# Causal Attribution of the Interannual Variability in Flood Peaks Through Bayesian Networks

J S Nanditha, Gabriele Villarini, Hanbeen Kim, Philippe Naveau

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To cite this version:

J S Nanditha, Gabriele Villarini, Hanbeen Kim, Philippe Naveau. Causal Attribution of the Inter-annual Variability in Flood Peaks Through Bayesian Networks. Water Resources Research, 2025, 61 (6), 10.1029/2024wr039385. http://dx.doi.org/10.1029/2024wr039385

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# Water Resources Research®

## RESEARCH ARTICLE

10.1029/2024WR039385

**Key Points:**

* The Bayesian network models can well reproduce the interannual variability in seasonal and annual flow peaks

* Precipitation drives floods in the eastern and southern US, while temperature influences peak flows in the western and northeastern US

* The causal link derived by the Bayesian Network models is consistent with the underlying physical mechanisms

**Supporting Information:**

Supporting Information may be found in the online version of this article.

**Correspondence to:**

J. S. Nanditha,  
nanditha.js@princeton.edu

**Citation:**

Nanditha, J. S., Villarini, G., Kim, H., & Naveau, P. (2025). Causal attribution of the interannual variability in flood peaks through Bayesian Networks. *Water Resources Research*, 61, e2024WR039385. https://doi.org/10.1029/2024WR039385

Received 15 NOV 2024
Accepted 8 MAY 2025

**Author Contributions:**

**Conceptualization:** J. S. Nanditha,  
Gabriele Villarini

**Data curation:** J. S. Nanditha,  
Hanbeen Kim

**Formal analysis:** J. S. Nanditha

**Funding acquisition:** Gabriele Villarini

**Investigation:** J. S. Nanditha

**Methodology:** J. S. Nanditha,  
Gabriele Villarini, Philippe Naveau

**Project administration:**  
Gabriele Villarini

**Software:** J. S. Nanditha

**Supervision:** Gabriele Villarini

**Validation:** J. S. Nanditha

**Visualization:** J. S. Nanditha

**Writing – original draft:** J. S. Nanditha

© 2025. The Author(s).
This is an open access article under the
terms of the Creative Commons
Attribution License, which permits use,
distribution and reproduction in any
medium, provided the original work is
properly cited.

# Causal Attribution of the Interannual Variability in Flood Peaks Through Bayesian Networks

<figure><img src="image_3.png" /></figure>

J. S. Nanditha¹,² ID, Gabriele Villarini¹,² ID, Hanbeen Kim¹,² ID, and Philippe Naveau³ ID

¹Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ, USA, ²High Meadows Environmental Institute, Princeton University, Princeton, NJ, USA, ³Laboratoire des Sciences du Climat et de l'Environnement, LSCE/IPSL, CNRS-CEA-UVSQ, Université Paris-Saclay, Gif-sur-Yvette, France

**Abstract** Classical regression models, due to the limited computational expense and good performance, can be used for the attribution of interannual variability in flood peaks. However, these models capture the relation between predictand (i.e., flood peaks) and predictors (i.e., climate variables), suffering from the disconnect between correlation and causation. Here, we utilize a causal Bayesian Network model to establish causal relationships between flood peaks and basin- and season-averaged precipitation and temperature, which were found to be useful predictors in previous regression-based attribution studies. We develop these models for seasonal flood peaks for 3,884 gauges across the conterminous United States, achieving a median Spearman's rank correlation above 0.7. By performing *do*-calculus intervention on the predictors, we found a strong causal relationship between seasonal maximum daily discharge and both concurrent and lagged season-precipitation and temperature, consistent with underlying physical processes across different basins. The Bayesian Network model effectively predicts the interannual variability in seasonal and annual peak discharges and establishes a causal link between them. The model identifies key drivers across different seasons and regions in CONUS and highlights that antecedent catchment wetness is particularly relevant for high magnitude flows, while precipitation is the dominant driver of medium flows. This study significantly expands our current knowledge on causal flood drivers and presents a novel approach to flood prediction and attribution.

**Plain Language Summary** Traditional regression models are found to be effective in modeling the year-to-year variability in flood peaks but fail to represent the causal relationship between flood and their drivers. To address this gap, we use a causal Bayesian Network model for modeling the interannual variability in flood peaks. We use basin- and season-averaged precipitation and temperature, which are climate variables commonly used in regression models as predictors and develop Bayesian Network models to describe the changes in flood peaks for 3,884 gauges across the conterminous United States and found that they work well. We also establish the causal relationship between flood peaks and their drivers using the causal framework. The causal relationship established in the study is consistent with the underlying physical mechanisms. The study results enhance our understanding of the key drivers of flooding and improve our ability to predict floods.

## 1. Introduction

Floods are a prominent natural disaster driven by the interplay of climatic, catchment and human-system drivers. The observed interannual variability in floods, caused by the interactions of these factors, has been closely monitored due to their significant impact on socio-economic systems and ecosystems. Previous studies across the globe highlight the prominent role of precipitation characteristics (e.g., Brunner et al., 2021; Nanditha & Mishra, 2024), antecedent catchment moisture (e.g., Berghuijs & Slater, 2023; Berghuijs et al., 2016; Berthet et al., 2009; Hettiarachchi et al., 2019; Nanditha & Mishra, 2022, 2024b), basin temperature, evapotranspiration, and canopy interception (e.g., Berghuijs et al., 2019; Blöschl et al., 2019; Sharma et al., 2018) in modulating flood peaks. Multiple studies have explored the potential reasons for the interannual variability in flood peaks across the conterminous United States (CONUS), with most of them pointing to climatic factors as the main drivers (Awasthi et al., 2022; Berghuijs et al., 2016; Ficklin et al., 2018; Kim & Villarini, 2023; Mallakpour & Villarini, 2015; Schlef et al., 2021; L. J. Slater & Villarini, 2017). Several regression models have been developed to correlate the interannual variability in flood peaks with both the climatic factors and human modifications in the catchments (Awasthi et al., 2022; Neri et al., 2019; L. J. Slater & Villarini, 2017; L. J. Slater et al., 2019; Schlef et al., 2021). However, while the existing regression models perform well in capturing the relationship between floods and potential drivers despite their simplicity and low computational time, they fall short of providing a
Writing – review & editing:

J. S. Nanditha, Gabriele Villarini,
Hanbeen Kim, Philippe Naveau

causal link between flood and the identified drivers. Establishing a causal link avoids spurious correlation, permits better interpretation of the results and improves the understanding of how changes in drivers affect flood peaks.

Seasonal precipitation and temperature have been largely used to predict flood peaks across CONUS since they not only capture well the interannual variability in seasonal and annual discharge peaks (Kim & Villarini, 2023; Slater & Villarini, 2017; Slater et al., 2019) but also have the advantage of being available in most global climate models (GCMs) for future periods (Kim & Villarini, 2024). Additionally, the use of seasonal aggregates as predictors eliminates the need to downscale GCM outputs, which are generally available at coarser spatial and temporal resolutions, especially for moderate to large catchments. Kim and Villarini (2023) developed a statistical prediction framework for seasonal flow peaks across CONUS using seasonal precipitation and temperature as predictors. They found climatic drivers alone explain the major changes in flow patterns, which is further supported in a global extension of the statistical framework (Kim et al., 2024), reflecting the role of climatic factors in driving flood changes in the observational period. However, the use of seasonal quantities does not guarantee a cause-effect relationship between discharge and its drivers as climate variables at the seasonal scale are not always the immediate flood trigger in a catchment, potentially limiting the robustness of the quantitative interpretation of the modeling results (Berghuijs et al., 2016; Nanditha & Mishra, 2022; Tramblay et al., 2021).

Establishing a cause-effect relationship helps in answering questions such as to what extent the change in the drivers or the absence of drivers affects the probability of discharge occurrence, which correlational association fails to provide (Pearl, 2009). Causal analysis makes such interpretations easier through direct intervention in the data. Causal inference methods like *do*-calculus intervene in the data by changing the possible states of the variables into counterfactual states, enabling the assessment of the corresponding changes in the joint probability distribution (Pearl, 1995a, 1995b). Causal belief in a driver increases when the effect occurs in the presence of the driver and fails to occur in its absence while the rest of the drivers are present. The *do*-calculus approach provides a mathematical formulation to derive a quantitative statement on the strength of the causal belief between the drivers and the target variables. For example, Hannart et al. (2016) leveraged causality to make a conceptual link between counterfactual theory and the field of climate Detection and Attribution.

The objective of the study is to complement classical regression-based attribution frameworks by addressing the existing disconnect between correlation and causation. In line with previous studies on attribution of flood peaks, we chose the simple predictors of seasonal precipitation and temperature since they are available in all climate model projections and would further aid in evaluating the causal mechanisms of projected changes in annual maximum flood peaks. We use Bayesian Networks (BNs) to assess the causal link between seasonal maximum daily discharge and its drivers. BNs are structural causal models that are used in the literature for predictions, assessing flood risk and identifying prominent land-ocean-atmospheric interactions in driving floods in previous studies (Das & Ghosh, 2020; Ossandón et al., 2021, 2022; Ravindranath et al., 2019). To our knowledge, BNs are not yet used for attribution in the hydrological literature, and we address this research gap by constructing causal BNs for peak flow prediction across 3,884 stream gauges in CONUS. We specifically explore the following research questions: (a) Does the BN model built on simple climatic predictors capture the interannual variability in seasonal and annual flow peaks? (b) Can we derive a reliable quantitative attribution statement on the relative role of the different drivers in altering flows across different seasons? (c) Is the derived cause-effect relationship consistent with the catchment-level physical processes?

## 2. Data and Methods

We used 3,884 United States Geological Survey (USGS) discharge gauges with daily streamflow measurements for at least 30 years from 1951 to 2020, retrieved using the R package, *dataRetrieval* (De Cicco & Hirsch, 2014). About half of the sites have over 55 years of streamflow record (Figure S1 in Supporting Information S1) and include both basins with minimal human intervention (reference gauges) and those with significant changes in basin hydrology (non-reference gauges). We extracted seasonal and annual maximum mean daily flows for each gauge as described in Kim and Villarini (2023). We follow the approach used by previous studies, which separately model floods in different meteorological seasons as proxies for various flood mechanisms (Kim & Villarini, 2023; Slater & Villarini, 2017); therefore we separately develop BN models for fall (September–November), winter (December–February), spring (March–May) and summer (June–August). Then we estimate
<figure><img src="image_4.png" /><figcaption>**Figure 1.** The representation of the Bayesian Network used in the study. Seasonal peak flow (Q) is the target variable (or the child node), while the four predictors, concurrent precipitation (P<sub>C</sub>), concurrent temperature (T<sub>C</sub>), lagged precipitation (P<sub>L</sub>) and lagged temperature (T<sub>L</sub>) are the parent nodes.</figcaption></figure>

the maximum modeled flow in each year as the maximum modeled seasonal peak flows from fall to summer of the following year (September–August).

To represent the flood drivers, we considered basin- and season-averaged precipitation and temperature for the current season (concurrent) as well as the previous season (lagged) as adopted in previous studies (Kim & Villarini, 2023; Kim et al., 2024; L. J. Slater & Villarini, 2017). For example, for modeling the flow peaks for fall of 2005, summer of 2005 is the lagged season and fall of 2005 is the concurrent season. Basin- and season-averaged precipitation and temperature for previous seasons are used as proxies for different hydrological processes like snowmelt, antecedent soil moisture, evapotranspiration, representing the major catchment storage or losses crucial in determining the peak flow (Berghuijs et al., 2016; Blöschl et al., 2019; Nanditha & Mishra, 2024b; L. Slater et al., 2021; Villarini & Slater, 2017; Wasko et al., 2020). We obtained the observed monthly total precipitation (mm) and monthly mean temperature (°C) from the Parameter-Elevation Regression on Independent Slopes Model (PRISM) data set, available at a resolution of ~4 km (Daly et al., 2002, 2008). The precipitation data include precipitation in both liquid (i.e., rain) and solid (i.e., snow) form.

## 2.1. Design of Bayesian Networks

Bayesian Networks (BNs) are probabilistic graphical models based on a Directed Acyclic Graph (DAG) that connects multiple nodes. Each node in the BN represents a random variable of a defined probability distribution. The process of deriving the structure of a BN is termed as learning, which involves structural and parametric learning. BNs can be trained to learn either from available data or using existing physical knowledge or both. In the structural learning process, the DAG (or the structure) of the BN is deciphered from the data, which is further used for parametric learning where the parameters of the regression relationship between various nodes of the network are estimated. Different algorithms are used for structure and parametric learning, which are in general classified into constraint-based, score-based and hybrid categories. Interested readers can refer to Scutari (2018) and references therein to learn more about the learning algorithms.

By building on the existing regression models that perform well in capturing the interannual variability in floods, we considered BN with concurrent and lagged, season- and basin-averaged precipitation and temperature as predictor (parent) nodes and seasonal peak flow as the predictand (child) node. Thus, we develop a simple Gaussian linear regression framework for peak flow prediction using four predictors. Our model is parsimonious since we adopt a linear regression approach which reduces the computational expense. We apply this BN model with five nodes (Figure 1) for predicting the seasonal peak flow time series for all four seasons across the gauges. We employ k-fold cross validation for training the BN model using streamflow observations. The maximum flow across all seasons within a year (September–August) is estimated as the annual maximum flow. Further, a do-calculus approach is considered for estimating the causal relationship between the predictors and peak flow.

## 2.2. Development of Causal Framework

We make use of do-calculus approach (Pearl, 1994, 1995a) to determine the causal relationship between the target and the four predictor variables. Do-calculus can be defined as any set of actions involving deliberate intervention in the data to set the probability of one random variable to any desirable state by eliminating all other possible states to assess the corresponding changes in the other interconnected random variables (Pearl, 1994). We followed the steps below:

1. We considered the timeseries of season- and basin-averaged precipitation (X<sub>1j</sub>) and temperature (X<sub>2j</sub>) and peak flow (X<sub>3j</sub>) at each gauge for the duration of the streamflow record for each season, j ∈ {1, 2, 3, 4}.

2. We transformed the timeseries (X<sub>ij</sub>) to standard normal quantiles (Z<sub>ij</sub>) after fitting an empirical cumulative distribution function (CDF) (Equation 1), since BN assumes Gaussian distribution at each node:

$$Z_{ij} = \Phi^{-1}[F_{ij}(X_{ij})] \qquad (1)$$
$i \in \{1, 2, 3\}$, indicates the timeseries of season- and basin-averaged precipitation and temperature, and flow peaks, respectively.

$F_{ij}$ (.) denotes the empirical CDF of the timeseries ($X_{ij}$) and $\Phi(\cdot)$ represents the CDF of the standard normal distribution with zero-mean and unit variance.

3. The bankfull discharge is considered as the flow threshold ($Q_{th}$) for determining flood occurrence, which we estimated as the median annual maximum flow at each gauge (e.g., Czajkowski et al., 2017; Gupta et al., 1994). This will ensure flood peaks identified in each season are comparable in magnitude to the annual flow peaks. In relatively dry (wet) seasons, $Q_{th}$ will be much higher (lower) than the individual season distribution, hence we limited the variate to a maximum of 2 (minimum of 0). For each season, we estimated the normal variate of $Q_{th}$ as,

$$z_{3j-\text{th}} = \begin{cases} \Phi^{-1}(F_{3j}(Q_{\text{th}})), & \text{if } 0 < \Phi^{-1}[F_{3j}(Q_{\text{th}})] < 2 \\ 2, & \text{if } \Phi^{-1}[F_{3j}(Q_{\text{th}})] > 2 \\ 0, & \text{if } \Phi^{-1}[F_{3j}(Q_{\text{th}})] < 0 \end{cases} \quad (2)$$

4. The prediction model based on Gaussian linear regression can be mathematically written as

$$Z_{3j} = [\theta_0, \theta_1, \theta_2, \theta_3, \theta_4] \begin{bmatrix} 1 \\ Z_{1j} \\ Z_{2j} \\ Z_{1j-1} \\ Z_{2j-1} \end{bmatrix} \quad (3)$$

where, $[\Theta]$ is the vector of regression coefficients.

5. Then, we perform do-calculus analysis. In do-calculus analysis, the probability distribution of predictor nodes (here, $Z_{ij}$ or $Z_{ij-1}$; $i \in \{1, 2\}$, the concurrent and lagged, season- and basin-averaged precipitation and temperature for each season) are altered to evaluate the change in these interventions on the probabilities of flood occurrence (i.e., $Z_{3j} > z_{3j-\text{th}}$). For this, we modify the probability distribution of each predictor node by adjusting it to two extreme ranges by truncating their normal distribution to intervals $\sim N[1, \infty]$ and $N[-\infty, -1]$ (Equations 4 and 5). Through this step, we effectively curtail the distribution of predictor nodes to higher and lower extremes. Thus, the probability of the predictor distribution below and above one standard deviation from the mean is set to zero through intervention. We chose a standard deviation of 1 as it effectively balances our assumption of extremes (with approximately 15% of data points lying above/below a standard deviation of $1/\sqrt{1}$ in a normal distribution) while ensuring sufficient overlap with the original data. This prevents the intervention from altering the predictor data range to an unrealistic extent.

The mean and variance of the truncated normal distribution are estimated as:

$$E(Z_{ij}|a < Z_{ij} < b) = \left[ \frac{\varphi(a) - \varphi(b)}{\Phi(b) - \Phi(a)} \right] \quad (4)$$

$$\text{Var}(Z_{ij}|a < Z_{ij} < b) = \left[ 1 - \frac{b\varphi(b) - a\varphi(a)}{\Phi(b) - \Phi(a)} - \left\{ \frac{\varphi(b) - \varphi(a)}{\Phi(b) - \Phi(a)} \right\}^2 \right] \quad (5)$$

where, $i \in \{1, 2\}$ and $j \in \{1, 2, 3, 4\}$

where $\Phi(\cdot)$ and $\varphi(\cdot)$ represent the CDF and probability density function, respectively, of the standard normal distribution, $a$ and $b$ correspond to the lower and upper limits of the truncation intervals ([a, b] is $[1, \infty]$) when
the probabilities below $z_{ij} = 1$ are curtailed [i.e., $P(z_{ij} < 1) = 0$], and $[-\infty, -1]$, when the probabilities above $z_{ij} = -1$ are curtailed [i.e., $P(z_{ij} > -1) = 0$].

6. To compute how these interventions in the predictor nodes influence the probability of flood occurrence ($Z_{3j} > z_{3j-\text{th}}$), we implement do-calculus intervention in the established BN prediction model (Figure 1, Equation 3) using the parameters of the truncated normal distribution. Then, we compute the conditional probability of flood occurrence for each season, after the interventions are performed successively at the four predictor nodes ($l$) (Equations 6 and 7).

The probability of flood occurrence in season, $j$, due to lower tail truncation at the predictor node, $l$ is

$$p_{l,j,LT} = P(Z_{3,j} > z_{3j-\text{th}} | do(Z_{i,j} \sim N[1, \infty]), Z_{i,j-1}, Z_{k,j}, Z_{k,j-1}) \quad (6)$$

Similarly, the probability of flood occurrence in season, $j$, due to upper tail truncation at the predictor node, $l$ is

$$p_{l,j,UT} = P(Z_{3,j} > z_{3j-\text{th}} | do(Z_{i,j} \sim N[-\infty, -1]), Z_{i,j-1}, Z_{k,j}, Z_{k,j-1}) \quad (7)$$

where, $i, k \in \{1, 2\}$, such that $i \neq k$ and

$l \in \{1, 2, 3, 4\}$ denotes the four predictor nodes.

7. Then we estimate the Bayes factor, which is the ratio of the likelihood of flood occurrence while the probability distribution of each predictor node is modified to the two extremes:

$$\text{BF} = \frac{P_{l,j,LT}}{P_{l,j,UT}} \quad (8)$$

A Bayes factor above 1 supports the hypothesis that higher extremes of predictors (e.g., $Z_{i,j} \sim N[1, \infty]$) are favorable to flooding, while values below 1 indicate that flood peaks have an inverse relationship with the predictor, and lower extremes of predictors (e.g., $Z_{i,j} \sim N[-\infty, -1]$) favor flooding. The absolute magnitude of the logarithm of Bayes factor corresponds to the strength of the causal relationship in either directions. To provide a qualitative judgment to the strength of the evidence, Jeffreys (1939) proposed a grading scale that classifies the Bayes factor into different categories based on its magnitude (Table S1 in Supporting Information S1). Here, we use Jeffreys' score and two of its modifications, the Lee and Wagenmakers' scale (Lee & Wagenmakers, 2014; Table S2 in Supporting Information S1), which represents a slight modification to the Jeffreys' score, and the Kass and Raftery's scale (Kass & Raftery, 1995; Table S3 in Supporting Information S1), a major modification to demonstrate the strength of the evidence that supports the two hypotheses (Taboga, 2017).

The magnitude of the Bayes factor provides evidence against the null hypothesis of no causal link between the predictor and flood peak. Bayes factor above 3 (below 0.3) provides substantial evidence to reject the null hypothesis of no causal link (Tables S1–S3 in Supporting Information S1). However, for a more quantitative assessment of the significance of the Bayes factor, we randomly disturbed the pairing of the predictor-predictand time-series and conducted causal inference. This results in a 1,000 bootstrapped sampling of the Bayes factors, whose confidence intervals are estimated by taking the 2.5th and 97.5th percentile values of the bootstrapped distribution. If the Bayes factor of the original data falls either above or below the 95% confidence intervals of the bootstrapped Bayes factor, the causal relation in the gauge is considered to be statistically significant.

We used the *bnlearn* package in R for developing the BNs and for conducting causal inference (Scutari, 2017).

We performed 1,000 iterations for each do-calculus intervention and took its mean and standard deviation to calculate the point estimate and 95% confidence intervals of the Bayes factor.

# 3. Results

## 3.1. Bayesian Network Prediction Model

The Bayesian Network prediction model based on Gaussian linear regression performs well across the 3,884 gauges considered in the study. The Spearman's rank correlation coefficient between the observed and predicted
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Spearman's rank correlation between observations and Bayesian Network model predictions of seasonal and annual peak discharge at 3,884-gauge stations across the conterminous United States. The median Spearman's rank correlation ($ρ$) is displayed in the top right corner of each panel. The black line demarcates the seven regions considered in the study, see Figure S2 in Supporting Information S1 for reference to the regions.</figcaption></figure>

seasonal and annual peak flow is above 0.7 at most stream gauges (Figure 2). Spring and summer are the major flood seasons across the study region, except in the U.S. Northeast, Southeast and Northwest, where winter is a major flood season instead of summer (Kim & Villarini, 2024; Villarini, 2016). The BN model captures the
<figure><img src="image_6.png" /><figcaption>**Figure 3.** The boxplots of the Spearman's rank correlation coefficient of observed and modeled discharge and the percentage of annual maximum discharge occurrence in each season. We determined the percentage of time the seasonal peak flow contributes to the annual maximum flow in each gauge and for each season. Then for each season, gauges lying within the specific percentage class (e.g., [0–20%], [20–40%]) are used for generating the boxplots. The box plots on the left (right) represent relatively low (high) flow values. The boxes correspond to 25th, 50th, and 75th percentile correlation coefficient and the whiskers denote the 5th and 95th percentile correlation, in each percentage class. The black points are the correlation coefficient of individual gauges jittered on boxplots for each percentage class.</figcaption></figure>

interannual variability, exhibiting comparable performance for all the four seasons (Figure 2). The model performance is relatively lower in winter across the Great Plains of the North where the median Spearman's correlation coefficient falls below 0.6 (Figure S3 in Supporting Information S1). However, winter is not a major flood season in the Great Plains of the North and hence it does not affect the model performance in predicting the annual maximum flood peaks (Figure 2 and Figure S3 in Supporting Information S1). In all other seasons across all regions, the median correlation coefficient is well above 0.7 (Figure S3 in Supporting Information S1). We also observe the model to capture peak flows in selected locations (Figures S4 and S5 in Supporting Information S1). In general, we find the BN model to perform well despite the coarse nature of their predictors in capturing the interannual variability in seasonal and annual flow peaks, consistent with previous studies that used different statistical approaches (Kim & Villarini, 2023; L. J. Slater & Villarini, 2017).

We also evaluate the model performance at high flow magnitudes for each season to assess the model capability to capture extremes (Figure 3). We compare the sensitivity of the Spearman's rank correlation coefficient with the percentage of time the annual maximum discharge occurs in each season. There is a general increase in the correlation with the percentage of annual maximum discharge occurrence in winter, while there is a slight reduction in spring and summer. As mentioned earlier, low flows in winter are the worst captured by the model, whereas the relatively low flow peaks in fall are more effectively captured. Most gauges show a correlation well above 0.6 when the percentage of annual maximum occurrence is in 40%–60% class. Despite the variability in correlation across different gauges in each season, the values of the correlation between observed and modeled annual maximum discharge show low variability, with a median correlation of 0.67 (Figure 1 and Figure S3 in Supporting Information S1) underlining the good performance of the BN model in capturing the interannual variability of annual peak discharge.
<figure><img src="image_7.png" /><figcaption>**Figure 4.** Bayes factor in the logarithmic scale for all seasons, showing the causative association between flow peaks and basin- and season-averaged precipitation and temperature in the concurrent and lagged seasons. A positive log₁₀(Bayes factor) implies an increase in the driver increases the chance of flooding and vice versa.</figcaption></figure>

In addition, we compared the performance of the BN model with that of the Generalized Additive Model in Location, Scale and Shape (GAMLSS) which uses the same predictors (Kim & Villarini, 2023; Figure S6 in Supporting Information S1). We observe similar spatial patterns in the Spearman's rank correlation coefficient across seasons. The BN model shows relatively better performance than the GAMLSS with higher correlation coefficient (Figure 2 and Figure S6 in Supporting Information S1).

## 3.2. Causal Inference

We estimate the Bayes factor for each season by successively intervening in the probability distribution of all four predictors. In general, concurrent precipitation has the greatest influence on the flood peaks, followed by lagged precipitation, a proxy for antecedent wetness (Figure 4, Figures S7 and S8 in Supporting Information S1). Altering the probability distribution of concurrent and lagged precipitation to the higher extreme enhances the flood probability. An increase in concurrent temperature generally decreases the flood probability in most regions and seasons, except for winter in the northern CONUS. In the northern CONUS, during winter, an increase in concurrent temperature favors flood peaks due to increased snowmelt. Similarly, an increase in lagged temperature reduces the flood probability at most gauges, as high temperature in the previous season can be a proxy for high evaporation rate and reduced antecedent soil moisture. The cause-effect relationship between the four drivers and peak flow shows considerable variability across different geographical regions and seasons (Figure 4).
<figure><img src="image_8.png" /><figcaption>**Figure 5.** Jeffreys' score for the four predictors across the four seasons and seven regions. The color scale shows the strength of the evidence against the null hypothesis of zero causal link between the flood peaks and the corresponding drivers for each season. The darker the color, the stronger the strength of the evidence. The proportion of each color corresponds to the number of gauges in each region under that evidence category.</figcaption></figure>

Further, we observe that the gauges showing substantial and stronger causal evidence are consistent with those identified through significant tests on the Bayes factor, by disturbing the pairing of the time series (Figures S9 and S10 in Supporting Information S1). Overall, the causal link derived in the study is consistent with the correlational pattern of the predictors with flood peaks reported in Kim and Villarini (2023).

In all seasons, an increase in concurrent precipitation increases flood probabilities at all gauges, especially across the eastern United States, western coastal regions, the U.S. Midwest, and the Great Plains of the South (Figure 4). In fall, concurrent precipitation has a significant positive influence on flood generation in the northern and southern Great Plains as well as the western coastal plains. In summer, concurrent precipitation plays a dominant role in the eastern half of CONUS, while it is less prominent in the western regions which receive relatively low precipitation during the season (Figure 4, Figures S11–S13 in Supporting Information S1). Similarly, the influence of concurrent precipitation is relatively lower in the northern Midwest and northern Great Plains during winter and spring, likely due to the snow accumulation in winter and most of these regions showing a stronger connection to lagged precipitation (Figure 4). The U.S. northern tier witnesses snow accumulation in late fall, leading to snowmelt contributions in winter (Villarini & Slater, 2017). These findings are consistent with the covariate relationship reported by Kim and Villarini (2023) between lagged precipitation and flow peaks in winter.

An increase in lagged precipitation, proxy for antecedent wetness, increases flood probabilities across CONUS. The causal link is particularly prominent in the western United States in spring and summer, and in the northern United States in fall and winter (Figures 4 and 5 and Figures S11–S15 in Supporting Information S1). Lagged precipitation plays a major role during spring in the western United States, a proxy for snow accumulation in winter. In fall, lagged precipitation has a substantial causal relation with flow peaks in the eastern United States,
which receives relatively high precipitation in summer (Figure S16 in Supporting Information S1), highlighting the prominence of antecedent wetness. In winter, lagged precipitation shows strong causal relations with flood peaks in U.S. Midwest and central United States (Figures 4 and 5, Figures S14 and S15 in Supporting Information S1).

Higher concurrent temperature increases flood probabilities at few gauges in the western and eastern United States during fall, across the northern and western United States during winter, and the northwestern United States in spring and summer, respectively (Figures 4 and 5). On the contrary, the increase in concurrent temperature reduces flood probabilities in the central United States during winter, across the Midwest and Great Plains of the North in spring, and in the southern United States during summer (Figures 4 and 5). A strong negative causal link exists between high concurrent temperature and flow peaks in the southern and southeastern United States in summer (Figure 4). Higher temperature increases snowmelt contribution during winter in the northern United States and during spring in the northwestern United States. Our findings are consistent with Slater and Villarini (2017), who reported the direct relationship between winter temperature and snowmelt in the U.S. Midwest.

An increase in lagged temperature, a proxy for snowmelt or reduced antecedent catchment moisture conditions, reduces flood probabilities at most gauges in all seasons. The tendency is particularly prominent in spring and summer in the northern United States, and during fall and winter in the west-central United States and Northern Great Plains (Figures 4 and 5). The inverse relationship between lagged temperature and flood probabilities in summer and fall can be attributed to the reduction in antecedent soil moisture (Figures 4 and 5). In contrast, the inverse relationship during winter and spring could be due to low snow accumulation in the prior season due to high temperatures. In the U.S. Northwest, the first snowpack falls by the middle of the fall season and lagged temperature shows a negative relation with flood probability in these regions during winter (Villarini, 2016).

We assessed the relative prominence of the four drivers in causing floods. Concurrent precipitation is the most decisive driver in fall season in all regions, as well as in summer in the U.S. Southeast, Northeast and Midwest (Figure 5 and Figures S14 and S15 in Supporting Information S1). Lagged precipitation and concurrent and lagged temperature are prominent predictors of flow peaks with substantial to decisive causal evidence during winter season in the Great Plains and Midwest. Lagged precipitation also plays a key role in summer and fall. Lagged temperature is further found prominent in the U.S. Northwest during summer, with a substantial to decisive causal link. We also compared the strength of the causal link in relation to the relative flow magnitude in each season with respect to the annual flow peaks for each gauge (Figure 6). We observe that concurrent precipitation is most decisive in driving flows for medium flood magnitudes. As the magnitude of the discharge increases, the relative strength of concurrent precipitation reduces and the role of lagged precipitation and lagged temperature increases, except during winter. This implies extreme flows occur as a combination of precipitation and wet antecedent conditions. The derived causal relation is consistent with previous studies that highlight the role of antecedent catchment moisture conditions in driving rare flows (e.g., Jiang et al., 2024; Nanditha & Mishra, 2022; Nanditha et al., 2022; Wasko et al., 2021). Seasonal discharge with low magnitudes is sustained by the antecedent precipitation in all seasons (Figure 6).

We also compared the effect of model performance on the causal relation (Figures S17-S19 in Supporting Information S1). We found that, the higher the correlation between model predictions and observations ($\rho$ > 0.5), the better the strength of the causal link is. However, there are a few locations where models with low correlations show good causal link with the predictors, implying that the existing model does not capture all the flood causing mechanisms; modifications of the existing model framework to incorporate other relevant flood mechanisms might be appropriate to improve model skill at these gauges. Furthermore, we investigated the influence of catchment area on the strength of the causal link between different drivers and flow peaks (Figure S20 in Supporting Information S1). We observe that in small and medium catchments (<5,000 km²), concurrent precipitation has a strong to decisive causal link, whose influence reduces with increase in catchment size. In larger catchments, lagged precipitation and concurrent and lagged temperature emerge as major drivers with substantial causal link, emphasizing the role of antecedent conditions in driving flows, consistent with existing literature (e.g., Nanditha & Mishra, 2022; Tramblay et al., 2021).

### 3.3. Physical Significance of the Causal Link

We identified the two most important predictors based on the absolute magnitude of the $log_{10}$(Bayes factor) across all sites (Figure 7). Concurrent precipitation is the primary predictor, followed by lagged precipitation at
<figure><img src="image_9.png" /><figcaption>**Figure 6.** Jeffrey's score showing the causative relationship between peak flow and four drivers. For each season, the x-axis shows bins representing the relative magnitude of seasonal flow compared to the annual maximum peaks across all 3,884 gauges. The number of gauges within each bin for each season is marked at the top of the left panel. The y-axis shows the percentage of gauges within different Jeffrey's score categories for each of the four drivers. For example, in the fall season, concurrent precipitation has a decisive role in driving seasonal peak flows of low relative magnitude ([0–10]% bin) in more than 50% of gauges (out of 128 gauges). At the same time, concurrent precipitation is decisive in less than 10% of gauges (out of 82 gauges) in the [71–80]% bin: this indicates that with higher magnitude flows, the causative role of the other three drivers becomes more apparent.</figcaption></figure>

most sites and for all the seasons. We found lagged and concurrent temperature also play a significant role in driving floods across the seasons, especially in fall and summer as well as in spring in the U.S. Northeast (Figures S21–S28 in Supporting Information S1). The major predictors for the top flood season in each region are summarized along with the physical processes driving the observed causal link (Table 1).

In fall, which is not a major flood season in any region, concurrent precipitation is the primary predictor with a decisive causal link in the U.S. Northeast, Midwest, Southeast and western coastal regions (Figure 7). Lagged precipitation is the second most important predictor in the central and eastern United States, followed by lagged temperature in the west-central United States with substantial to strong causal links (Figure S28 in Supporting Information S1). The high precipitation in summer (i.e., lagged season) increases antecedent wetness and enhances flood probabilities in the eastern United States, while the high temperature reduces flood probabilities in the west-central part of the domain (Figures S21–S28 in Supporting Information S1).

In winter, concurrent precipitation shows a very strong to decisive causal link to floods in the eastern United States, with the exception of the upper Midwest and Great Plains. In the northern-tier region, which receives early snow spells, lagged precipitation is the major flood driver with strong to decisive causal link (Figure 7). Similarly, high temperature that triggers snowmelt is as well identified as the primary or secondary predictor with substantial to very strong causal link in the Northern Plains, the upper Midwest, the U.S. Southwest and Northwest (Figure 7 and Figures S21–S29 in Supporting Information S1). In the west-central United States, all four drivers exhibit a substantial to decisive role in driving floods, with high lagged temperature reducing the flow probabilities, while an increase in the rest of the predictors triggers floods (Figure 7). Winter is the main flood season in the U.S. Northwest, with concurrent precipitation being the primary driver with substantial to
<figure><img src="image_10.png" /><figcaption>**Figure 7.** Maps showing the gauge locations where each driver emerges as the top predictor. The color scale shows the strength of the causal relation using Jeffrey's score. The shades of red (green) indicate negative (positive) relationship of predictors with discharge. Only the gauges where at least substantial evidence is available against the null hypothesis of no causal link are shown in the figure.</figcaption></figure>

decisive causal link (Figure S21 in Supporting Information S1). Moreover, lagged precipitation and high con-current temperature favor flooding accounting for the early snow accumulation in fall and snowmelt in winter.

In spring, the major flood season in all regions except for the U.S. Northwest, high concurrent precipitation drives floods with strong to decisive causal relation in the eastern half of CONUS, with the exception of the upper Midwest and the Northern Plains (Figures 5 and 7, Figures S28 and S29 in Supporting Information S1). Lagged precipitation, implying the role of snow accumulation in winter, drives flood occurrence in the northern tier and west-central United States, with a substantial to very strong causal link (Figure 7). While high lagged temperature reduces the flow contributions in the northeast regions (in more than 60% gauges) and northwest coast, high concurrent temperature increases floods at a few locations in the northwest, with a substantial causal link (Figures S21 and S29 in Supporting Information S1). It is noteworthy that concurrent precipitation is the sole driver of floods in most gauges in the southeastern United States, with no significant role of other predictors during spring (Figures S26 and S27 in Supporting Information S1).

In summer, floods in the eastern United States are prominently driven by concurrent precipitation as the primary predictor, followed by lagged precipitation (Figures 5 and 7, Figures S26 and S27 in Supporting Information S1). Further high concurrent temperature in the southeastern United States and high lagged temperatures in the eastern United States substantially reduce flood peaks due to a reduction in antecedent wetness at more than 10% gauges (Figure S29 in Supporting Information S1). In the northwestern United States, where spring is a major flood season, high lagged temperature has a strong to decisive role in reducing flood probabilities and emerges as the
<table><caption>**Table 1**
Summary of the Top Two Predictors in the Seven Regions During the Two Major Flood Seasons in Each Region and the Physical Significance of the Derived Causal Relationship</caption><thead><tr><th>Region</th><th>Main flood season</th><th>Top predictor</th><th>Second predictor</th><th>Inference</th></tr></thead><tbody><tr><td rowspan="2">Great Plains of North</td><td>Spring</td><td>Lagged and concurrent precipitation. Lagged precipitation is particularly prominent in the northwest</td><td>Low concurrent temperature and lagged temperature</td><td>Lagged precipitation indicates snow accumulation and low lagged temperature favors this process</td></tr><tr><td>Summer</td><td>Concurrent and lagged precipitation. Lagged precipitation prominent in the west and concurrent precipitation in the east. Low lagged temperature in the northwest and eastern regions</td><td>Low lagged temperature and high lagged-precipitation in the interior and eastern regions</td><td>Low lagged temperature indicates high antecedent soil moisture</td></tr><tr><td rowspan="2">Great Plains of South</td><td>Spring</td><td>Concurrent precipitation</td><td>Lagged precipitation</td><td>High antecedent soil moisture leads to higher flood peaks</td></tr><tr><td>Summer</td><td>Concurrent precipitation, and low concurrent temperature in the central and north- central regions</td><td>Low concurrent and lagged temperature and high lagged precipitation</td><td>High antecedent soil moisture conditions</td></tr><tr><td rowspan="2">Midwest</td><td>Spring</td><td>Concurrent precipitation in the south and lagged precipitation in the north</td><td>Low lagged and concurrent temperature, high lagged precipitation in the south</td><td>High antecedent snow in the north</td></tr><tr><td>Summer</td><td>Concurrent precipitation</td><td>Lagged precipitation and low lagged temperature</td><td>High antecedent soil moisture conditions</td></tr><tr><td rowspan="2">Northeast</td><td>Spring</td><td>Concurrent precipitation, and low lagged temperature in the central northeast</td><td>Low lagged temperature and high lagged precipitation</td><td>Snow accumulation in winter and low lagged temperature favor high flows in spring</td></tr><tr><td>Winter</td><td>Concurrent precipitation and lagged precipitation in the north</td><td>Lagged precipitation along the east coast and high concurrent temperature in the north, and low concurrent temperature in the south</td><td>Snow accumulation along the east coast, and high concurrent temperature lead to melting in the north</td></tr><tr><td rowspan="2">Southeast</td><td>Spring</td><td>Concurrent precipitation</td><td>Lagged precipitation and low lagged temperature</td><td>High antecedent soil moisture along the east coast</td></tr><tr><td>Winter</td><td>Concurrent precipitation</td><td>Lagged precipitation and low concurrent temperature</td><td>High antecedent soil moisture and snow in mountainous regions</td></tr><tr><td rowspan="2">Northwest</td><td>Winter</td><td>Concurrent precipitation</td><td>High concurrent temperature and lagged precipitation, and low lagged temperature in the southeast</td><td>Antecedent snow in the northern parts, heavy precipitation along coast and high concurrent temperatures supporting snowmelt</td></tr><tr><td>Spring</td><td>Concurrent precipitation along the coast and lagged precipitation inland</td><td>High concurrent temperature, high lagged precipitation and low lagged temperature along the coast, and concurrent precipitation inland</td><td>Antecedent snow accumulation and high concurrent temperature enhancing melting</td></tr><tr><td rowspan="2">Southwest</td><td>Spring</td><td>Concurrent precipitation along the coast and lagged precipitation inland</td><td>Lagged precipitation along the coast and concurrent precipitation inland</td><td>Snow accumulation inland and melting along the western and southern coasts</td></tr><tr><td>Summer</td><td>Concurrent precipitation in southeast, high lagged precipitation and low lagged temperature in the northeast and west coast</td><td>Low lagged temperature inland and along coast, and low concurrent temperature in the southcentral parts</td><td>High antecedent soil moisture conditions</td></tr></tbody></table>

Note. Flood seasons are identified based on Kim and Villarini (2024). See Figures S21-S29 in Supporting Information S1 for supporting evidence for the inference presented in the table below.
primary predictor, with decisive causal link in 40% of the gauges (Figures S21 and S29 in Supporting Information S1). Lagged precipitation as well plays a substantial to decisive role in driving floods in the western coastal and west-central United States, where spring is the major flood season indicating the role of antecedent moisture in driving floods in these regions. Similarly, in the northern Great Plains, where spring is the major flood season, lagged precipitation and temperature have a substantial role in causing floods in 40% and 10% of the gauges, respectively (Figures 5 and 7, Figures S28 and S29 in Supporting Information S1).

In general, in the eastern United States and Great Plains of the South, irrespective of the season, concurrent precipitation is the primary predictor at most gauges. In the western CONUS, the primary predictor varies according to the season, with lagged temperature emerging as a decisive primary predictor in the northwest regions in summer. We observed the causal link derived by the BN model is consistent with the physical mechanism that overall drives floods across CONUS. The model encapsulates physical relationships and provides the most relevant predictor for each gauge. Furthermore, we could also identify the gauges where the four selected predictors fail to explain the causal relationship to floods (gauges not represented in Figure 7 and Figure S28 in Supporting Information S1). This could be partially due to the poor correlation in certain locations (Figures S17–S19 in Supporting Information S1), linked to other physical and human driven processes not captured by the present model.

## 4. Discussion and Conclusions

We applied a BN causal model for identifying the prominent flood driver across different seasons at 3,884 USGS streamflow gauge stations across CONUS. Our model performs well at all gauges, with a median Spearman's rank correlation coefficient above 0.7 for all seasons. The *do*-calculus intervention applied to the parent nodes in the BN model identifies the prominent flood drivers, with substantial to decisive role in altering the flood probabilities. Concurrent season precipitation emerges as the prominent predictor at most gauges, especially in the eastern CONUS and during the fall season. Lagged precipitation and lagged and concurrent temperature are also identified as relevant predictors with substantial to decisive causal links in various seasons. The causal model highlights regions and seasons where the land-atmospheric interactions that sustain high antecedent catchment wetness, snow accumulation and snowmelt are crucial in altering flood probabilities. Further, the model also identifies a few gauges where the coarse predictors are not sufficient in explaining the flow variability and where floods could be driven by other physical or human-driven processes. The results also highlight the relevance of the combination of predictors in driving high magnitude flows as well as flows in large (>5,000 km²) catchments, while concurrent precipitation alone could drive medium-magnitude flows. We found the relationship derived by the model is consistent with the physical processes. Overall, climatic factors drive floods in most river gauges, and we identify the major predictors for these gauges.

We also conducted the study separately for reference and non-reference gauges across the CONUS and we did not observe any significant difference in the observed pattern of cause-effect relationships, underlying the relatively muted role of human interventions in altering seasonal peak flow volumes as highlighted in previous studies (Ficklin et al., 2018; Kim et al., 2024; Mallakpour & Villarini, 2015; Schlef et al., 2021). Since the predictors used in the study mostly explain flood variability and the results are consistent across the reference and non-reference sites, our study satisfies the test for inconsistency of flood changes due to other drivers which are not included in the model (Merz et al., 2012). Further, the causal inference method applied in the study provides ample evidence for the strength of the causal link between flood peaks and relevant drivers satisfying the consistency test for attribution studies. The success of the BN model in explaining the observed flood variability can be extended to future flood projections.

In this study, we developed separate BN models for the four seasons, since season-lagged precipitation and temperature are used as proxies for antecedent wetness and snow processes. We acknowledge that the rigid seasonal boundaries used uniformly for all gauges may limit physical interpretations in certain gauges where flood generation processes are not bound within the seasonal boundaries. To address this, we classified the gauges into seven regions such that the timeline of flood processes is uniform within these regions. Nevertheless, spatial variability remains due to topographical differences and relative dominance of snow and rain within these regions. However, our interest here is to develop a broad understanding of the causal links between the climatic drivers and seasonal peak flows, and to provide insights to undertake regional studies. Future research can refine this
approach by classifying the gauges into snow- and rain- dominated regimes to get a better picture of the un-
derlying causal drivers.

We derived the structure of the BN used in the study from prior knowledge and existing studies related to the United States. However, BN models can learn the model structure, that is, the number of nodes and the direction of arcs from the data and this skill could be made use of in extending the study to other regions where other physical or human drivers are relevant in driving floods. Further, possible confounding factors like sea surface temperatures, atmospheric circulations, soil moisture measurements can be considered as predictors. We demonstrated the application of the BN in determining the causal relationship between floods and the four coarse climate predictors in a *do*-calculus perspective. The causal inference method used in the study can be applied to more complicated Bayesian Networks that incorporate other physical or human-driven drivers.

The major findings of the study are:

1. We developed a parsimonious Bayesian Network causal model for predicting seasonal and annual flow variability using concurrent and lagged season- and basin-averaged precipitation and temperature as predictors. Despite the simplicity of the model, we found it to perform well in capturing the streamflow variability in all seasons as well as the annual maximum flow, with a median Spearman's rank correlation coefficient around 0.7.

2. We applied the *do*-calculus intervention strategy by successively altering the probability state of each predictor and determined the impact of these interventions in altering the flood probabilities. When the predictors increase in magnitudes, we observed the flood probabilities to increase in response to concurrent and lagged precipitation in all seasons in most of the regions. On the other hand, flood probabilities reduced in the western and west-central United States during fall and winter and across the northern United States during summer in response to increased lagged temperatures. An increase in concurrent temperature is found to increase flood probabilities in fall and winter in the northern United States, while it reduces flood probabilities in the southeastern United States in summer.

3. We determined the Bayes factor quantifying the strength of the causal links between the predictors and seasonal peak flow, the target variable. We also used different scales such as Jeffreys' score, Lee and Wagenmakers' scale and Kass and Raftery's scale, providing a qualitative assessment of the strength of these relationships. Further, we conducted significance tests and found that the qualitative scales used are significant at the 95% confidence level. Concurrent precipitation emerged as the most decisive predictor in most of the gauges, especially in the eastern and southern United States, followed by lagged precipitation, lagged temperature and concurrent temperature.

4. We observe that concurrent precipitation is the dominant driver of medium floods, as well as floods in small and medium catchments (<5,000 km²). In contrast, lagged precipitation and temperature—proxies for antecedent catchment wetness—play a major role in driving large-magnitude flows, and floods in larger catchments.

5. We assessed the physical consistency of the observed cause-effect relationship. We found that the physical processes of accelerated snowmelt caused by increased temperature in winter in the northern United States, the role of early snow accumulation in the northern tier increasing the flood probability in the winter season, reduction in antecedent moisture due to increased temperature in the concurrent and lagged seasons in summer and fall, the role of wet antecedent moisture in driving floods in the summer and spring in the western United States are identified by the causal model.

Conflict of Interest

The authors declare no conflicts of interest relevant to this study.

Data Availability Statement

The data used in the study is freely available for download. The USGS streamflow data can be downloaded from
USGS Daily Data for USA_ Stage and Streamflow. The R package to retrieve the data can be obtained from
https://cran.r-project.org/web/packages/dataRetrieval/vignettes/dataRetrieval.html. Monthly precipitation and
temperature data is available for download from https://prism.oregonstate.edu/.
Acknowledgments

We acknowledge the availability of USGS streamflow gauge observations. This work was in part funded by Princeton University. Part of Philippe Naveau's work was supported by European H2020 XAIDA (Grant agreement ID: 101003469) and the French national programs: Agence Nationale de la Recherche (ANR): the ANR EXSTA, and the PEPR TRACCS programme under Grants ANR-22-EXTR-0005, the PEPR IRIMA (IRIMONT) and the Mines Paris/INRAE chair Geolearning. The authors acknowledge the assistance by Marco Scutari in implementing the BN causal framework in R. We also acknowledge the comments by the three anonymous reviewers, the editor, and the associate editor, which helped improve the manuscript.

References

Awasthi, C., Archfield, S. A., Ryberg, K. R., Kiang, J. E., & Sankarasubramanian, A. (2022). Projecting flood frequency curves under near-term climate change. *Water Resources Research*, 58(8). https://doi.org/10.1029/2021WR031246

Berghuijs, W. R., Harrigan, S., Molnar, P., Slater, L. J., & Kirchner, J. W. (2019). The relative importance of different flood-generating mechanisms across Europe. *Water Resources Research*, 55(6), 4582–4593. https://doi.org/10.1029/2019WR024841

Berghuijs, W. R., & Slater, L. J. (2023). Groundwater shapes North American river floods. *Environmental Research Letters*, 18(3), 034043. https://doi.org/10.1088/1748-9326/acbecc

Berghuijs, W. R., Woods, R. A., Hutton, C. J., & Sivapalan, M. (2016). Dominant flood generating mechanisms across the United States. *Geophysical Research Letters*, 43(9), 4382–4390. https://doi.org/10.1002/2016GL068070

Berthet, L., Andréassian, V., Perrin, C., & Javelle, P. (2009). How crucial is it to account for antecedent moisture conditions in flood forecasting? Comparison of event-based on continuous approaches on 178 catchments. *Hydrology and Earth System Sciences*, 13(6), 819–831. https://doi.org/10.5194/hess-13-819-2009

Blöschl, G., Hall, J., Viglione, A., Perdigão, R. A. P., Parajka, J., Merz, B., et al. (2019). Changing climate both increases and decreases European river floods. *Nature*, 573(7772), 108–111. https://doi.org/10.1038/s41586-019-1495-6

Brunner, M. I., Swain, D. L., Wood, R. R., Willkofer, F., Done, J. M., Gilleland, E., & Ludwig, R. (2021). An extremeness threshold determines the regional response of floods to changes in rainfall extremes. *Communications Earth & Environment*, 2(1), 1–11. https://doi.org/10.1038/s43247-021-00248-x

Czajkowski, J., Villarini, G., Montgomery, M., Michel-Kerjan, E., & Goska, R. (2017). Assessing current and future freshwater flood risk from North Atlantic tropical cyclones via insurance claims. *Scientific Reports*, 7(1), 41609. https://doi.org/10.1038/srep41609

Daly, C., Gibson, W. P., Taylor, G. H., Johnson, G. L., & Pasteris, P. (2002). A knowledge-based approach to the statistical mapping of climate. *Climate Research*, 22, 99–113. https://doi.org/10.3354/cr022099

Daly, C., Halbleib, M., Smith, J. I., Gibson, W. P., Doggett, M. K., Taylor, G. H., et al. (2008). Physiographically sensitive mapping of climatological temperature and precipitation across the conterminous United States. *International Journal of Climatology*, 28(15), 2031–2064. https://doi.org/10.1002/joc.1688

Das, M., & Ghosh, S. K. (2020). Standard Bayesian network models for spatial time series prediction. *Studies in Computational Intelligence*, 858, 11–22. https://doi.org/10.1007/978-3-030-27749-9_2

De Cicco, L. A., & Hirsch, R. M. (2014). *The dataRetrieval R package* (pp. 1–26). US Geological Survey.

Ficklin, D. L., Abatzoglou, J. T., Robeson, S. M., Null, S. E., & Knouft, J. H. (2018). Natural and managed watersheds show similar responses to recent climate change. *Proceedings of the National Academy of Sciences of the United States of America*, 115(34), 8553–8557. https://doi.org/10.1073/pnas.1801026115

Gupta, V. K., Mesa, O. J., & Dawdy, D. R. (1994). Multiscaling theory of flood peaks: Regional quantile analysis. *Water Resources Research*, 30(12), 3405–3421. https://doi.org/10.1029/94WR01791

Hannart, A., Pearl, J., Otto, F. E. L., Naveau, P., & Ghil, M. (2016). Causal counterfactual theory for the attribution of weather and climate-related events. *Bulletin of the American Meteorological Society*, 97(1), 99–110. https://doi.org/10.1175/bams-d-14-00034.1

Hettiarachchi, S., Wasko, C., & Sharma, A. (2019). Can antecedent moisture conditions modulate the increase in flood risk due to climate change in urban catchments? *Journal of Hydrology*, 571, 11–20. https://doi.org/10.1016/j.jhydrol.2019.01.039

Jeffreys, H. (1939). *Theory of probability* (3rd ed.). Oxford University Press.

Jiang, S., Tarasova, L., Yu, G., & Zscheischler, J. (2024). Compounding effects in flood drivers challenge estimates of extreme river floods. *Science Advances*, 10(13), 4005. https://doi.org/10.1126/sciadv.adl4005

Kass, R. E., & Raftery, A. E. (1995). Bayes factors. *Journal of the American Statistical Association*, 90(430), 773–795. https://doi.org/10.2307/2291091

Kim, H., & Villarini, G. (2023). On the attribution of annual maximum discharge across the conterminous United States. *Advances in Water Resources*, 171, 104360. https://doi.org/10.1016/j.adwatres.2022.104360

Kim, H., & Villarini, G. (2024). Higher emissions scenarios lead to more extreme flooding in the United States. *Nature Communications*, 15(1), 237. https://doi.org/10.1038/s41467-023-44415-4

Kim, H., Villarini, G., Wasko, C., & Tramblay, Y. (2024). Changes in the climate system dominate inter-annual variability in flooding across the globe. *Geophysical Research Letters*, 51(6). https://doi.org/10.1029/2023GL107480

Lee, M. D., & Wagenmakers, E. J. (2014). Bayesian cognitive modeling: A practical course. Cambridge University Press.

Mallakpour, I., & Villarini, G. (2015). The changing nature of flooding across the central United States. *Nature Climate Change*, 5(3), 250–254. https://doi.org/10.1038/nclimate2516

Merz, B., Vorogushyn, S., Uhlemann, S., Delgado, J., & Hundecha, Y. (2012). Hydrology and Earth System Sciences HESS Opinions “More efforts and scientific rigour are needed to attribute trends in flood time series”. *Hydrology and Earth System Sciences*, 16(5), 1379–1387. https://doi.org/10.5194/hess-16-1379-2012

Nanditha, J. S., & Mishra, V. (2022). Multiday precipitation is a prominent driver of floods in Indian River basins. *Water Resources Research*, 58(7), e2022WR032723. https://doi.org/10.1029/2022WR032723

Nanditha, J. S., & Mishra, V. (2024). Projected increase in widespread riverine floods in India under a warming climate. *Journal of Hydrology*, 630, 130734. https://doi.org/10.1016/j.JHYDROL.2024.130734

Nanditha, J. S., & Mishra, V. (2024b). Drivers of widespread floods in India. *Journal of Hydrometeorology*. https://doi.org/10.1175/JHM-D-23-0168.1

Nanditha, J. S., Rajagopalan, B., & Mishra, V. (2022). Combined signatures of atmospheric drivers, soil moisture, and moisture source on floods in Narmada River basin, India. *Climate Dynamics*, 58(9–10), 2831–2851. https://doi.org/10.1007/s00382-022-06244-x

Neri, A., Villarini, G., Slater, L. J., & Napolitano, F. (2019). On the statistical attribution of the frequency of flood events across the US Midwest. *Advances in Water Resources*, 127, 225–236. https://doi.org/10.1016/j.adwatres.2019.03.019

Ossandón, Á., Nanditha, J. S., Mendoza, P. A., Rajagopalan, B., & Mishra, V. (2022). A Bayesian hierarchical framework for postprocessing daily streamflow simulations across a river network. *Journal of Hydrometeorology*, 23(6), 947–963. https://doi.org/10.1175/JHM-D-21-0167.1

Ossandón, Á., Rajagopalan, B., Lall, U., Nanditha, J. S., & Mishra, V. (2021). A Bayesian hierarchical network model for daily streamflow ensemble forecasting. *Water Resources Research*, 57(9), e2021WR029920. https://doi.org/10.1029/2021WR029920

Pearl, J. (1995a). Causal diagrams for empirical research. *Biometrika*, 82(4), 702–710. https://doi.org/10.1093/biomet/82.4.702

Pearl, J. (1995b). From Bayesian networks to causal networks. In G. Coletti, D. Dubois, & R. Scozzafava (Eds.), *Mathematical models for handling partial knowledge in artificial intelligence* (pp. 157–182). Springer US. https://doi.org/10.1007/978-1-4899-1424-8_9

Pearl, J. (2009). *Causality*. Cambridge University Press.
Pearl, J. (1994). A probabilistic calculus of actions. In R. Lopez de Mantaras & D. Poole (Eds.), *Proceedings of the Tenth Conference on Uncertainty in Artificial Intelligence (UAI-94)* (pp. 454–462). Morgan Kaufman.

Ravindranath, A., Devineni, N., Lall, U., Cook, E. R., Pederson, G., Martin, J., & Woodhouse, C. (2019). Streamflow reconstruction in the upper Missouri River basin using a novel Bayesian network model. *Water Resources Research*, *55*(9), 7694–7716. https://doi.org/10.1029/2019WR024901

Schlef, K. E., François, B., & Brown, C. (2021). Comparing flood projection approaches across hydro-climatologically diverse United States river basins. *Water Resources Research*, *57*(1). https://doi.org/10.1029/2019WR025861

Scutari, M. (2017). Bayesian network constraint-based structure learning algorithms: Parallel and optimized implementations in the bnlearn R package. *Journal of Statistical Software*, *77*(1). https://doi.org/10.18637/jss.v077.i02

Scutari, M. (2018). Dirichlet Bayesian network scores and the maximum relative entropy principle. *Behaviormetrika*, *45*(2), 337–362. https://doi.org/10.1007/s41237-018-0048-x

Sharma, A., Wasko, C., & Lettenmaier, D. P. (2018). If precipitation extremes are increasing, why aren't floods? *Water Resources Research*, *54*(11), 8545–8551. https://doi.org/10.1029/2018WR023749

Slater, L., Villarini, G., Archfield, S., Faulkner, D., Lamb, R., Khouakhi, A., & Yin, J. (2021). Global changes in 20-year, 50-year, and 100-year river floods. *Geophysical Research Letters*, *48*(6), e2020GL091824. https://doi.org/10.1029/2020GL091824

Slater, L. J., & Villarini, G. (2017). Evaluating the drivers of seasonal streamflow in the U.S. midwest. *Water*, *9*(695), 695. https://doi.org/10.3390/w9090695

Slater, L. J., Villarini, G., Bradley, A. A., & Vecchi, G. A. (2019). A dynamical statistical framework for seasonal streamflow forecasting in an agricultural watershed. *Climate Dynamics*, *53*(12), 7429–7445. https://doi.org/10.1007/s00382-017-3794-7

Taboga, M. (2017). *Lectures on probability theory and mathematical statistics* (3rd ed.). CreateSpace Independent Publishing Platform.

Tramblay, Y., Villarini, G., El Khalki, E. M., Gründemann, G., & Hughes, D. (2021). Evaluation of the drivers responsible for flooding in Africa. *Water Resources Research*, *57*(6). https://doi.org/10.1029/2021WR029595

Villarini, G. (2016). On the seasonality of flooding across the continental United States. *Advances in Water Resources*, *87*, 80–91. https://doi.org/10.1016/j.advwatres.2015.11.009

Villarini, G., & Slater, L. J. (2017). *Climatology of flooding in the United States*. Oxford Research Encyclopedia of Natural Hazard Science. https://doi.org/10.1093/acrefore/9780199389407.013.123

Wasko, C., Nathan, R., & Peel, M. C. (2020). Changes in antecedent soil moisture modulate flood seasonality in a changing climate. *Water Resources Research*, *56*(3). https://doi.org/10.1029/2019WR026300

Wasko, C., Nathan, R., Stein, L., & O'Shea, D. (2021). Evidence of shorter more extreme rainfalls and increased flood variability under climate change. *Journal of Hydrology*, *603*, 126994. https://doi.org/10.1016/j.jhydrol.2021.126994