Original Article Quarterly Journal of the Royal Meteorological Society Joshua
Dorrington PhD, IMK-TRO, KIT, Karlsruhe, 76133, Germany
<EMAIL_ADDRESS>Karlsruhe Insitute of Technology; German Research
Foundation (DFG), Grant/Award Number: FR4363/1-1; Transregional Collaborative
Research Centre, Project T2, Helmholtz Young Investigator Group ‘Sub- Seasonal
Predictability: Understanding the Role of Diabatic Outflow’ (SPREADOUT; Grant
VH-NG-1243)

# Domino: A new framework for the automated identification of weather event
precursors, demonstrated for European extreme rainfall.

Joshua Dorrington  Institute of Meteorology and Climate Research (IMK-TRO),
Department Troposphere Research, Karlsruhe Institute of Technology (KIT),
Karlsruhe, Germany  Christian Grams  Institute of Meteorology and Climate
Research (IMK-TRO), Department Troposphere Research, Karlsruhe Institute of
Technology (KIT), Karlsruhe, Germany  Federico Grazzini  Ludwig-Maximilians-
Universität, Meteorologisches Institut, München, Germany  ARPAE-SIMC, Regione
Emilia-Romagna, Bologna, Italy  Linus Magnusson  ECMWF, Shinfield Park,
Reading, RG2 9AX, United Kingdom  Frederic Vitart  ECMWF, Shinfield Park,
Reading, RG2 9AX, United Kingdom

###### Abstract

A growing number of studies have investigated the large-scale drivers and
upstream-precursors of extreme weather events, making it clear that the
earliest warning signs of extreme events can be remote in both time and space
from the impacted region. Integrating and leveraging our understanding of
dynamical precursors provides a new perspective on ensemble forecasting for
extreme events, focused on building story-lines of possible event evolution.
This then acts as a tool for raising awareness of the conditions conducive to
high-impact weather, and providing early warning of their possible
development. However, operational applications of this developing knowledge-
base is limited so far, perhaps partly for want of a clear framework for doing
so. Here, we present such a framework, supported by open software tools,
designed for identifying the large-scale precursors of categorical weather
events in an automated fashion, and for reducing them to scalar indices
suitable for statistical prediction, forecast interpretation, and model
validation. We demonstrate this framework by systematically analysing the
precursor circulations of daily precipitation extremes across 18 regional- to
national-scale European domains. We discuss the precursor rainfall dynamics
for three disparate regions, and show our findings are consistent with, and
extend, previous findings. We provide an estimation of the predictive utility
of these precursors across Europe based on a linear statistical model, and
show that large-scale precursors can usefully predict heavy rainfall between
two and six days ahead, depending on region and season. We further show how
for more continental-scale applications the regionally-specific precursors can
be synthesised into a minimal set of indices that drive heavy precipitation.
We then provide comments and guidance for generalisation and application of
our demonstrated approach to new variables, timescales and regions.

###### keywords:

_extreme precipitation_ , large-scale precursors, predictability

##  1 Introduction

In recent years, the conditioning of surface extreme weather on the large-
scale flow, both locally and upstream, has been explored with the hope of
obtaining dynamical insights that can identify windows of opportunity and
improved guidance for the forecasting of extreme events. High rainfall over
North Italy is driven by Rossby wave trains with north American origin [23],
while Atlantic wave-breaking favours Alpine heavy precipitation [35, 4], and
atmospheric regime variability modulates both alpine serial cyclone clustering
[5, 54] and European atmospheric river occurrence [42]. Localised temperature
and wind extremes are also often associated with hemispheric variability, as
revealed by recent studies on compound heatwaves [57], links between cold
spells and storm damage [38], and the well-documented link between sudden
stratospheric warmings and surface cold extremes, modulated by the
tropospheric flow [7].

Such large-scale extreme precursors are particularly interesting because they
hold the promise of extending forecast predictability. Even in state-of-the-
art forecast systems, skill in predicting extreme precipitation, for example,
rarely extends even a week ahead [16, 32] placing limits on the mitigation
strategies available to policy makers and industry stakeholders. However, as
forecast skill takes longer to decay at larger spatial scales [33], large-
scale ‘event-prone’ precursors – which may take the form of particular
tropospheric circulation patterns or anomalies in slow-varying earth system
modes – can be expected to be more predictable than the extreme events
themselves. As small-scale processes, long-range teleconnections, and
orography are imperfectly represented in models, there is no guarantee that a
model will always successfully convert the large-scale conditioning into a
surface extreme. However, if the conditioning of an extreme event on a large-
scale precursor is sufficiently large – that is,
$P(\text{event}|\text{precursor})/P(\text{event})>>1$ – then such precursors
could still be used to offer quantitative improvements in forecast skill by
using forecasts of precursors to statistically infer the surface extreme.

Complementarily, from a qualitative perspective a dynamical precursor
perspective raises awareness of the necessary conditions for an extreme event
rather than focusing on raw model output. This is useful for building forecast
story-lines, highlighting important dynamical steps during the unfolding – or
not – of an extreme even, a perspective which is already popular in the domain
of climate projections [52]. As these precursors often occur well before the
extreme itself, and may be tied to predictability barriers [17, 40], such an
approach can help explain ‘jumpiness’ in situations of high forecast
uncertainty, increasing end-user trust [48].

Despite these advantages, there has been relatively little operational use of
large-scale flow precursors thus far to improve operational forecasts. One
notable exception is the use of regime approaches [14], which have been shown
to boost precipitation forecast skill [37]. Here, the approach is typically
top-down; regimes are identified in large-scale fields such as geopotential
height, and the impacts on surface weather in each regime are diagnosed [18,
55]. Other large-scale modes such as the MJO, ENSO or stratospheric polar
vortex are also integrated into statistical modelling, especially at seasonal
lead times but are less commonly employed in the extended range. Bottom-up
regime approaches exist, as in Bloomfield et al. [8] which look for the
dominant flow regimes that explain the most variability in variables of
interest (in ibid, energy generation). Of course, there is no guarantee that
any pre-established dominant variability modes cleanly project upon a generic
extreme event. In fact, if such an event is comparatively rare, as an extreme
is by definition, then any predictability coming from dominant variability
modes must necessarily be highly diffuse and associated with only minor
modifications in occurrence probability. AghaKouchak et al. [3], in the
context of drought prediction, advocated for the bottom-up, event-centric
approach of identifying the precursors of specific event types, rather than
‘thinking platonically’, i.e. thinking in terms of idealised modes of
variability.

In this paper we lay the groundwork needed to make bottom-up event-centric
precursor analysis a routine technique in the toolbox of the operational
meteorologist, the applied scientist or the forecast end-user. In section 2 we
introduce our analysis framework for identifying useful precursors,
‘Domino’111For the curious, the name ‘Domino’ was inspired by the imagery of a
row of dominoes falling in causal sequence, which serves as a (admittedly
optimistically deterministic) metaphor for the sequence of meteorological
developments that increase the likelihood of a weather event of interest.,
supported by a corresponding Python implementation, a fully-documented api and
clear worked examples of different use-cases.

In the rest of the paper we apply this new framework to analyse the precursors
of daily rainfall extremes across Europe. Extreme weather events in Europe are
responsible for billions of euros in economic damage and the loss of hundreds
of lives every year [29, 11], with 40% of these deaths between 1950-2006 a
result of flash floods. Such short-timescale and spatially-localised rainfall
extremes are particularly challenging to forecast well, especially in regions
with complex topography. Our analysis serves as a demonstration of the
precursor approach, but also, by providing the first systematic analysis of
national-scale precipitation drivers, as an end in its own right.

In section 3 we discuss the data and event definitions used, the eighteen
spatial regions we choose, and our selection of Domino parameters. In section
4 we will discuss the precursor dynamics of three archetypal regions: North
Italy, West Iberia, and North Europe which cover Mediterranean, Atlantic and
continental climates respectively, and demonstrate consistency with prior
knowledge as well as note a number of more novel observations. In section 5,
we will analyse the large-scale precursor patterns in wind, geopotential
height, and vapour transport derived for all 18 regions, and discuss
commonalities and variations. We will estimate the predictive skill of these
precursors using logistic regression, and investigate regional and seasonal
differences. In section 6 we show how the regionally-specific precursor
indices we have computed can be synthesised for larger scale considerations
into a minimal set of European rainfall modes’ using partial least squares
regression, and we use these to characterise regional variations in rainfall
dynamics. Finally in section 7, we discuss our conclusions and provide
suggestions for the generalisation and extension of this work, and how the
framework can be best used for directly applied work.

##  2 Domino

Figure 1: A schematic diagram of the Domino workflow. Inputs are shown in
green boxes, key analysis steps in white diamonds, output datasets in blue
boxes, and different applications that could make use of output data are
indicated by red ovals. See accompanying main text for full description.
Graphics show visual examples of each data type.

The aim of the Domino framework is to automatically identify the predictive
precursors of categorical, meteorological events. Comprehensive documentation
and discussion of the full range of workflow parameters and customisation
options can be found in the Domino github repository
`https://github.com/joshdorrington/domino`

In section 3 we define concretely the parameters and analysis used in this
paper for our proof-of-concept application to European extreme rainfall, while
in this section we summarise the general purpose and motivation behind each
analysis step.

The framework, sketched schematically in figure 1, is based on two types of
input: ‘events’, defined as a categorical time-series indicating the class of
event at each time-step, and ‘variables’: time-series of candidate precursors
which may play a role in influencing the event and which may be time series of
scalars, vectors, fields, or higher-dimensional variables. Our overarching
goal is to provide a simple workflow that can replicate, semi-automatically,
previous analyses of extreme event precursors, and convert the results into a
form that can be used for event prediction. In this way we aim to
operationalise current academic insights, and streamline the process of
obtaining new insights.

1\. Compute lagged composites

Implemented via the LaggedAnalyser class, time-lagged composites of the input
variables are calculated with respect to each event category. Lag-dependent
background-states and/or seasonal cycles can be computed and subtracted.
Synthetic event time-series can be provided or generated automatically to
serve as the basis for bootstrap significance testing. As such compositing
approaches are a valuable analytical tool used in many studies of extreme
event precursors, long-range teleconnections and atmospheric dynamics, we
anticipate the clean, efficient implementation provided here will be of
interest to many geoscientists.

2\. Refine precursor patterns

Implemented via the PatternFilter class, a number of filters can be applied to
precursor composites of multi-dimensional variables, iteratively building a
Boolean mask which can be used to focus on features of interest. The
motivation is to replicate in an automated way the subjective assessment of
interesting features a scientist might make when looking at a composite field.
Provided functions include filters for large-amplitude, spatially extended and
statistically significant patterns.

3\. Generate precursor indices

Implemented via the IndexGenerator class, each precursor pattern can be
reduced to scalar time-series of ‘precursor activity’, by projecting anomaly
fields onto the masked pattern. Such indices are the automatically-generated,
event-centric equivalents of, for example, the well-known NAO, QBO or nino3.4
indices defined as spatial averages or differences of spatial averages over
particular regions, and which provide a valuable scalar summary of important
Earth-system variability. As well as computing precursor activity indices from
scratch, updating an index using new variable time series and precomputed
composites is supported, making it easy to calculate online updates of index
time series, or project indices into a different dataset, such as in model
data. Although not explored in this work, there is potential for using these
indices to validate the event-centric dynamics of models much in the same vein
as model assessments of ENSO [15] and the NAO [53].

4\. Evaluate index predictivity

Implemented via the PredictionTest class, this step provides an estimate of
the predictive power of scalar indices for a categorical event. Uni- or multi-
variate combinations of predictor indices can be specified, as can the
statistical model to use, the cross-validation strategy and the skill score of
interest. This can be used for feature selection, allowing for a subset of
most predictive indices to be taken forward for additional analysis, or as in
this paper, to compare the predictability of precursors in different regions
and seasons.

5\. Reduce index dimensionality

Implemented via the PLSR_Regression class, a set of scalar indices can be
reduced to a lower-dimensional set via partial-least-squares regression, a
statistical model which maximises the covariance between the reduced modes and
a set of target variables. In this way, we can reduce the dimensionality of
our scalar indices in an event-focused way which is well suited to prediction
problems. This auxilliary functionality is useful in specific cases where
precursors are desired which balance predictive power over different event
types or across regions, as discussed in section 6. The output PLSR modes can
then also be passed to PredictionTest, just as can any other scalar index.

The software implementation is built on top of the popular Python package
xarray [28], which provides support for meta-data rich, multi-dimensional
array manipulation. Domino makes heavy use of its DataArray and Dataset
classes to store and process data, which support conversion both to and from
Pandas Dataframes and iris Cubes, providing cross-compatibility with a range
of data pipelines. Implementations of logistic regression and PLSR are taken
from scikit-learn [43]. Via xarray, much of the workflow supports parallel and
lazy computation via Dask [49], allowing large datasets to be processed
quickly.

In this paper we focus on the case where the events have two classes, e.g.
case the occurrence or not of extreme rainfall, but all the algorithmic steps
of Domino equally apply to multi-class applications; terciles, deciles, events
of multiple type, etc. Here we only apply Domino to precursors which are 2D
continuous lat-lon fields, but arbitrary dimensioned continuous precursors can
be handled, as can categorical scalar precursors. In the latter case,
occurrence frequency for each category is automatically computed instead of a
mean composite.

##  3 Data and Methods

###  3.1 Data

Both events and precursors are identified in ERA5 reanalysis data [25] over
the 43-year period 1979-2021. Precipitation data was downloaded as 3-hourly
accumulations at 0.25 deg. resolution, and then accumulated to daily data.
Four candidate precursor fields were chosen to capture the large-scale
circulation: 500 hPa geopotential height (Z500), 850hPa zonal wind (U850),
300hPa meridional wind (V300), and the magnitude of the column integrated
vapour transport (IVTmag). These were downloaded as 12:00 instantaneous values
at 1 deg. resolution.

###  3.2 Region definition

Spatially-averaged daily rainfall was defined as the cosine-latitude weighted
averages of precipitation over each of the eighteen European regions shown in
figure 2. The boundaries for these regions were based on the HydroSHEDS
drainage-basin product [31], which avoids the mixing of hydrologically
distinct areas that would result from a simple box-averaging approach. Basins
were in some cases merged or split in a subjective fashion based on
consideration of meteorological factors such as the position of mountain
ranges and prevailing winds, or to merge very small regions.

The areas, chosen to demonstrate the Domino workflow, are not of equal area or
population and so are therefore not optimised for direct application. Rather,
they serve to collectively cover the majority of Europe in a way that respects
the underlying hydrometeorology, and the analysis of their precursors can then
inform further studies of particular refined regions.

For each season and region, a categorical heavy rainfall event time series was
computed. Events were defined as daily accumulated rainfall exceeding the
region’s seasonal 90th percentile. This relative definition ensures an equal
number of samples in all samples and regions, with the drawback that in some
cases such events will not always be high-impact. Supplementary figure 1 shows
the exact threshold for each case and region.

Figure 2: 18 regions used in this paper to define spatially-aggregated
precipitation extremes. Regions are based on catchment data taken from Lehner
and Grill [31], with modifications to account for orographic impacts on the
prevailing circulations.

###  3.3 Precursor computation

The specifics of how precursor patterns and activity indices were computed are
detailed here, and are accompanied by schematic figure 3. Each of the
precursor variables (Z500, IVTmag, U850, and V300) were restricted to the
extended north Atlantic region [20N-80N,140W-100E], and deseasonalised at a
gridpoint level, by calendar date: the mean value of each variable on the same
day of the year over the period 1979-2021 was smoothed with a 30-day rolling
mean, and subtracted from the daily fields.

For each season and region, lagged mean composites of the precursor variables
were computed at daily lags from 7 to 0 days before the corresponding heavy
rainfall event series. As heavy rainfall events were defined as exceedance of
the 90th percentile threshold in each case, this represents composites of
385-395 individual days.

In order to assess the statistical significance of precursor anomalies, a
bootstrap approach was used. To account for any autocorrelation in heavy
rainfall days, 400 synthetic event indices were generated by fitting a Markov
chain to the observed event series and resampling from it. Composites of each
variable were computed with respect to each synthetic event series, and
significance was defined at a gridpoint basis as a value below/above the 10th
lowest/highest bootstrap composite (2-sided p=0.05). A Holm-Bonferroni
correction was then applied to each variable and lag separately [27], to
reduce the false-positive rate. Composites at all time lags were compared to a
single set of bootstrap composites computed at lag 0 for reasons of
computational efficiency. This amounts to assuming an approximately stationary
climate over a 10-day period, which is reasonable.

Lagged composites are transformed into ‘precursor patterns’ by applying a
number of filters to the Boolean significance field, and then using it to mask
the composite. This procedure results in statistically robust spatial anomaly
patterns with large amplitudes, smooth edges and without isolated unmasked
points. Concretely, in addition to masking gridpoints with statistically
insignificant anomalies, gridpoints with anomalies <0.25x the gridpoint
climatological standard deviation were masked. Groups of unmasked gridpoints
that comprise a connected area of less than 250,000 km2 (i.e. equivalent to 20
1x1 deg equatorial gridpoints) were also masked. Finally, all unmasked
gridpoints were convolved with a 5-point lat-lon square: all gridpoints within
2 degrees of an unmasked gridpoint were unmasked.

These precursor patterns are then used to compute precursor activity indices.
A dot-product is computed between each deseasonalised variable time series and
each corresponding precursor pattern, and then spatially summed, using cosine-
latitude weighting. This produces a scalar index which is standardised to 0
mean and standard deviation 1.

Figure 3: A schematic indicating how lagged composites are transformed into
precursor patterns and then into precursor indices, demonstrated for SON
IVTmag precursors 2 days prior to North Italian extreme rainfall events. The
initial lagged composite is masked based on statistical significance, anomaly
amplitude, and spatial extent of unmasked regions, then padded with a 5-point
2D-convolution. A dot-product between this pattern and the daily IVTmag
anomaly field discriminates between days that closely resemble the precursor
flow and those that do not. When applied to all days in the IVTmag dataset, a
daily precursor activity index is produced. The three daily fields used as
examples are marked with red dots in the resulting precursor activity time
series.

###  3.4 Skill assessment

To characterise the potential predictability of heavy rainfall precursor
activity indices, we use a logistic regression approach. Stratifying by
season, region and precursor variable, in each case our target variable is the
corresponding ERA5 Boolean heavy rainfall event index, where we aim to predict
the probability of an event occurring. The schematic figure 4 illustrates how
this is done. For each time lag, season, region and large-scale variable we
have one precursor pattern (fig 4a). Each index is offset in time according to
its lag, i.e. a 4-day precursor index calculated from data for 1st Jan 2000
will be used to predict for 5th Jan 2000. We train a model for each lag $l$
between 7 and 0 by using all precursor indices that have a lead time $\leq l$:
thus the 7-day model is a univariate model using only the lag 7 precursor
index, while a 1-day forecast uses all precursors except for the lag 0
precursor. Each logistic regression model was cross-validated: each year of
data was sequentially excluded, and the model was fitted on the remaining 42
years, then used to predict the excluded year. The probabilistic ROC Area
Under the Curve (ROC AUC) skill score [24] was computed for each cross-
validation, and the mean value is reported. This provides a much more
realistic estimate of the potential predictability achievable by a perfect
forecast than simply fitting a model to the entire event time series.

Figure 4: A schematic illustrating how the predictive power of precursor
activity indices is estimated in a reanalysis framework. For each time lag,
region, season and variable a precursor pattern is defined (a). Each pattern
corresponds to a time-varying activity index which captures to what degree the
current flow resembles the pattern. These indices co-vary with rainfall to a
greater or lesser extent (b)). We fit a logistic regression model with
precursor indices for all lags >=$l$ for a given season, region and variable,
in order to predict categorical heavy rainfall occurrence, and quantify skill
using the ROC AUC score.

###  3.5 Partial least squares regression

Dimensionality reduction is a common task in atmospheric data analysis, with
principal component analysis (PCA) being the most popularly used approach. PCA
aims to find the N-dimensional linear subspace of an initially M-dimensional
dataset which captures the greatest degree of variance. Partial least squares
regression (PLSR) is similar, in that it aims to reduce dimensionality while
maximising explained variance, but now finding the N-dimensional linear
subspace of an M-dimensional dataset $X$ which maximises the cross-covariance
with a second L-dimensional dataset $Y$. That is, it estimates the low-
dimensional projection of $X$ which is best at predicting the various target
variables of $Y$. For the mathematical formalism see Abdi [2] and for
algorithmic implementation see skl [1]. It is closely related to canonical
correlation analysis, which has been previously applied to meteorology in,
e.g., Maldonado et al. [34]. For our example, we demonstrate how PLSR can be
applied to SON Z500 precursor patterns for different regions and time lags, to
obtain a minimal set of modes that balance week-ahead heavy rainfall
predictability across Europe. This is described in more detail in section 6.

##  4 Dynamics of rainfall precursors

Having discussed our approach and methodology in general terms, we now apply
precursor analysis to European daily heavy rainfall extremes. We begin, in
this section, by analysing the multivariate precursor patterns for North
Italy, West Iberia and North Europe. Figure 5 shows precursor patterns for the
three regions, 6, 4, 2 and 0 days before heavy rainfall events and across
seasons. For each of our 4 large-scale variables we have a unique precursor
pattern for each combination of time lag, season and region. Z500 precursors
indicate the anomalous trough and ridge structure associated with events, V300
indicate the upper-level wave dynamics, U850 the near surface jet, and IVTmag
the role of anomalous moisture transport. Collectively they therefore capture
most of the important mid-latitude driving dynamics and allow a holistic
understanding of each event type.

Figure 5: Precursor patterns 6, 4, 2 and 0 days before heavy rainfall events
in West Iberia (a,), North Italy (b) and North Europe (c), as a function of
season and computed from ERA5 reanalysis data. Red-to-green shading indicates
Z500 anomalies as shown by the colorbar. Grey contours show full field Z500
for levels with 1e4 spacing between 5.2e5 and 5.7e5 $m^{2}/s^{2}$. Gold
contours show V300 anomalies at 4$m/s$ intervals ranging from -16 to 16 $m/s$,
and excluding the 0 contour. Blue-purple shading shows positive IVT magnitude
anomalies at the 50, 80, 110,and 140 kg$m/s$ levels. Arrows indicate anomalies
of the U850 wind, and should be interpreted qualitatively.

###  4.1 West Iberia

The ‘West Iberia’ region consists of Portugal and coastal Northern Spain, as
well as the inland basins of the Douro and Tagus rivers. Precursors extend
further back in time in this region than any other considered, as we will
discuss in section 5. In this mountainous domain, the majority of rainfall
occurs in Autumn and Winter [39], and while Summer rainfall is less frequent
and generally dominated by meso-scale processes, the majority of JJA coastal
extreme rainfall events are driven by atmospheric rivers [45]. An important
flood driver in MAM which we do not consider here, is snow-melt. In general,
heavy daily precipitation in this region is known to be dominated by the
latitudinal shifting of the jet stream which, when in its southerly state, can
guide cyclones and atmospheric rivers to this area [46], and is associated
with the negative phase of the NAO [12, 50] and cut-off lows to the West [13].

This pre-existing knowledge is reaffirmed by the large-scale West Iberian
precursor patterns shown in Figure 5a). Robust precursors extend back 6 days
in all seasons, demonstrating a substantial conditioning of heavy rain events
here on the upstream Atlantic state. In DJF, precursor in the Z500 and U850
fields indicate a southerly jet deflection, accompanied by a negative-NAO-like
geopotential height anomaly, is typical 6 days before a rainfall event. Both
wind and geopotential anomalies strengthen through to lag 0. This extended
persistence of the precursor flow is consistent with the persistent, trimodal
regime dynamics of the wintertime Atlantic jet [58, 41]. From lag 4 onwards a
clear zonally oriented IVT precursor emerges to the west of Iberia, indicating
increased atmospheric river frequency, and the need for not only a displaced
jet but also anomalous moisture transport to induce an extreme. While the U850
anomaly is initially confined to the east Atlantic, a secondary trough forms
over the south-east US from lag 6, which induces its own local U850 anomaly
and a diagonally oriented IVT precursor that suggests a secondary source of
moisture flowing from the west to the east Atlantic. While the early
precursors are predominately zonal, large V300 anomalies emerging from lag 4
onward indicate the formation of an upper-level trough over the domain and a
possible role for wave breaking dynamics for triggering local trough formation
as discussed in [51]. In SON, dynamics are similar, but now less zonally
symmetric with clearer upper-level wave features, and a more confined trough
and associated deep ridge over the mid-Atlantic. Here the role of the
secondary cyclone is comparatively more pronounced, perhaps a requirement
given the overall weaker moisture transport associated with a weaker low-level
jet. In MAM and JJA there are less pronounced large-scale precursors, but a
substantial amount of non-local conditioning can still be seen including low
pressure over the US, high pressure over Greenland and East Atlantic jet
anomalies as well as the high IVT anomalies indicative of increased
atmospheric river frequency. The lag 0 Z500 precursor indicates cut-off lows
are dominant in Summer, while troughs are more common in Autumn, in agreement
with Pino et al. [44], which analysed the synoptics of the 24 most severe
Iberian floods in the past 150 years. The upstream trough over the Panhandle,
visible in U850 and Z500 precursors for SON and DJF 2-4 days prior to heavy
precipitation, is also associated with a known dynamical pathway uncovered
only very recently: Leeding et al. [30] have showed DJF cold air outbreaks
over this same US region are associated with increased Iberian precipitation
0-5 days later, and that this trough pattern is associated with further
southward jet deflection. Our results suggest there may also be a role for
this teleconnection in SON, something not explored in ibid.

###  4.2 North Italy

The North Italian domain suffers from extreme flash flooding events, and is
well known to possess substantial conditioning on the large-scale flow and
Atlantic Rossby wave activity [23, 20]. Highly mountainous, and shielded from
direct westerly flows by the French alps, southerly winds are needed to bring
large moisture transport into the region, necessitating some level of upstream
wave activity. The precursor patterns (Figure 5b)) reveal a sharp distinction
between the non-local conditioning of SON and DJF events, which stretch back
to lag 6, and MAM and JJA events, which only emerge at lag 2. Despite this,
precursor patterns for the non-winter seasons are qualitatively rather
similar, all showing zonally oriented wave trains with large-scale moisture
transport over Southern Europe. For SON, a pre-existing weak ridge over the
west Atlantic interferes constructively with the upstream Rossby wave, which
becomes clear by lag 4, and is centred at $\sim$45N. From here the wave
amplifies and shows slow eastward movement. High IVT at lag 2 in the west
Atlantic may indicate the increased occurrence of extratropical cyclones
recurving north-east, which can amplify wave growth and induce Mediterranean
cyclone development [19, 47]. The trough to the west of Italy is fully
developed by lag 2, with associated southern European IVT anomaly. Even in
JJA, when convective rainfall would be expected to dominant, high IVT over
North Italy is seen, likely as a result of unstable moist air from the
Mediterranean advecting and rising over the steep orography, as described by
[22]. In DJF, the picture is quite different, with precursors showing a
Greenland blocking/NAO- precursor pattern emerging, interacting with an
upstream wave train, and leading to a more tripolar high latitude circulation
anomaly up until lag 2. High geopotential heights around the European trough
indicate a cutoff structure, and there is a sharp decline in the amplitude of
the Greenland blocking anomaly on the day of the North Italian extreme
rainfall. This break down of the blocking and associated regime transition
implies elevated levels of wave breaking over western Europe [36], which may
aid in deepening and localising the trough, and so amplifying southerly
moisture transport. For the North Italian case, the Z500, V300 and IVTmag
precursor indices have been shown in Grazzini et al. [21] to increase the
predictive power of a hybrid dynamical-statistical prediction model,
indicating a capability to not only reproduce the known dynamics, but to
transform those into a usable source of skill.

###  4.3 North Europe

The ‘North Europe’ domain consists of the catchments of the Ems and Weser
rivers, as well as the northern portion of the Rhine and Meuse catchments,
including large parts of Germany, eastern Belgium and the northern
Netherlands. In contrast to the other two domains considered in this section,
this domain which excludes the German alps and the Ore mountains to the south
and east respectively, features relatively flat terrain and exhibits a more
continental climate. Although the heavy rainfall threshold is comparable in
all seasons ( 6mm/day), there are notable seasonal variations in the precursor
patterns (figure 5)c). Without strong orographic constraints or direct coastal
exposure to Atlantic variability, significant precursors only begin to emerge
4 days before rainfall events during DJF, and only 2 days before in JJA and
MAM. In DJF, an initial sub-tropical ridge off the coast of Morocco develops
into a meridional dipole, with low geopotential height over the North sea and
a intensified central low-level jet 4 days prior to heavy rainfall. By lag 2
this anomaly strengthens, with high IVT anomalies coincident with the jet,
indicating an influx of Atlantic moisture into the region. At the same time,
upper-level wave activity becomes prominent, with some possible indications of
upstream forcing from a weak wave-like geopotential anomaly over the US. By
the day of the event itself, the V300 field shows a quadrapole structure
supporting strong moisture convergence into the region. The low geopotential
heights in the precursor pattern are more diffuse than for the other regions
considered. As the domain is more isotropic, the exact placement of weather
features over the domain is less stronglyy constrained, resulting in a greater
‘blurring’ of the precursor pattern. Indeed, cyclones impacting this region
are known to possess many disparate paths of evolution [26]. A qualitatively
similar, but less robust series of precursors is visible for MAM and SON, with
less large-scale conditioning.

Surprisingly, JJA rainfall events have quite extended and remote precursor
patterns with a similar North American low appearing 4 days prior as was seen
for SON West Iberian rainfall events. This is followed by the formation of a
localised low over the UK, displaced South compared to other seasons with an
accompanying upper-level wave visible at 2 days prior, possibly as a result of
downstream development. There is no accompanying southerly high geopotential
height as in other seasons indicating a less zonal flow and a more localised
cut-off. The high pressure precursor over Italy at 1 day prior suggests a
Mediterranean source of moisture is relevant, and capturing the footprint of
the rare but impactful ‘Vb’ cyclones [26].

We have shown for three disparate European regions that our automated
precursor detection approach is able to reproduce key insights from previous
analyses of regional extreme precursors, and via its systematic application we
have been able to discuss interseasonal variability fairly comprehensively,
and in a way that enables intercomparison between regions. The purported value
of this approach is that by conversion of precursor patterns into precursor
activity indices we will be able to likewise convert these dynamical insights
into improved event predictability.

In the next section, we will validate this claim, zooming out from detailed
consideration of a few case study regions to characterise the predictive power
of precursor indices for all 18 European regions. Equivalent plots to figure 5
for all regions can be found in the supplementary material.

##  5 European heavy rainfall precursors and prediction

Figure 6: a) Predictive skill of precursor indices, as measured by cross-
validated ROC AUC score averaged across seasons for each of 500hPa
geopotential height (Z500), 850hPa zonal wind (U850), 300 hPa meridional wind
(V300) and the magnitude of column-integrated vapour transport (IVTmag), with
the skill for each region shown by a black line and the mean across both
seasons and regions shown in red. b) ROC AUC now averaged across regions and
conditioned on season. b) ROC AUC averaged across seasons

In this section we turn our attention to the precursor activity indices that
correspond to each precursor pattern, and show that they have operational
utility; that is, the potential to improve the predictability of heavy
rainfall events. Without analysing forecast data, which will be the subject of
future work, we estimate a lower-bound on predictive power by assessing the
skill of logistic regression models fitted on reanalysis precursor indices for
the prediction of heavy rainfall probability in each region, as discussed in
section 3.1. Heuristically we consider a ROC AUC score >0.6 to indicate some
useful amount of predictive power in a precursor.

By using a multivariate or nonlinear model, it is likely considerably higher
skill could be reached, while in a hybrid dynamical-statistical prediction
framework the predictive value will be closely tied to the forecast model’s
ability to identify precursors at extended lead times, as will be discussed in
further detail in section 7. In fact, for the specific case of North Italy, a
hybrid dynamical/machine-learning model leveraging these non-local precursors
has already been found to boost skill out to day 10 [21].

For now, we proceed by analysing the univariate logistic-regression based ROC
AUC of precursor indices in the European regions shown in figure 2. Figure 6a)
shows the precursor skill averaged over season (for all seasons for which a
precursor pattern is defined), and showing each region separately. As previous
studies on extreme event precursors have focused on targeted regions and are
often directed by specialist intuition that precursors should exist, it is by
no means obvious that extremes over a generic region will or should have
predictive long-range precursors. Through our systematic multi-region
analysis, we are now able to comment on this point. There is substantial
inter-regional spread in precursor skill; approximately double that of the
seasonal spread. Differences are largest at short lead times, with precursor
skill between regions converging at lags 3-5, and generally falling below 0.6
during this period.

It is interesting to consider the worst case scenarios, that is the regions
with the lowest skill for each lead time and variable. The regions with the
least skilful precursors nevertheless show ROC AUC above 0.7 for day 0
precursors, and maintain non-negligible skill for Z500, U850 and V300 out to
lag 2. These least predictable regions are in all cases one of CZ and
NE_Europe (see SI figure 2 for a more detailed presentation), both of which
are isolated from the upstream Atlantic flow. Many regions show much greater
precursor skill and in the best cases there are indications that even lag 5
precursors have useful levels of skill.

Summarising another way, figure 6b) shows the ROC AUC for precursors based on
each large-scale predictor variable, conditional on season and averaged across
the 18 regions. For all variables, precursor skill is highest in DJF,
generally followed by SON, with MAM and JJA showing similar, lower skill, with
a total inter-seasonal spread of approximately 1 day of predictability. This
corresponds to the known increased importance of large-scale forcing in DJF,
and the increased role of convective processes in JJA. This skill difference
may then be in part derive from choice of precursor variables, which
emphasises the large-scale flow. While early experiments including CAPE and
SST fields did not reveal skilful precursor patterns, it is possible a more
detailed consideration of such thermodynamic variables might reveal better
summertime precursors.

IVTmag precursors show the highest skill at short timescales but decay
rapidly, with ROC AUC dropping below 0.6 at lags 3 and 4 for non-winter and
wintertime precursors respectively. Short-lag Z500 precursors are
comparatively less skilful but maintain predictive power out to lag 5 on
average, which is consistent with the general large-scale wave-like precursors
often seen for Z500, rather than the more isolated streaks of moisture flux
seen in IVTmag. V300 and U850 precursors fall in between these two extremes,
although U850 is notably more skilful in DJF, in keeping with the dominant
role of DJF Atlantic trimodal jet variability [58].

For simplicity, we are working within a categorical event context here, both
to identify our precursor indices, and to test their predictive skill. However
it is interesting to consider whether the precursor indices are able to assess
the amplitude of an extreme event. This is primarily left to future work but
in supplementary figure 3 we show the value of precursor indices for the most
extreme daily rainfall total in each region and season, according to ERA5.
There are several notable cases where these extremes are associated with index
values – usually for IVTmag – >6 standard deviations above the norm,
suggesting that this is indeed the case.

Figure 7: Z500 precursor ROC AUC for each spatial region, as a function of lag
and season.

The role of spatial variability in shaping precursor skill can be seen more
clearly in figure 7 for Z500 precursors. Equivalent skill maps for the other
large-scale variables can be found in the supplementary data. In general, the
Z500 precursor patterns considered represent either upstream wavetrains or
blocking events at long lead times, and localised troughs at shorter lead
times. We see that coastal regions show the longest range precursors, at lags
of 6, both on the Atlantic coast, as for Iberian and UK regions, but also
along the Eastern Mediterranean, where Italy and the Adriatic coast show early
precursors especially in DJF. Most regions start to show precursors, often
skilful ones, from day 4 onwards, often associated with the first indications
of local trough formation. More mountainous domains tend to show higher
precursor skill, a consequence of the greater constraints placed on the large-
scale flow to allow substantial vapour transport into the domain. There is
certainly no simple overall relation between location and precursor skill
however: even neighbouring regions can have very different precursor skill and
different seasonal variability. This motivates exactly the sort of spatially
differentiated approach we have taken here.

##  6 Synthesising precursors across regions

In the previous sections we have demonstrated the ability of automated
precursor detection to reveal the dynamical drivers, and aid the prediction,
of extreme events across a large number of European domains. For many
applications such a regional perspective is ideal, but there are also
applications for which a broader perspective is useful. For example,
operational centres responsible for an entire continent (such as ECWMF or NWS)
might wish to monitor the risk of extreme events in geographically and
administratively distinct regions of their domain, but without tracking
individual precursors for all regions. Entirely equivalently, a national
forecaster whose domain features great climactic diversity may wish to apply
the precursor approach outlined above to sub-regions or individual stations,
but then also to synthesise extreme event risk holistically. To address this
use-case we now demonstrate how precursors for different regions can be
synthesised down to a minimal set of indices that balance predictability in
different regions, using PLSR regression.

Our task is simplified in many cases by the fact that precursor patterns for
extremes in neighbouring regions can often be very similar. This is
demonstrated for lag 2 Z500 precursors in figure 8, which is representative of
many precursor patterns, and which suggest that it should be possible to
reduce the dimensionality of an initially large number of precursor activity
indices. By doing so, we aim to obtain a few general modes relevant for many
regions. Concretely, in our case we see that seasonal differences for
particular regions are often very pronounced, while qualitative similarities
between regional precursor patterns for a given season are easy to see in
figure 8: the positive NAO-like pattern characteristic of DJF precursors in
north-west domains and the negative NAO-like patterns in Mediterranean
domains, the JJA south-easterly wave visible in a large number of regions, and
the more zonally propagating wave that dominates SON central and southern
Europe precursors with only slight phase differences. Further, as can be seen
in figure 5, the precursor patterns for sequential time lags can also be very
similar.

Figure 8: Lag 2 Z500 anomaly precursor patterns for each region and season.
Clear similarities can be seen between many neighbouring regions. Figure 9:
Z500 anomaly patterns corresponding to each PLSR mode of SON Z500 precursors
across lags 5 to 0. b) The mean composite of each PLSR mode relative to
occurrence of a SON heavy rainfall event in each region.

Without carrying out an exhaustive synthesis for all variables and seasons, we
demonstrate how a minimal set of precursor indices can be produced for Z500
SON precursors, with a method that can be easily applied to any other season
or variable of interest. Z500 precursors in SON generally show high predictive
power at long lags, while large-scale SON dynamics are not as comprehensively
studied as those of DJF, making this a natural combination to trial our
approach on.

We apply partial-least-squares regression (PLSR) as discussed in section 3,
searching for the $n$ orthogonal linear combinations (PLSR modes) of our
predictor indices that explain the most variance in our event variables. As
our predictors, we take SON Z500 precursor indices spanning from lags 5 to 0
for each of the 18 regions (108 variables total, reduced to 88 after
discarding blank precursor patterns). For the target variables we use the
heavy rainfall event index for each of the 18 regions, but as we seek modes
that capture precursors at different lead times, i.e. we want to include lag-
covariance, we duplicate each event index and offset it in time between 0 and
5 days, producing a corresponding 108-dimensional event index as our target
variable.

Figure 9a) shows Z500 anomaly patterns corresponding to the first eight PLSR
modes. As for empirical orthogonal functions (EOFs), the sign of the anomaly
patterns is arbitrary, each representing a symmetric mode of variability.
Unlike EOFs of Z500 however, which tend to be dominated by large-scale dipole
patterns, the resulting PLSR modes clearly project onto small scale flow
features that are important for extreme European precipitation events. Larger
scale trough/ridge patterns are visible in PLSR modes 1 and 3, zonal wave
trains in modes 2, 4 and 5, and cutoff lows/highs in modes 7 and 8.

In exchange for reducing 88 initial precursors to 8 modes, the interpretation
of the PLSR modes’ impact on rainfall becomes more complex: positive values of
any given mode will increase heavy rainfall occurrence in some regions and at
certain lead times over the following five days, and decrease it in others.
Furthermore, as the PLSR indices are uncorrelated to eachother by
construction, the covariability of the indices prior to an event carries
additional information. This can be seen in figure 9b) which shows the average
value of each PLSR index in the days preceding and following SON heavy
rainfall events in each region. For North Italy as an example, pronounced high
or low values of almost all modes occur in the lead up to a heavy rainfall
event: high modes 1 and 6 combining to approximate the lag 4 wavetrain
precursor seen in figure 5b), low values of mode 2 indicating the movement of
the wave eastward by lag 2, and strong amplification of mode 4 as the wave
reaches its final position just to the west of Italy.

The decrease of straightforward interpretability that comes from PLSR
reduction perhaps makes them most suitable for use in statistical prediction
models rather than manual operational monitoring. Indeed we find that the
predictive power of the PLSR indices compares favourably to the predictive
power of the regionally-specific precursor indices considered previously.
Figure 10a) shows the ROC AUC of statistical predictions of SON regional heavy
rainfall using the same cross-validated logistic regression approach as in
section 3.4 but now using different numbers of PLSR indices as predictive
variables. The difference between the PLSR-based skill and the skill found for
SON Z500 precursors for each region is shown in figure 10b). In general we
find that the regional indices are more skilful than predictions based on <5
PLSR indices, with little improvement seen from using more than 8 PLSR
indices. This is particularly the case when looking at lag 0 patterns, where
the trough driving rainfall in each region is most localised and so least well
approximated by general modes. However, overall we see that the degrees of
freedom in our precursors can be reduced by a factor of 10, without reducing
statistical skill. We therefore suggest PLSR reduction may be a useful post-
processing step for multi-domain applications to simplify workflows and for
deep learning approaches.

Figure 10: ROC AUC skill score for SON Z500 PLSR modes used to predict SON
heavy rainfall via logistic regression (a) and the difference between PLSR
skill and the corresponding SON Z500 precursor activity index skill (b) in
each region as a function of region and number of modes used. In b), red
values indicate regional indices are more predictive, while blue values
indicate the PLSR modes are more predictive and lag-region combinations with
no defined Z500 SON precursor pattern are greyed out.

##  7 Discussion and conclusion

The ‘quiet revolution’ of steadily improving weather forecast skill has been
achieved through the gradual synthesis and operational application of improved
meteorological understanding [6]. In this paper we have presented a novel
framework, termed ‘Domino’, for identifying the large-scale precursor patterns
of extreme weather events, and reducing those to data-efficient, predictive
scalar activity indices. In doing so we provide a starting point for the
operationalisation of research into the conditioning of extreme events on
large-scale dynamics.

###  7.1 Summary of results

We have tested our ‘Domino’ precursor identification framework by analysing
regional daily heavy precipitation across western and central Europe, and
across seasons. Even estimating predictive skill using only a simple
univariate regression framework, we have shown that in all cases at least some
degree of usable predictability (ROC AUC>0.6) is present in large-scale
precursors based on geopotential height, wind fields and vapour transport
patterns. This conditioning is in general lowest in JJA when meso-scale
dynamics are more dominant, and in flatter, more continental domains. Even in
these cases, predictive precursors are seen three days ahead, while in DJF,
where conditioning on the large-scales is strongest, some domains indicate
borderline-useable skill 6 days ahead. We have demonstrated that the precursor
patterns identified are consistent with the previously known dynamical
precursors for three diverse regions, and by virtue of our systematic
approach, are able to comprehensively characterise them. Finally, we have
demonstrated that many precursor patterns are similar between different
domains; that is to say, the same large-scale dynamics favour rainfall in
multiple regions. We show how this can be used to simplify the analytical
framework without giving up regional specificity by synthesising the precursor
indices for different regions together into a lower-dimensional set of
predictive modes, using partial least squares regression. For the specific
case of SON Z500 precursors, we show a 10x reduction in dimensionality can be
achieved with only minor decrease in predictive skill and a more complex
physical interpretation, a trade-off that may be useful for certain
applications. This synthesis is enabled by the use of precursor patterns to
generate scalar precursor indices, and indicates the value of this novel
approach.

###  7.2 Caveats

Naturally, there are limitations of the Domino framework which must be
discussed.

Most obviously, as a bottom-up data-driven method the precursor patterns will
be sensitive to changes in the event time series, with longer time series
providing more robust results. The convolution filter applied during pattern
computation counteracts this to an extent, by smoothing the pattern mask. In
this paper we have used O(300) events for each region, which was found in
early tests to provide qualitatively identical results for different data
subsamples.

Secondly, as the framework is based on lagged composites of potentially
predictive variables over all events, it is not possible to distinguish
distinct dynamical pathways leading to the same event; instead the precursor
pattern will contain a superposition of both. In the pathological case where
two precursor pathways have opposite circulations then they would cancel,
producing no significant composite. This can be mitigated by subsetting the
extreme events if such pathways are suspected a priori (e.g. by splitting by
season as done here), or by choice of variables (e.g. wave envelope amplitude
vs V300 to prevent phase cancellation). Another option is to compare the
variance of the composite to the climatological value, an analysis supported
by Domino, but which has not been included in this work for simplicity. A
related point is the implicit stationarity assumption at play. Non-stationary
dynamics can essentially be understood as multiple pathways occurring
preferentially during different time periods. Applying Domino to rolling time
windows of the event time series could reveal such underlying non-
stationarity.

Finally, the motivating hypothesis of our approach is that the predictive
precursors we identify have potential operational use. Considering the time-
scale of our precursors (i.e. days 2-6 ahead) and the quality of modern NWP
systems, it is unlikely a purely statistical prediction based on precursors
will outperform dynamical forecasts in terms. Instead quantitative value may
come from hybrid forecasting; using the dynamical model to predict the
precursor patterns at extended range lead times, and then infer rainfall
probability even further ahead from the precursors. For this to be relevant,
it requires that at some lead time the dynamical model has skill at predicting
the large-scale precursors, but no longer has skill at small-area rainfall. We
consider this highly plausible, and Grazzini et al. [21] has recently shown
the assumption to hold for the North Italy region, but showing this is
generally the case is an important question left for future work.

In addition, the operational analysis of precursors has narrative value for
forecasters, helping to understand the range of possibilities and developing
dynamical event precursors in a forecast, and so in general boosting their
awareness of plausible weather beyond the limits of formalised skill. In a
sense, analysing the unfurling preconditions to extreme events can serve to
precondition the forecaster, allowing them to more quickly interpret and act
on probable extreme events as they emerge at shorter lead times.

###  7.3 Extension

In this section, we present ideas and methodological advice for extensions and
generalisations of the work done in this paper to different events, regions
and timescales.

In this paper we have aimed to provide a relatively straightforward proof-of-
concept application, and there are multiple refinements that could be made for
real-world flood-prediction applications. Firstly, rain gauge data or a local
high-resolution gridded rainfall dataset could be easily used in place of the
ERA5 precipitation data used to define events here. Spatial averaging can be
avoided entirely to provide extra information on local extremes: precursor
indices could be computed for each rain gauge or grid-point independently, and
then synthesised via the PLSR approach described above. For a consideration of
true flood risk instead of simply high precipitation risk hydrological
discharge data be used to define events. In this case, including land-surface
precursor variables such as soil saturation and snow accumulation are likely
to be of value.

Extending beyond the logistic regression used here, the scalar precursor
indices are well-suited for input into more advanced statistical and machine-
learning models, already providing a low-dimensional summary of the flow.
While raw data sometimes proves just as effective as physically refined data
for deep-learning applications, at the least, precursor analysis can assist
variable selection by highlighting the nonlocal regions and timescales that
are relevant for a particular event. In this case, it may be useful to lower
the predictive threshold used to define a ’useful’ precursor, in order to
include more marginal or conditional sources of skill.

Relatedly, the predictability of events we report here should not be
understood as inherent, but also as a result of the precursor variables we
have chosen. For summer rainfall, predictive thermodynamic precursors may
exist for some regions, based on variables such as CAPE, static stability, and
T2m, although the predictability of such processes is inherently much more
limited than the large-scale dynamics we focus on here.

Generalisation to different regions is trivial, simply requiring different
event data. However relevant precursor variables will of course be quite
different in tropical or sub-tropical regions where the thermodynamic
variables mentioned above may contribute more significantly, and in strongly
continental climates, where land-surface variable are likely more relevant. As
a coastal, mid-latitude region, the variables considered here may extend most
straightforwardly to the West coast US, perhaps also with the addition of
Pacific SSTs.

It is similarly easy to move beyond rainfall extremes; wind extremes or
extreme waves are both impactful on daily timescales, and could be treated in
much the same way as rainfall. For longer timescale events such as heatwaves,
or even relevant non-extremes such as predicting monthly rainfall terciles,
the main methodological differences come as a result of reduced sample size.
When computing the pattern mask, a reduced amplitude mask and increased
convolution filter may serve to increase sensitivity to weak sources of S2S
predictability while helping to maintain robust patterns. Variables such as
sea-ice extent and SSTs, and stratospheric wind are natural candidates for
such a study.

Finally, beyond prediction, automated precursor analysis has value both for
the fundamental study of dynamics and for model validation purposes. Just as
circulation regimes are used to explore the dynamics of surface
teleconnections [9], sources of model bias [56] and to judge the performance
of climate models [10], so event-centric precursor patterns and indices could
be used to summarise the representation of different dynamical processes in
models. Vice-versa, events could be defined in terms of dynamical features –
such as sudden stratospheric warmings, regime transitions or wave breaking
events – and their precursor and ‘postcursor’ indices can be used to help
understand the variability between events by, for example, clustering in the
space of indices.

In our own continuing work, we aim to demonstrate how precursors can be used
to summarise and digest the behaviour of large forecast ensembles, and to
develop an operational early-warning system for extreme events.

## scripts and datasets

ERA5 data used in this study is freely available from the Copernicus Data
Store. Software tools required to duplicate or extend this work are available
as part of the Domino package: `https://github.com/joshdorrington/domino`. A
compressed folder of further supplementary figures is available at
`https://doi.org/10.5281/zenodo.8087482`

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