1. Introduction
Extreme rainfall is one of the major hazards associated with tropical cyclones (TCs), and the resultant flooding has caused large numbers of fatalities and vast economic damage (Rezapour and Baldock 2014). Recent severe TC rainfall cases include Hurricane Harvey in 2017, which set the record for extreme rainfall exceeding 1000 mm, and Hurricane Georges (1998) with estimated rainfall accumulations of over 990 mm on the island of Hispaniola, which resulted in extreme mudslides and flooding that led to 589 fatalities across the island (Guiney 1999).
Within a TC, the region of maximum wind (located at the radius of maximum wind, Rmax) is kept in near–thermal wind balance by a strong secondary circulation, which in turn produces eyewall clouds and precipitation. Outside of the eyewall, precipitation is primarily associated with rainbands, which have a spiral geometry as opposed to the quasi-circular geometry of the eyewall. Distant rainbands, outside the region dominated by vortex dynamics, are located far from the storm center and are composed of buoyant convection aligned with confluence lines in the large-scale low-level wind field spiraling into the hurricane vortex.
TC rainfall in these features can be broadly categorized into two forms—convective and stratiform rainfall—with the former generally associated with extreme rainfall rates (≥5 mm h−1) and the latter with lower rain rates (≤4 mm h−1) (Liu et al. 2012; Tu et al. 2021; Houze 2010). These two types of precipitation reflect sharply contrasting dynamical processes controlling vertical motion and microphysical processes controlling the growth of precipitation particles. Convective rainfall is mainly found in the eyewall and rainbands; stratiform rainfall generally covers larger nonconvective areas surrounding these features.
Several mechanisms have been found to be important for TC rainfall. Over-ocean TC rain rates have been shown to be a function of TC intensity, with greater rates generally linked to stronger TCs (Lonfat et al. 2004). Vortex stretching, being the effect of storm intensification and weakening associated with vortex spinup and spindown, can also affect TC rainfall (Lu et al. 2018; Rogers et al. 2020). Vertical wind shear has been found to influence the intensity and distribution of convection and precipitation (Wang and Holland 1996), with the primary location for TC rainfall in the Northern (Southern) Hemisphere being downshear left (right) (Chen et al. 2006). Extratropical transitioning storms can also drastically influence TC rainfall, particularly in mid- and high-latitude regions (Atallah and Bosart 2003; Atallah et al. 2007; Liu et al. 2020). At landfall, frictional effects are of major importance for producing heavy rainfall, primarily through convergence and vertical motion, and have been found to have a strong impact on the location of convection and storm rainfall asymmetry, particularly where there are changes in surface roughness (e.g., at a land–sea interface) (Li et al. 2014; Zhang et al. 2018). Topographic forcing can significantly influence TC rainfall, especially in mountainous regions (Lin et al. 2001; Cheung et al. 2008; Huang et al. 2020).
Multiple studies have analyzed rain fields using Fourier-decomposed rainfall wavenumber asymmetries—with studies primarily focusing on wavenumber-1 asymmetries when the TC is over the ocean (Lonfat et al. 2004; Chen et al. 2006; Pei and Jiang 2018). These wavenumber-1 asymmetries have been attributed primarily to two physical mechanisms: (i) the movement of the storm through the surrounding atmosphere and (ii) the wind shear of the large-scale environment (Houze 2010). The first reason for wavenumber-1 asymmetry is because the eyewall is closely associated with the cyclone’s strong circulation, which is inertially stable and resistant to the surrounding airflow. The mean vertical velocity pattern and associated convective precipitation of a moving TC is inherently asymmetric due to wind field convergence (Houze 2010). The second reason for the wavenumber-1 asymmetry is vertical wind shear in the environment; although shear is unfavorable to the development and intensification of TCs, these storms usually exist within environments having some degree of shear. Regardless of the strength of the environmental shear, the maximum in eyewall rainfall is on the downshear left side of the storm in the Northern Hemisphere. These two factors acting together lead to a variety of wavenumber-1 asymmetries of the eyewall structure and intensity of TC rainfall.
Despite considerable research on the processes governing TC rainfall, the literature on TC rainfall modeling is limited and only a few methods have been employed. High-resolution climate and weather forecasting models enable the explicit simulation of TC rainfall (Bacmeister et al. 2018; Liu et al. 2019); however, it is currently computationally inefficient to model enough TC rainfall events for risk assessment or probabilistic forecasting by running these models. To overcome this constraint, simplified TC rainfall models have been developed to efficiently simulate TC rainfall, with existing models falling broadly into two categories: physics- and statistics-based models.
Langousis and Veneziano (2009) developed one of the first physics-based TC rainfall models using basic thermodynamics and boundary layer theory. The model accounts for the azimuthal asymmetries in surface friction related to storm motion, but it does not account for the interaction of the storm with its environment (e.g., wind shear, topography). Zhu et al. (2013) first described a physics-based TC rainfall model (TCRM) that estimates the vertical velocity and upward vapor flux from a time-evolving, axisymmetric storm wind field and the environmental winds. The TCRM takes into account the major rainfall generation mechanisms including frictional, topographic, baroclinic (shear), and vortex stretching effects (Lu et al. 2018). TCRM has been shown to have high accuracy in simulating TC rainfall patterns and climatology in the North Atlantic (Lu et al. 2018; Xi et al. 2020) and has been used for TC rainfall risk assessment (Emanuel 2017; Gori et al. 2022).
In contrast to physics based approaches, Tuleya et al. (2007) developed a statistical model to simulate TC rainfall profiles (RCLIPER) as a function of TC intensity with an axisymmetric model that mimics the radius profile of TC rainfall developed based on the satellite observation of TC rainfall. The input for RCLIPER is the maximum sustained wind, and no environmental or surface information is used. Lonfat et al. (2007) advanced this model by adding parameters that account for rainfall asymmetry caused by topographic and shear effects (the Parametric Hurricane Rainfall Model). More recently, Villarini et al. (2022) developed an ensemble generator for rainfall based on the observations of 12 storms that made landfall along the Louisiana coastline between 2002 and 2017. Their model, composed of a deterministic and a stochastic component, was largely able to capture key patterns of rainfall, but their results were sensitive to the storms used to train the model, particularly the role of Hurricane Katrina.
Due to the computational expense associated with modeling precipitation and many mechanisms that contribute to precipitation, existing models have a number of limitations. Primarily, these models cannot model rainbands and many do not account for wavenumber-1 asymmetries. These models also simulate rainfall mechanisms of TCs with tropical features only and therefore lack an explicit account of the effect of extratropical transition (Evans et al. 2017). Precipitation estimates of statistical and physics-based models are also sensitive to the wind field model used (Lu et al. 2018; Xi et al. 2020): evaluating modeled estimates of the TCRM, Xi et al. (2020) found that the peak azimuthal mean rainfall rate for category-3–5 historical (1999–2018) North Atlantic TCs almost doubled between the Chavas et al. (2015) and Holland (1980) wind field models.
Increases in computational power and in the sophistication of machine learning techniques have led to recent advances in data-driven modeling of weather and climate extreme events (Reichstein et al. 2019), including in the estimation and prediction of TC characteristics and associated hazards (Brunner et al. 2020; Lockwood et al. 2022). Xue et al. (2022) used a multitask convolutional neural network (CNN) to estimate near-real-time global TC precipitation using HURSAT-B1 data and satellite data. Their CNN model could accurately model the spatial pattern of rainfall, but systematically underestimated extreme rainfall. Loridan et al. (2017) developed a TC wind field model by applying unsupervised (principal component analysis) and supervised (quantile random regression forests) machine learning methods to historical TC events modeled using the Weather Research and Forecasting (WRF) Model in the western North Pacific. By sampling the conditional distributions of the principal component weights, their model could produce probabilistic maps for the likelihood of attaining given wind speed thresholds, greatly increasing the range of plausible wind field patterns for TC risk assessments compared to traditional parametric wind models.
Key goals and outline for this study
The clouds and precipitation of a TC are practically always distributed asymmetrically around the storm. To model these rainfall asymmetries, we develop a machine learning–based rainfall model following the framework of Loridan et al. (2017), using rain-rate maps instead of wind speed maps. The goal of this paper is to model rain-rate maps of mature TCs at 1-hourly time steps conditioned on TC characteristics (e.g., intensity, translation speed, location) and local monthly environmental variables (e.g., surface temperature, vertical wind shear), such that the model may be deployed in flood hazard risk assessments and probabilistic forecasting (Fig. 1). We refer to the model as the Machine Learning Tropical Cyclone Rain Model (MLTRM). We develop the model for the North Atlantic basin using WRF simulations and historical satellite observations. We then compare the MLTRM estimates during cross validation to the WRF simulations and to historical satellite observations. We emphasize that we introduce an overwater model here that neglects the impact of surface roughness, terrain, and landfalling on TC precipitation. These factors may be incorporated using relevant parameters identified by previous studies (e.g., Langousis and Veneziano 2009; Lonfat et al. 2004; Liu et al. 2018).
Schematic overview of the (a) rain-rate shape model, (b) maximum rain-rate model, and (c) quantile random forest models. The outputs for (a) and (b) are the targets for the quantile random forest models in (c).
Citation: Journal of Hydrometeorology 25, 1; 10.1175/JHM-D-23-0065.1
2. Methodology
a. Model architecture and input data
We train, deploy, and validate the model in the North Atlantic basin. Briefly, given a set of hourly track characteristics and local monthly environmental conditions (described below), the model outputs two separate components (Fig. 1):
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Rain-rate maps which are normalized by the maximum rain rate (i.e., all values are between 0 and 1). Specifically, this model is trained to predict rain-rate patterns that are derived from historical events modeled using the WRF Model (Fig. 1a). The shape variability is identified using principal component analysis (PCA).
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A maximum rain-rate value at hourly time steps. This transfers the normalized rain-rate map in step 1 to an absolute rain-rate map (Fig. 1b). We train this model using historical satellite precipitation measurements and International Best Track Archive for Climate Stewardship (IBTrACS) data.
To model both components, we use supervised learning algorithms, in which both input (predictors) and outputs (predictands) are provided. Of these algorithms, decision trees are widely used as they are intuitive and allow for simple and useful interpretation. Random forests were introduced as a machine learning tool as a way of averaging multiple deep decision trees, trained on different parts (known as bagging) of the same training set, with the goal of reducing high variance that can be common in simple decision trees (Breiman 2001). This modeling technique is known as an efficient model of ensemble learning, as it ensures high predictive precision, flexibility, and immediacy (Aria et al. 2021). For regression, random forests give an accurate approximation of the conditional mean of a response variable. However, the conditional mean does not provide a complete summary of the distribution, so in order to estimate the associated uncertainty, quantile random forest (QRF) models have been developed that provides an estimate of the conditional distribution at different quantile values. The key difference between QRF and random forests is as follows: for each node in each tree, random forests keep only the mean of the observations that fall into this node and neglect all other information. In contrast, QRF keeps the value of all observations in this node, not just their mean, and assesses the conditional distribution based on this information (Meinshausen and Ridgeway 2006).
To gain an understanding of the uncertainty of our rain map predictions, we use QRF models and we focus throughout the paper on the 0.2, 0.5 (median), and 0.8 quantile levels. The median can serve as a prediction of the center of the distribution, and for example, the 95th percentile represents the value of the response with 95% of the predicted points below. In other words, for each QRF model and under the conditions specified by all input features, 60% of the predictions should fall in the range between the 0.2 and 0.8 quantiles.
We train the QRF models to predict rain-rate maps conditioned on both the physical characteristics of the TC and local environmental characteristics. Table 1 lists the predictors that will be used to train the QRF models. Specifically, we use nine input parameters related to the physical characteristics of the TC, including the maximum wind speed (Vmax), Rmax, bearing, latitude, longitude, translation speed, and central pressure. As rapid intensity changes often occur in conjunction with rapid reorganization of the cyclone’s mesoscale cloud and precipitation structures, we also include the 6-hourly change in central pressure and Vmax. To capture the local environmental conditions we also use as inputs to the QRF models monthly ERA5 data (Hersbach et al. 2020) at the grid cell closest to the TC center (the data have a resolution of 0.25°). We include as predictors to the QRF models from ERA5 the monthly mean vertical wind shear, mean surface pressure at 2 m, convective available potential energy (CAPE), and mean surface temperature at 2 m. We use CAPE to quantify atmospheric instability or the positive buoyancy that would be experienced by a lifted parcel, with increases in CAPE resulting mainly from increased low-level specific humidity (Chen et al. 2020). We use monthly data so we can capture the interannual variability rather than modeling the day-to-day weather scale variability. Additionally, we set choose to model using monthly data such that the model is deployable on a large range of climate change scenarios (e.g., CMIP6 model data) that is predominately monthly data.
List of TC characteristics used to train the supervised machine learning module. Change is calculated from the 6 h before.
b. QRF output data
Above we introduced the architecture of the QRF models and the variables used as inputs to the QRF models, which include the physical characteristics of the TC and local monthly environmental characteristics. Next, we introduce the output (predictands) datasets used to train the QRF models. To reconstruct the overwater TC rain-rate shape for a given TC, we use QRF models to predict the PC weights for five PCA shapes [section 2b(1)]. Using these predicted PC weights, we can reconstruct the spatial variability of rainfall. A sixth QRF model is used to predict the maximum rain rate, which transforms the normalized maps to absolute rain-rate maps—this model is trained with the satellite rain-rate values [section 2b(2)].
The input and output data for each QRF model were randomly divided into 70% for training, 10% for validation, and 20% for testing. We note that the rain-rate model is trained by randomly dividing the data 70%–10%–20%, thus similar events are likely located with both the training and test datasets. To explore the impact of this on model performance, we perform an out-of-sample test, training the QRF model on years 2000–17, before testing its accuracy on years 2018 and 2019 (Fig. S1 in the online supplemental material). We note that the accuracy does not significantly alter. We used the Scikit-Learn RandomizedSearchCV package to determine the best set of QRF hyperparameters by optimizing a set of hyperparameters by cross-validated search over parameter settings. The function minimizes the mean-square error evaluated on the validation dataset, including the number of tree regressors to train (n_estimators) and number of features to split on each node (mtry). The hyperparameters tested and chosen are shown in Table S1.
1) Overwater TC rainfall shape model
A total of 26 historical TCs were simulated using the WRF Model (Skamarock et al. 2008) to create a database of rain-rate field targets in the North Atlantic basin. Table 2 summarizes the historical TCs simulated to develop a training data source. The database has a total of 2073 target time steps that comprise of two-dimensional maps of storm rain rates when the TC is over the ocean. To select the storms, we sampled 26 events within each intensity category 1–5 such that the selected events closely matched the intensity distribution of the historical (1849–2019) IBTrACS climatology. Figure 2 shows the tracks and characteristics of the modeled WRF events compared to historical (1849–2019) IBTrACS climatology (Knapp et al. 2018). The WRF events slightly overrepresent high-latitude and lower-longitude storms in the North Atlantic compared to climatology (Figs. 2a,b). The subset of historical events modeled using WRF very well captures the distribution of maximum winds, central pressure, radius of maximum wind, and translation speeds compared to climatology in the North Atlantic basin (Figs. 2d–g).
Historical North Atlantic TCs (26) modeled with WRF.
(a) Tracks of the 26 historical events modeled with WRF. (b)–(g) Normalized frequency plots of the characteristics of the TCs for the WRF runs (blue) and the IBTrACS (red) dataset.
Citation: Journal of Hydrometeorology 25, 1; 10.1175/JHM-D-23-0065.1
For each simulation, the WRF Model is run with three nested domains. The outer domain covers the whole track of the TC and is static with a horizontal resolution of 9 km. The two inner domains are vortex-following domains with 3- and 1-km horizontal resolution, respectively. The details of the modeling and parameterizations can be found in supplemental text S1. Since these WRF runs are model realizations rather than observations, the data will not exactly match the historical events due to the potential limitations associated with the WRF Model, such as a low bias in storm intensities and sensitivity of rain rate to microphysical scheme (Yang et al. 2019; Talbot et al. 2012). Our intention when using the WRF Model is not to exactly recreate the historical events, but rather to create a dataset with realistic variability in TC rainfall shapes that can be learned by a machine learning model. We compare our WRF simulations to historical satellite rainfall observations (SRO), which consists of the Tropical Rainfall Measuring Mission (TRMM), which started service in 1988, and the Global Precipitation Mission (GPM), which replaced the TRMM after 2015 (Skofronick-Jackson et al. 2017). We find that our WRF simulations generally overestimate the rain rates and storm accumulation (Figs. S2 and S3) compared to satellite data; however, the WRF runs show good ability in capturing the shape of TC rain rates compared to the satellite observations (Fig. S3). As a result of the overestimation of rain rates and accumulation in the WRF simulations, we only use the WRF simulations to model rainfall spatial variability. We do this by normalizing each time step by the storms maximum rain rate at that time step, such that all values are between 0 and 1; we do not use the rain rates modeled in WRF.
We homogenize the WRF rain-rate maps for the PCA analysis following Loridan et al. (2017) such that the precipitation fields are rotated and interpolated into a standard reference grid with dimensions 500 × 500 (i.e., 250 points in each direction starting from the storm center). The convention adopted is to rotate the precipitation field so that the wind maximum (Vmax) ends up at point (X = 0, Y = 25) and this gives rise to a target-dependent grid resolution of Rmax divided by 25, covering a 20Rmax × 20Rmax domain. Figure S4 provides an example of a precipitation target before and after rotation and interpolation.
2) Maximum rainfall-rate data
As discussed above, our WRF simulations show good ability in capturing the spatial patterns of rain rate over the oceans, but generally overestimate the rain rates and accumulation compared to satellite observations. We therefore develop a maximum rain-rate model based on SRO that can be used to transform the normalized rain-rate maps derived using PCA to absolute rain-rate fields (Fig. 1b). The SRO is used as the primary data source to train our maximum rain-rate model. The data are available with a 0.25° × 0.25° horizontal resolution and 3-h temporal resolution; we use spline interpolation to transform the 3-hourly rain values to 1-hourly values. Specifically, we identify in IBTrACS data all TC events in the North Atlantic basin between 2000 and 2020. We extract a single maximum rain-rate value within 600 km of the TCs center from the SRO. Uncertainties in SRO are reported in previous research, and it is likely that SRO biases toward underestimating heavy rainfall and overestimating light rainfall (Prakash et al. 2016). The distribution of the maximum rain-rate values, conditional on the storm and environmental characteristics, are the target (output) data for the sixth QRF model.
c. Model deployment and validation
To compare the model to SRO, we deploy the MLTRM for all category-1–5 TC events identified in IBTrACS data in the North Atlantic basin for the historical period from 2000 to 2020, with Rmax from extended best track. The deployment of MLTRM requires the input of angle of maximum wind to regrid the PCA output to a latitude–longitude grid. As data for the location of Rmax are not available, following Loridan et al. (2015), we assume that in the Northern Hemisphere the angle of maximum wind is 90° to the right of the storm bearing. We acknowledge that this assumption is idealized, and because the storm track is an integral of the three-dimensional flow the storm is embedded within at any instant in time, and the relationship between the angle of maximum wind and bearing is not a constant relationship. When a storm changes direction or there are strong changes of the vertical wind shear, the Earth-relative location of the maximum wind relative to the instantaneous motion of the storm can be out of phase by as much as 90°–120° (Houze 2010). Based on the WRF simulations, there is large variability in the angle of maximum winds (Fig. S5). However, the highest frequency of angle of maximum winds is found between 100° and 80° from the bearing of the storm, giving us some confidence that this assumption is generally valid.
3. Results
a. QRF prediction and PCA dimension reduction
To model rainfall asymmetries, the historical normalized rain-rate targets modeled using WRF are decomposed using PCA and our focus is on the first five PCA shapes (Fig. 3), which collectively explain 45% of the rainfall variability in the data. We remind the reader that the 2073 WRF rain-rate maps input into the PCA are normalized by the maximum rain rate at each time step and are regridded and rotated such that the Rmax is located at (0, 25). Each PCA shape has a probability density function of the PC weights (Fig. 4).
(a) Average and (b)–(f) first five principal component shapes for the 26 WRF runs in the North Atlantic basin.
Citation: Journal of Hydrometeorology 25, 1; 10.1175/JHM-D-23-0065.1
Density plots for the mean weights associated with the first five principal components.
Citation: Journal of Hydrometeorology 25, 1; 10.1175/JHM-D-23-0065.1
As noted above, we use QRF models to predict the distributions of the five PC weights and maximum rain-rate values, using the same set of input features (Table 1). We first present the performance of the QRF model predictions on the test datasets for the maximum rain-rate prediction and PC weights (Fig. 5). To assess the importance of each input variable to the QRF models in prediction of the PC weights and maximum rain-rate values (Fig. 6), we use the Breiman importance metric (Breiman 2001). The Breiman importance function probes which inputs are most predictive, as measured by changes in a combination of both the mean squared error (MSE) and r2 when each input variable is removed from the prediction. Large importance values indicate that the input variable strongly impacts the performance of the QRF prediction.
Scatter of QRF predictions for the (a) maximum rain-rate value and (b)–(f) the first five PC weights.
Citation: Journal of Hydrometeorology 25, 1; 10.1175/JHM-D-23-0065.1
Breiman importance metric for the (a) maximum rain-rate model and (b)–(f) first five PC weight predictions.
Citation: Journal of Hydrometeorology 25, 1; 10.1175/JHM-D-23-0065.1
The QRF prediction of maximum rain rate has a correlation coefficient of 0.71 (Fig. 5a). For maximum rain rate, the most important input variables are the TC latitude and longitude, local monthly surface temperature, local monthly surface pressure, and CAPE (Fig. 6a). The QRF predictions of the five PC weights have r2 values and correlation coefficients ranging from 0.4 to 0.75 and from 0.64 to 0.86, respectively (Figs. 5b–f).
The mean rain-rate field derived from PCA shows a maximum approximately at the location of Rmax, with a distinct quasi-circular geometry associated with the expected location of TC eyewall (Fig. 3a). We note that each PCA shape is multiplied by the predicted PC weight, both of which can be positive or negative, thus both the positive and negative PCA values in Fig. 3 may contribute to increases or decreases in the overall rain rate depending on the sign of the PC weight. The first PCA shape explains approximately 13% of rain-rate shape variability across the 2073 rain-rate maps (Fig. 3b). This shape exhibits a pattern similar to that of the primary rainbands in a canonical TC, with a large spiral rainband that moves inward from the outer reaches of the cyclone until it becomes roughly tangent to the eyewall (Houze 2010). The proximate cause of the principal rainband has not been firmly established (Houze 2010), although Houze (2010) suggests that the rainbands have dynamics somewhat in common with eyewalls related to the storms’ Vmax and Rmax. Indeed, the Rmax, central pressure, maximum wind, and CAPE are the strongest predictors of the QRF PC weights for this PCA shape, as measured by the Breiman importance metric (Fig. 6b). The second PCA shape, which explains 10% of shape variability, also has a quasi-circular geometry associated with the expected location of TC eyewall. For PCA shape two, the central pressure, Rmax, maximum wind, CAPE, and monthly vertical wind shear are of primary importance for the QRF prediction of the weights (Fig. 6c). The third PCA shape, which explains 9% of shape variability, shows wavenumber-1 asymmetries with a clear dipolar structure possibly associated with either the motion of the storm through the environment or vertical wind shear (Houze 2010; Lonfat et al. 2004; Pei and Jiang 2018). The TC longitude, translation speed, and storm bearing are the most important predictors of PCA shape three, followed by surface temperature and vertical wind shear. Although it is unclear what the primary cause of this asymmetry is, the high importance of the translation speed and bearing suggest that the motion of the storm through the environment may be a dominating factor for this rainfall asymmetry (Fig. 6d). The fourth PCA shape, which explains 8% of the spatial variability, shows a pattern similar to the mean eyewall rainfall pattern (Fig. 3a), but with the maximum precipitation rotated clockwise from the location of the Rmax. For this PCA shape, the strongest predictor is the maximum wind speed followed by TC central pressure (Fig. 6e).
Finally, the fifth PCA shape explains approximately 5% of rainfall variability (Fig. 3f). This shape has a notable asymmetry to the north of the location of maximum rainfall outside of the eyewall region. These asymmetries on the outer circulation may imply lower-tropospheric frontogenesis, which is commensurate with extratropical transition as the TC moves poleward over lower sea surface temperatures (Klein et al. 2000). In particular, this shape shows similarities to stages 2 and 3 of the conceptual ET model of Klein et al. (2000), with the outer asymmetry indicative of ascent that produces cloud bands wrapping around the storm center and the inner asymmetry indicative of dry-adiabatic descent that occurs close enough to the circulation center to produce erosion of eyewall convection as the TC evolves into a baroclinic storm. For this PCA shape, the strongest predictors are the TC Rmax, latitude, and translation speed (Fig. 6e).
b. Comparison to WRF and observations
To further investigate the ability of the model in capturing the spatial variation of TC rainfall, we compare the shapes predicted by MLTRM to the WRF simulations. To do this we use a leave-one-out procedure (cross validation) where the shape models are repeatedly trained using 25 of the WRF historical events simulated (the training set) and then deployed to predict the left-out storm (the testing set whose data are unseen during model building). We remind the reader that we focus on rain map predictions for three quantile levels, 0.2, 0.5, and 0.8, which correspond to the rain maps at which 20%, 50%, and 80% of the distribution falls below that level, respectively.
Figure 7 shows the event-averaged rain-rate fields predicted by MLTRM and the corresponding WRF simulations for five randomly chosen events (see the data availability statement for the 21 other events). MLTRM shows good ability in capturing the shape and location of the maximum rain rates as compared to the WRF simulations, with a clear rainfall maximum at the Rmax and rainfall variability commensurate with rainbands and wavenumber asymmetries. For TCs Fabian (2003) and Dennis (1999) (Figs. 7a–d,e–h), MLTRM shows little variability across the quantile predictions and high correlation coefficient and r2 values between 0.8 and 1, suggesting that the model is able to capture these patterns with high confidence. The predictions for TCs Sandy (2012) and Marie (2017) show large shape variability across the predicted quantiles, suggesting that there is large uncertainty in this shape prediction (Figs. 7i–k,m–p). Despite this variability across predictions for TC Sandy, it is clear that the upper (0.8) MLTRM quantile prediction shows a similar pattern to that modeled in WRF with a correlation coefficient and r2 value of 0.7 and 0.5, respectively (Fig. 7k). In particular for the upper (0.8) MLTRM quantile prediction for TC Sandy (2012), MLTRM predicts large rain asymmetries on the outer circulation of the TC that are commonly found during extratropical transition (Klein et al. 2000).
Event averaged rain-rate fields (shown in the first three columns) predicted by MLTRM model as compared to the WRF simulations (shown in the last column). The results show the leave-one-out cross validation, where the shape models are repeatedly trained using 25 of the WRF historical events simulated (the training set) and then deployed to predict the left-out storm. Supplementary file 1 (see the data availability statement) shows the results for the other 21 storms.
Citation: Journal of Hydrometeorology 25, 1; 10.1175/JHM-D-23-0065.1
To evaluate the performance of MLTRM more generally, we further analyze the statistics of storm total rainfall compared to SRO. We model all category-1–5 TC events over the ocean identified in IBTrACS data in the North Atlantic basin for the historical period between 2000 and 2020, totaling 92 events. The MLTRM 0.5 quantile captures the observed satellite storm total rainfall with high accuracy, with a correlation value of 0.91 and r2 value of 0.96 (Fig. 8a). A quantile–quantile plot (Q–Q plot) is used to examine the model’s performance of reproducing the correct statistical distribution of STR, and the behavior in the tail region is of particular interest from risk perspectives. The Q–Q plot determines if two datasets come from populations with a common distribution; if the estimates fall on the dotted line, then they are likely from the same distribution. Figure 8 shows the Q–Q plots for the 0.2, 0.5, and 0.8 quantiles for simulated and observed storm total rainfall. The STR levels generated by MLTRM show a slightly heavier tail compared to the satellite observations (Fig. 8b).
(a) Scatterplot and (b) Q–Q plot of the storm total rainfall estimates derived from MLTRM and from satellite observations for North Atlantic TCs between 2000 and 2020. The plot shows the predictions and Q–Q analyses for the 0.2 (green), 0.5 (blue), and 0.8 (red) QRF quantile predictions plotted against the satellite observations.
Citation: Journal of Hydrometeorology 25, 1; 10.1175/JHM-D-23-0065.1
To investigate the ability of MLTRM in capturing the magnitude and spatial variation of TC rainfall, we compare rain-rate fields predicted by MLTRM to satellite observations. Figure 9 shows a comparison of MLTRM and SRO rain field maps for five randomly chosen time steps in the IBTrACS data (see the data availability statement for a comparison for all other time steps). We find that the MLTRM is generally well able to model observed rain-rate asymmetries within the distribution of predictions (Fig. 9). In particular, MLTRM is able to model rain-rate variability in the outer circulation with well-defined rainbands and outer rainfall asymmetries (Fig. 9). Additionally, MLTRM clearly models rain-rate variability in the inner circulation with predictions that include both symmetrical and highly asymmetrical inner core rain patterns (Fig. 9).
Comparison of rain rates predicted by MLTRM (shown in the first three columns) as compared to satellite observations (shown in the last column). The five events are randomly chosen with supplementary file 2 (see the data availability statement) showing the results for the other storms identified in the IBTrACS dataset between 2000 and 2020.
Citation: Journal of Hydrometeorology 25, 1; 10.1175/JHM-D-23-0065.1
Finally, we investigate the potential application of MLTRM in studying TC-generated flooding, where MLTRM can provide simulated rainfall time series to drive hydrologic modeling. Figure 10 shows event hourly STR time series for the MLTRM and SRO for five randomly chosen events (see data statement for a comparison for other events). MLTRM is generally well able to model the temporal evolution of the total rainfall within the distribution of predictions with high correlation coefficient values ranging from 0.4 to 0.9 and RMSE values between 0.1 and 0.5 × 106 mm.
Hourly time series of total rainfall derived from MLTRM and from satellite observations for six North Atlantic TCs.
Citation: Journal of Hydrometeorology 25, 1; 10.1175/JHM-D-23-0065.1
4. Discussion
In an attempt to better characterize uncertainty in TC precipitation modeling the potential of machine learning methods was explored as an alternative to canonical statistical or physics based models. In particular, quantile regression modeling approaches were employed as they have the virtue of allowing simulation of the full conditional distribution of a response variable. This approach was implemented to simulate the conditional distributions of the weights associated with the first five PCs derived from a set of high-resolution WRF precipitation field simulations. These five PCA shapes were able to capture approximately 45% of the rainfall variability in the WRF dataset. The PCA model can be used to predict various rainfall asymmetries, notably shapes consistent with rainbands, wavenumber asymmetries, and potentially extratropical transition.
Using a similar approach in the western North Pacific to model uncertainty in TC wind fields, Loridan et al. (2017) were able to explain over 66% of wind speed variability using three PC shapes. The considerable reduction in variability explained in our approach for rain rate compared to wind speed as in Loridan et al. (2017) may result from the multitude of complex processes involved in TC rainfall modeling as discussed in section 1.
Multiple studies have analyzed rain fields using Fourier-decomposed rainfall wavenumber asymmetries—with studies primarily focusing on wavenumber-1 asymmetries that have been attributed primarily to the motion of the storm through the environment and vertical wind shear (Lonfat et al. 2004; Chen et al. 2006; Pei and Jiang 2018). In these studies, the TC is rotated relative to the motion of the storm. Here we use PCA instead of Fourier decomposition to assess rainfall asymmetries as the PCA produces an estimated variance explained for each asymmetry. Additionally, we use PCA as we can use the PC weights to predict rain-rate maps in risk assessment and probabilistic forecasting. We rotate each TC relative to the Rmax instead of the bearing of the TC as we seek to model primarily the region of maximum rain rate that is often collocated with the Rmax and the TC rainbands that often originate at the Rmax (Houze 2010).
The purpose of this study was to develop a framework to model variability of mature TC’s rainfall, such that the model may be deployed in flood hazard risk assessments or probabilistic forecasting. Although our training data span a realistic range of TCs compared to historical climatology (Fig. 2), we emphasize that our results may differ with the addition of other events. Given the stochastic nature of these events, future work could examine a wider range of historical events or idealized events, providing a more comprehensive view of rainfall hazard. The deployment of MLTRM requires the input of angle of maximum wind to regrid the PCA output to a latitude–longitude grid. In this study following Loridan et al. (2015), we assume that in the Northern Hemisphere the angle of maximum wind is 90°. We acknowledge that this assumption is idealized; in further studies a separate machine learning model could be developed to predict the angle of maximum wind conditioned on TC characteristics and local monthly environmental variables.
While the machine learning structures are not directly derived from physical mechanisms governing precipitation fields, future work can make use of physics-informed machine learning to improve the predictive ability (Karniadakis et al. 2021). We also envision a joint physics–machine learning framework in which the machine learning model may be used to predict rainfall variability such as rainbands and extratropical transition that current physics based models are unable to model.
Climate change is expected to alter hurricane characteristics into the future, including their translation speeds, maximum wind speeds, and rain rates (Knutson et al. 2020). Multiple studies have suggested that TC rainfall rates are expected to exceed the increase in saturation specific humidity predicted by the Clausius–Clapeyron relation (∼7% °C−1; super CC) due in part to concurrent changes in maximum wind speed, which can also affect moisture convergence and resultant rainfall rates (Liu et al. 2019; Xi et al. 2023). These expected changes in rain rate with global temperature change may prove to be a challenge for predictions made by machine learning models, as models that are trained on one population (e.g., historical events) generally have poor accuracy on different temporal and spatial populations (e.g., future climates) (Chattopadhyay et al. 2020). To address this issue related to out of training data predictions, we include local environmental characteristics such as local surface temperature and CAPE and storm characteristics such as wind speed that may facilitate predictions of super CC rain-rate changes in future climates, and these characteristics may be obtained from climate projections or their downscaling (Emanuel et al. 2008). In further studies, the training set could be expanded to include training data from future climate simulations.
Finally, we emphasis that we have introduced an overwater model here that neglects the impact of surface roughness, terrain, and landfalling on TC precipitation. The inclusion of these components is the subject of future development. While we have focused on the North Atlantic basin, a global TC model is also the subject of future work.
5. Conclusions
In this study, we develop a probabilistic overwater machine learning–based TC rain model (MLTRM) based on a set of 26 historical events modeled in WRF and satellite data in the North Atlantic Basin. The model includes two components, a rain shape model and a rain-rate intensity model. The model allows for the easy generation of rainfall maps through repeated sampling of the modeled conditional distributions, and as such are able to produce probabilistic representations of rainfall shape and intensity.
We use a leave-one-out cross validation to evaluate the robustness of the model as compared to the WRF simulations. We find that the distribution of shapes and asymmetries are well captured by the model, notably the model predicts rain shape variability associated with five PCA shapes commensurate with TC rainbands, wavenumber asymmetries and possibly extratropical transition. The cross-validation exercise confirmed the ability of this modeling framework to reproduce major variability in the WRF simulations. Additionally, we compare the MLTRM estimates to historical satellite rainfall observations, with the predictions well able to capture storm total rainfall compared to satellite observations with a correlation coefficient of 0.91 and r2 value of 0.96.
The model framework developed in this study can accommodate fast predictions for the probabilistic rainfall estimates given limited storm characteristics and environmental data and thus it may also be used to predict flood area and extent, and compound flooding (Gori and Lin 2022). To broaden the developed model’s application to risk assessment over land, it is important to mention that our modeling is exclusively conducted over oceanic regions, and the model will need to be expanded to take into account the effects of surface roughness, terrain, and landfall on TC precipitation. Finally, our framework may also be able to predict probabilistic rainfall estimates for coastal communities and support stakeholders and communities through efficient risk assessment and emergency response management operations, in near term forecasting and adaptation planning.
Data availability statement.
Supplemental files can be found at https://zenodo.org/record/7813832#.ZDQyny9w30o. Code used in this analysis can be obtained at https://zenodo.org/record/8286523. You can access the ERA5 monthly data can be accessed from Hersbach et al. (2020). The original precipitation data sets used in this study are available for research purposes and can be obtained freely from Joseph Lockwood or Thomas Loridan.
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