariates improve seasonal predictions?

How does the explicit representation of spatial dependencies improve prediction performance?

To what extent are seasonal projections for longer lead times still skillful?

We propose a space–time modeling framework for the prediction of seasonal streamflow extremes that has three components, namely (i) a hierarchical model structure, (ii) nonstationary margins, and (iii) a spatial dependence model. Each of these model components, their estimation strategies, and the estimation of ensembles of seasonal streamflow extremes are described below.

We conduct a nonstationary frequency analysis of seasonal streamflow extremes at

A conceptual sketch of the BHM is shown in Fig.

Conceptual sketch of the Bayesian hierarchical model. Blue boxes denote the data, orange boxes the GEV distribution parameters, gray boxes the models, and red circles the inferred quantities. Model boxes correspond to the data layer (Gaussian copula and GEV marginal distributions) and the latent layer (time regressions of GEV parameters).

The first two GEV parameters (location and scale) are modeled as linear functions of time-dependent, large-scale climate variables and regional mean variables from the previous season, while the shape parameter is considered to be stationary, as follows:

Copulas are a flexible tool for modeling multivariate random variables since they can represent dependence independently of the choice of marginal distributions

Let

The copula dependence matrix,

By definition, the dependency between a streamflow gauge and itself is unity, so the diagonal elements of

There are two main approaches to estimate the unknown parameters of the conditional copula

The inference of the model parameters is done in a Bayesian framework, which can account for parameter uncertainties. The posterior distributions of the model parameters,

The predictive posterior distribution of the spring maximum streamflow (ensembles) for the

If the posterior PDFs of the slope coefficients of the GEV parameters were found to be significant, compute the GEV parameters (

Then, simulate the marginal cumulative distributions,

Thereafter, compute the spring maximum streamflow for each streamflow gauge

As a final step, repeat steps 2–3 for each gauge and year of the record.

We demonstrate the utility of the framework proposed in the previous section by applying it to project 3 d spring maximum (May–June) streamflow at seven gauges in the Upper Colorado River basin (UCRB) with 0- to 2-month lead time (Fig.

Streamflow gauges in the Upper Colorado River basin (UCRB) considered in this study. Light blue squares correspond to the snow gauges (18) and purple triangles to the meteorological stations (3) considered in this study.

Daily spring, May through June, streamflow data were obtained from the U.S. Geological Survey (USGS) using the R package dataRetrieval

Figure

Basic data corresponding to the streamflow gauges in the Upper Colorado River basin (UCRB) considered in this study.

Pairs plot (lower triangular matrix) and Kendall's rank correlation coefficients (upper triangular matrix) of spring 3 d maximum streamflow between the seven stations in the UCRB. Kendall's rank correlation is significant (

It has been shown in previous studies that the snow water equivalent (SWE) accumulated until the beginning of spring is the most skillful predictor of spring–summer seasonal streamflow across mountainous regions, such as the western USA

We test the usefulness of both snow-related and climatic drivers within the BHM framework by using SWE and different climatic indices as potential covariates of the marginal GEV distributions. Specifically, we tested the following covariates for modeling the temporal nonstationarity of the GEV parameters (see Eqs.

Schemes of nonstationary models considered for three different lead times. The dark turquoise box denotes the model for a 0-month lead time (projections are released on 1 May), the turquoise box the model for a 1-month lead time (projections are released on 1 April), and the light turquoise box the model for a 2-month lead time (projections are released on 1 March). The same color scheme will be considered for the Sect. 4.

The potential covariates – monthly ENSO, PDO, and AMO climate indices, monthly accumulated SWE from November to April, and daily mean temperature for April – were computed using the following data sources:

ENSO, PDO, and AMO climate indices are from the National Oceanic and Atmospheric Administration (NOAA;

SWE is from the Natural Resources Conservation Service (NRCS) Snow Course Data (

April's daily mean temperature is from the Global Historical Climatology Network

For SWE and daily mean temperature, we only considered snow gauges (18) and meteorological stations (3) located inside the UCRB with full records for the period of interest (1965-2018; see Fig.

Time series of normalized 3 d spring maximum streamflow, best covariate, and the spatial average covariate of

We assessed the strength of the relationship between the covariates and spring maximum streamflow by computing the Spearman's rank correlation coefficient, which is shown in Fig.

Spearman's rank correlation coefficient between 3 d spring maximum streamflow and potential covariates for a 0-month lead time.

We checked the suitability of the GEV distribution as a marginal distribution and of the Gaussian copula as a spatial dependence model using their maximum likelihood estimates for 3 d spring maximum streamflow.

To check the validity of the GEV distribution as a marginal distribution, we fitted a stationary GEV distribution at each gauge using maximum likelihood. Then, we ran two goodness-of-fit tests, i.e., the Cramér–von Mises and Anderson–Darling tests

For testing the suitability of the Gaussian copula, we ran three multivariate normality tests using marginal transformations based on pseudo-observations from the tests of

Pairwise dependence structure between UCRB2 and UCRB4

The specific structure of the BHM for the UCRB incorporates the covariates described in Sect.

We considered noninformative priors for the covariance matrix of the GEV regression coefficients by setting

The model was implemented in R

To test the out-of-sample predictability of the model, we performed a leave‐1‐year‐out cross-validation by dropping 1 year from the record (1965–2018) and fitting the BHM using the remaining years, which are also known as the calibration years. The fitted model is applied to provide estimates for the 1 validation year. This cross‐validation procedure was repeated 54 times.

As the goal of this study is to provide seasonal streamflow extremes projections for risk-based flood adaptation, we also implemented this leave‐1‐year‐out cross-validation just for high-flow years in which all the station gauges exceeded their 60th percentile.

We computed the energy skill score (ESS) as our verification metric, since we are interested in capturing the spatiotemporal dependence of the data. The energy score (ES) assesses the probabilistic forecasts of a multivariate quantity, as follows

For each lead time, different candidate BHMs were calibrated for the period 1965–2018, and the best BHM was selected based on the lowest DIC value.
Table

DIC values for different candidate BHMs for a 0-month lead time and the best model for the other two lead times. For each model, the same covariates for the location parameter are considered at all gauges. Candidate BHMs are sorted from the lowest to the highest DIC value for a 0-month lead time. Scale and shape parameters are considered stationary. All candidate BHMs consider a Gaussian copula to model spatial dependence.

To assess the ability of the BHM in capturing the spatiotemporal dependence through nonstationary covariates and a Gaussian copula, we compared the best model for the 0-month lead time selected in the previous section (i.e., SASWE and SAAMT as covariates with a Gaussian copula) against three models that do not consider a Gaussian copula (i.e., stationary, nonstationary with SASWE as covariate, and nonstationary with SASWE and SAAMT as covariates).

Figure

Energy skill score (ESS) distribution for four different BHM versions for a 0-month lead time for the calibration period (1965–2018). Higher values of the ESS indicate better model performance. The whiskers show the 95 % credible intervals, boxes show the interquartile range, and horizontal lines inside the boxes show the median. Outliers are not displayed. The light gray box denotes a stationary model, the gray box a nonstationary model with SASWE as a covariate, the dark gray box a nonstationary model with SASWE and SAAMT as covariates, and the dark turquoise model adds a Gaussian copula to model spatial dependence.

To further highlight the ability of the Gaussian copula to capture the spatiotemporal dependence, we computed the spatial joint threshold non-exceedance probabilities for the seven stations and compared them to non-exceedance probabilities derived from three models without the copula, i.e., spatially independent models for each streamflow gauge. In Fig.

Distribution of the probability that all gauges do not exceed their

In order to assess the spatial performance of the BHM, Fig.

Time series of average projected spring maximum specific streamflow over all seven gauges of the UCRB (

To define the extent to which seasonal streamflow extremes projections for longer lead times can be skillful, we assess the performance of the BHM at different lead times for the calibration and the leave‐1‐year‐out cross-validation, including cross-validation focusing on extremes (60th percentile). The covariates considered for each lead time were presented in Sect.

Figure

Energy skill score (ESS) distribution for different lead times from the

To assess at-site performance, we computed the continuous rank probability skill score

Figure

Time series of average projected spring maximum specific streamflow (millimeters per day; hereafter mm d

Compared to operational forecast models that consider short lead times and seasonal streamflow forecast models that are useful for reservoir operation with a focus on water availability during the dry season, the BHM proposed here has the following benefits:

It allows for the consideration of potential climate change effects by modeling the margins in a nonstationary setting using suitable covariates.

It allows one to capture the spatiotemporal dependence by including a Gaussian copula. Consequently, the spatial BHM captures observations that are not captured by the average projection of spring maximum specific streamflow of a BHM without a copula.

It provides average projections of spring maximum specific streamflow for up to 2 months in advance by relying on the predictive skill of snow accumulated during the winter season.

The following question comes to mind: how can the proposed modeling framework be used to deliver interpretable seasonal average projections? It might be difficult for decision-makers to make decisions based on the average spring maximum specific streamflow over all seven gauges of the UCRB (see Fig.

To illustrate this system, Fig.

The nonstationary and spatial BHM framework proposed here was applied to the UCRB using 3 d maxima. However, the framework is flexible and can be applied to other types of maxima, such as 1 d maxima, be implemented in other regions, be applied to other types of extremes such as droughts, or be used under future climate conditions. In order to apply the framework in another variable or region, the choice of covariates has to be reconsidered and potentially adjusted. In the application presented here, we only modeled the location parameter as nonstationary. If the framework is applied to another basin, this modeling choice has to be reconsidered. It is advisable, as a first step, to do an initial run of the model for defining which parameters should be considered nonstationary.
If one wishes to apply the framework to predict another type of extreme such as low flows, one needs to reconsider distribution choice and to identify suitable covariates. In addition, the framework is not limited to projections; it can also be adapted for simulation purposes by considering real-time covariates. In addition, it can be easily adjusted such that it represents future climate conditions if future projections of the covariates are available. However, the predictive skill of the model fitted and applied to the UCRB may change under future climate conditions. The relative importance of snowmelt and precipitation in causing flood events may change in the future, with precipitation becoming relatively more important. Consequently, the model's predictive skill, which heavily relies on SWE as a covariate, might slightly decrease in future. However, in the case of headwater basins in mountainous regions, such as the one considered in this study, snowmelt will remain the dominant flood generation process in the future, as shown by climate change projections in the region

In this study, we presented a Bayesian hierarchical model (BHM) to project seasonal streamflow extremes for several catchments in a river basin for several lead times. The streamflow extremes at a number of gauges in a basin are modeled using a Gaussian elliptical copula and generalized extreme value (GEV) margins with nonstationary parameters. These parameters are modeled as a linear function of suitable covariates from the previous season.

We applied this framework to project 3 d spring maximum (May–June) streamflow at seven gauges in the Upper Colorado River basin (UCRB) network, at 0-, 1-, and 2-month lead times. As potential covariates, we used indices of large-scale climate teleconnections, i.e., ENSO, AMO, and PDO, regional mean snow water equivalent, and temperature from the preceding winter season.

From the analysis of different models for a 0-month lead time, we conclude the following:

The spatial average snow water equivalent (SASWE) accumulated during fall and spring is the most skillful predictor of spring season maximum streamflow across the UCRB.

The increase in BHM performance is low when adding other climatic indices such as PDO.

Including a copula in the BHM enables us to capture the spatiotemporal dependence of streamflow extremes, which is not fully possible with independent marginal models.

The comparative analysis for three different lead times revealed that increasing the lead time from 0 to 2 months only weakly decreases model skill. This finding implies that the framework proposed could be useful for the early implementation of flood risk adaptation and preparedness strategies. We propose an alternative to guide decision-making by providing the average projections of spring maximum specific streamflow as the first three quartiles of the ensembles of the average projection of the spring maximum specific streamflow along with past observed specific streamflow values as reference. Such a communication strategy could help decision-makers to implement adaptation strategies that address the spatial dimension of flooding.

The dataset used in this study, which consists of the time series of potential covariates and 3 d spring maximum streamflow for the seven station gauges, can be downloaded from HydroShare

The supplement related to this article is available online at:

The idea and setup for the paper were jointly developed by the four co-authors. The model implementation and analysis were performed by AO and discussed with the other co-authors. AO wrote the first draft of the paper, which was revised and edited by MIB, RB, and WK.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This project has been funded by the National Science Foundation (grant no. 1243270). We also acknowledge the support from the Fulbright Foreign Student Program and the Comisión Nacional de Investigación Científica y Tecnológica (CONICYT) Scholarship Program (grant no. DOCTORADO BECAS CHILE/2015-56150013) for the first author. The second author was supported by the Swiss National Science Foundation via a PostDoc.Mobility grant (grant no. P400P2_183844). Partial support from the Monsoon Mission Project of the Ministry of Earth Sciences, India, for the first and third authors is thankfully acknowledged. The fourth author has been supported by the NSF (grant nos. DMS-1811294 and DMS-1923062).

This research has been supported by the National Science Foundation (grant nos. 1243270, DMS-1811294, and DMS-1923062), the Comisión Nacional de Investigación Científica y Tecnológica Scholarship Program (grant no. DOCTORADO BECAS CHILE/2015-56150013), the Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (Postdoc.Mobility; grant no. P400P2_183844), and the Ministry of Earth Sciences, India (grant the Monsoon Mission Project).

This paper was edited by Stacey Archfield and reviewed by two anonymous referees.