# Using Explainable AI and Transfer Learning to understand and predict the
maintenance of Atlantic blocking with limited observational data

###### Abstract

Blocking events are an important cause of extreme weather, especially long-
lasting blocking events that trap weather systems in place. The duration of
blocking events is, however, underestimated in climate models. Explainable
Artificial Intelligence are a class of data analysis methods that can help
identify physical causes of prolonged blocking events and diagnose model
deficiencies. We demonstrate this approach on an idealized quasigeostrophic
model developed by Marshall and Molteni (1993). We train a convolutional
neural network (CNN), and subsequently, build a sparse predictive model for
the persistence of Atlantic blocking, conditioned on an initial high-pressure
anomaly. Shapley Additive ExPlanation (SHAP) analysis reveals that high-
pressure anomalies in the American Southeast and North Atlantic, separated by
a trough over Atlantic Canada, contribute significantly to prediction of
sustained blocking events in the Atlantic region. This agrees with previous
work that identified precursors in the same regions via wave train analysis.
When we apply the same CNN to blockings in the ERA5 atmospheric reanalysis,
there is insufficient data to accurately predict persistent blocks. We
partially overcome this limitation by pre-training the CNN on the plentiful
data of the Marshall-Molteni model, and then using Transfer Learning to
achieve better predictions than direct training. SHAP analysis before and
after transfer learning allows a comparison between the predictive features in
the reanalysis and the quasigeostrophic model, quantifying dynamical biases in
the idealized model. This work demonstrates the potential for machine learning
methods to extract meaningful precursors of extreme weather events and achieve
better prediction using limited observational data.

JGR: Machine Learning and Computation

Courant Institute of Mathematical Sciences, New York University Department of
Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of
Technology Department of the Geophysical Sciences, University of Chicago

Jonathan<EMAIL_ADDRESS>
Given sufficient training data, convolutional neural networks can predict the
maintenance of Atlantic blocking from an initial blocked state.

Transfer learning from an idealized model to reanalysis data enables
predictive skill in the low data regime of the observational record.

Feature importance analysis reveals the influence of upstream flow on blocking
persistence and quantifies biases in the idealized model.

## Plain Language Summary

Blocking events are an important cause of extreme weather, especially long-
lasting blocking events that trap weather systems in place. The duration of
blocking events is, however, systematically underestimated in climate models.
Using data generated by a simplified atmospheric model we demonstrate that,
given sufficient training data, convolutional neural networks can predict the
maintenance of Atlantic blocking from an initial blocked state. Next, we show
that first training the neural network on data from the simplified model and
then fine tuning the training using real world weather data enables prediction
even with few examples of long-lasting blocking events in the observational
record. Subsequent feature analysis of the resulting neural networks
identifies the input variables that most strongly impact their predictions,
revealing that areas of high pressure in certain parts of North America and
the North Atlantic Ocean are important for predicting long-lasting blocking
events and quantifying biases in the idealized model relative to real weather.

##  1 Introduction

Blocking events are high-amplitude, quasi-stationary anticyclonic high-
pressure anomalies that give rise to prolonged abnormal weather conditions in
the mid-to-high latitudes [Rex (1950), Woollings . (2018), Lupo (2021)].
Blocking events can lead to regional extreme weather by disrupting the usual
westerly flow for extended periods [<]e.g.,¿[]Woollings2022, causing extreme
heatwaves, floods, and winter storms [<]e.g.,¿[]lupo2012dynamic.

The predictive skill of numerical weather models has improved dramatically,
but they still cannot accurately forecast important aspects of blocking
events. Blocking frequency and duration are generally simulated poorly by
climate models [Davini D’Andrea (2020)], and even by numerical weather
prediction models in medium-range forecasts [Woollings . (2018), Matsueda
(2009), Ferranti . (2015)]. Several possible contributing factors have been
proposed, including the accuracy of the model’s mean flow [Scaife . (2010)] or
synoptic eddies [Berckmans . (2013), Zappa . (20141)], the model’s resolution
[Davini D’Andrea (2016)] and subgrid-scale parameterizations [d’Andrea .
(1998)], and even the choice of blocking index itself [Tibaldi Molteni (1990),
Dole Gordon (1983), Pelly Hoskins (2003)].

Two commonly used blocking indices [Tibaldi Molteni (1990), Dole Gordon
(1983)] highlight two essential features of a blocking _event_ : (i) a large
positive anomaly of geopotential height that displaces the midlatitude jet,
“blocking” the flow, that (ii) persists for longer than typical synoptic
variability. Often a 5 day threshold is invoked, but the longer the flow
remains in a blocked state, the more severe the implications, either for
extended cold/hot conditions or an increased likelihood of compound storm
events [<]e.g., back-to-back storms, which can dramatically increase the
potential for damage;¿[]Woollings2022. The persistence of blocking is the
focus of our study: given the onset of a blocked state, what is the likelihood
that the flow will remain blocked for an extended period, 5 days for a
standard event, or up to 9 days for more extreme cases? We take a data-driven
approach, training a convolutional neural network to identify persistent
blocks at the onset of a blocked state.

To understand blocking, various low-order models have been formulated to
identify essential features. In an influential early work, charney1979multiple
modeled blocking as one of two equilibrium states of a set of dynamical
equations for a highly truncated barotropic channel model. Others used low-
order models to propose that the positive feedback of synoptic-scale eddies on
the blocking structure contributes to the long-time maintenance of blocks
[Hoskins . (1983), Shutts (1983), McWilliams (1980)]. While these low-order
models have provided useful physical insight, realistic land-sea interactions,
topography, and other factors present in the real world limit their
application. Comprehensive models, on the other hand, are becoming skillful in
simulating realistic blocking, but their complexity makes it challenging to
isolate the essential mechanism(s), and expensive to simulate numerous events.

To strike a balance between complexity, transparency, and statistical
robustness from abundant data (model output), we begin with the Marshall-
Molteni (MM) model [Marshall Molteni (1993)], a three-layer quasigeostrophic
(QG) approximation of the atmosphere that has previously been used to study
blocking events [<]e.g.,¿[]lucarini2020new. The MM model captures the main
features of the northern hemisphere atmosphere reasonably well. For example,
michelangeli1998dynamics found that an enhanced baroclinic wavetrain traveling
across the North Atlantic is necessary to trigger the onset of the Euro-
Atlantic blocking in both this simple model and reanalysis. They also pointed
out that wave-wave interactions and wave-mean interactions dominate local
amplification and the propagation of anomalies, respectively.

The MM model allows us the freedom to develop and test methods in a data-rich
setting. How well can a data-driven method identify persistent events as a
function of the input data you allow it? Following work by explanableAIBarnes
and RAMPAL2022100525, can so-called Explainable Artificial Intelligence (XAI)
techniques provide physical insight into both the AI methods and the model
itself? We show that Shapley Additive ExPlanation (SHAP) analysis reveals key
regions upstream of the blocking center that enable prediction, and use this
to construct low-order models the can be interpreted in the context of prior
work.

Our ultimate goal, however, is to forecast and understand the maintenance of
blocks in our atmosphere, for which we shift the focus to ERA5 reanalysis
[Hersbach . (2020)]. For the most extreme case of a 9-day block in the North
Atlantic, only 18 have occurred in the historical record (See Tab. 3). What
chance does a data-driven approach have? To address the problem of limited
data, we apply transfer learning: first we train a convolutional neural
network on the MM model to learn the basic features of blocking, and then we
re-train it on the limited ERA5 data to calibrate it for the real atmosphere.
We find that pre-training on the MM model yields a better predictor than when
we train the same network on ERA5 alone, proving the efficacy of the transfer
learning approach.

The remainder of this paper is organized as follows. Section 2 introduces the
Marshall-Molteni (MM) model. Sections 3 and 4 define our choice of blocking
index and blocking event criteria, and formulate an objective function for
machine learning. Section 5 discusses our convolutional neural network
structure and training details. We first focus exclusively on the MM model in
sections 6 and 7, applying XAI techniques to visualize the important features
for prediction and testing the results by building a sparse model with
features guided by the XAI. We also suggest physical interpretations for these
predictive features. Finally, we turn to the ERA5 data set in Section 8,
applying transfer learning to improve the prediction of persistent blocks in
ERA5, especially for more extreme events. SHAP analysis shows how transfer
learning has modified the CNN to adapt to the new data set, but preserves the
use of key upstream regions for prediction.

##  2 Marshall-Molteni Model

marshall1993toward developed a 3-layer model of the atmosphere to study
atmospheric low-frequency variability. We use a Northern Hemisphere only
version of the model developed by lucarini2020new with 6210 degrees of
freedom. We refer the reader to that paper for a complete description, but
review key details here. The Marshall-Molteni (MM) model state is specified by
potential vorticity $q_{j}$ in three layers of the atmosphere, $j=1,2,3$,
corresponding to pressure levels 200, 500, and 800 hPa. $q_{j}$ evolves
according to quasi-geostrophic dynamics as

$\displaystyle\partial_{t}q_{j}+J(\psi_{j},q_{j})=-D_{j}+S_{j}$ (1)

where $\psi_{j}$ is the streamfunction in layer $j$, related to $q_{j}$ as

$\displaystyle q_{1}$
$\displaystyle=\Delta\psi_{1}-(\psi_{1}-\psi_{2})/R_{1}^{2}+f$ (2)
$\displaystyle q_{2}$
$\displaystyle=\Delta\psi_{2}+(\psi_{1}-\psi_{2})/R_{1}^{2}-(\psi_{2}-\psi_{3})/R_{2}^{2}+f$
(3) $\displaystyle q_{3}$
$\displaystyle=\Delta\psi_{3}+(\psi_{2}-\psi_{3})/R_{2}^{2}+f(1+h/H_{0}).$ (4)

Here, $\Delta$ is the horizontal Laplacian operator, $R_{1}=761$ km and
$R_{2}=488$ km are the Rossby deformation radii in layers 1 and 2,
$f=2\Omega\cos\phi$ is the latitude-dependent Coriolis parameter, and $h$ is
the orography of the surface, rescaled by the constant $H_{0}$. The operator
$D_{j}$ combines all dissipative terms, including radiative damping, surface
friction and hyper-diffusion to crudely parametrize small scale diffusion, but
is also necessary for numerical stability:

$\displaystyle\begin{split}-D_{1}=&(\psi_{1}-\psi_{2})/(\tau_{R}R_{1}^{2})-R^{8}\Delta^{4}q_{1}/(\tau_{H}\lambda_{max}^{4})\\\
-D_{2}=&-(\psi_{1}-\psi_{2})/(\tau_{R}R_{1}^{2})+(\psi_{2}-\psi_{3})/(\tau_{R}R_{2}^{2})-R^{8}\Delta^{4}q_{2}^{\prime}/(\tau_{H}\lambda^{4}_{max})\\\
-D_{3}=&-(\psi_{2}-\psi_{3})/(\tau_{R}R_{2}^{2})-EK_{3}-R^{8}\Delta^{4}q_{3}^{\prime}/(\tau_{H}\lambda_{max}^{4}).\end{split}$
(5)

The forcing, $S_{j}$ is computed from observed data to inject energy into the
system and give the model a realistic mean state:

$\displaystyle S_{j}=\overline{J(\psi_{j},q_{j})}+\overline{D}_{j}$ (6)

The data to construct $S_{j}$ were drawn from the 1983–1992 winter (DJF)
climatology of the ERA40 reanalysis provided by ECMWF. The model is run with
T31 horizontal resolution (corresponding to 90 longitude $\times$ 23 latitude
gridpoints across the northern hemisphere). All model output fields, as well
as the reanalysis used later, are averaged daily. The climatology of Marshal-
Molteni model is shown in the supplemental materials, and we compare its
blocking statistics with ERA5 reanalysis in the next section.

##  3 Blocking index

In this study, we use the “DG” index [Dole Gordon (1983)] to define blocking
events. This is an anomaly-based blocking index, but has been shown to capture
the same essential features of blocking as other indices, e.g., that of
tibaldi1990operational.

We compute this index by transforming the spherical harmonic representation of
$\psi$ into approximate geopotential height, $Z$, on a Gaussian grid for
latitude and a uniform grid for longitude. The approximation is the choice of
a fixed Coriolis parameter $f_{0}$ to convert from $\psi$ to $Z$, which leads
to minimal distortion over our midlatitude area of focus. A blocking event is
said to occur at a specific location when $Z$ stays above a tunable
geopotential height anomaly threshold, $M$, for at least 5 consecutive days.
In their paper, dole1983persistent tested statistics for varying $M$ values,
ranging from 50 m to 250 m, with subsequent studies adopting different
thresholds (Pedramblockingindices, Tab. 2). For our investigation, we
calibrated $M=100$ m for our MM model simulation to roughly match the blocking
fraction computed from ERA5 reanalysis data, where we used the threshold
$M=150$ m as in mullen1987transient.

Fig. 1 shows the blocking event statistics during the simulation. For
comparison, blocking event statistics computed from ERA5 reanalysis data from
1959-2021 are also shown. In this study, we focus on North Atlantic blockings
indicated by the white rectangle in Fig. 1. We pick this region because it has
a relatively high blocking frequency, and for its important influence on
western Europe. We use $Z_{B}$, the mean 500 hPa geopotential height anomaly
in this target region over the North Atlantic, to define blocked states and
blocking events.

Figure 1:  (a) blocking fraction (the percent of days with $T\geq 5$ days) for
MM model data with $M=100$ m. (b) total blocking event counts for MM model
data during the simulation. (c) blocking fraction for ERA5 reanalysis data
with $M=150$. (d) total blocking event for ERA5 reanalysis data with $M=150$m.
In all subfigures, the region we focus on is indicated by the white rectangle
centered at $0^{\circ}$E and $62^{\circ}N$ (approximately spanned by 3
longitude points covering $4^{\circ}W-4^{\circ}E$, and 2 latitude points
covering $60^{\circ}N-64^{\circ}N$)

##  4 Probabilistic forecasting and event definition

We aim to study the _maintenance_ of blocks rather than their _onset_.
Precisely, we formulate the question as the classification problem posed in
Fig. 2: given a nascent blocked state, i.e., the state on a day that
geopotential height anomalies over the North Atlantic first exceed the
threshold $M$, can we immediately predict whether the flow will remained
blocked for 5 or more days – evolving into a blocking _event_ – or will the
flow return back towards the climatological state before 5 days have passed?
In the MM model, nascent blocked states evolve into 5-day persistent blocking
events approximately 1/5th (21%) of the time on average, more often fading
back towards climatology. Given only the state at the time of blocking onset,
can a data-driven method accurately identify the rarer cases that will persist
for more than 5 consecutive days?

Figure 2: Left: The blocking persistence problem: given a nascent blocked
state, the goal is to forecast whether it will persist into a long-lasting
blocking event, or quickly return to climatology. The percentile represents
the climatological probability. Right: A sample trajectory of $Z_{B}(t)$, the
anomaly of geopotential height defined in Sec. 3. The vertical dashed lines
indicate new blocked states ($T=1$). The red shading indicates the duration of
the block. The label $Y=1$ indicates that the blocked state persisted 5 days
to constitute a blocking event, while $Y=0$ indicates that it did not.

To formulate this classification problem mathematically, we denote the full
model state by $\boldsymbol{X}$ and further introduce a variable $T$ for the
running duration of a blocked state:

$\displaystyle T=\\{\text{days since }Z_{B}<M\\}.$ (7)

Note that $Z_{B}(t)$ is determined by the state vector $\boldsymbol{X}(t)$ at
any time $t$, but $T(t)$ retains some memory of previous states and thus is
not fully determined by $\boldsymbol{X}(t)$. For example, as shown in Fig. 2,
suppose $Z_{B}(t)$ first rises above $M$ on day $t=16$ and dips back below $M$
on day $t=18$. Then, $T(t)=0$ for all days through $t=15$, $T(16)=1$,
$T(17)=2$, and $T(18)=0$. With this notation, we can say that
“$\boldsymbol{X}(t)$ is the beginning of a blocking event” if

$\displaystyle T(t)=1\quad\text{and}\quad T(t+D-1)=D.$ (8)

The condition $T(t+D-1)=D$ only holds when there are at least $D$ consecutive
days with $Z_{B}(t)\geq M$ starting from $t$. We can see an example of this in
Fig. 2 at day 24, for both a block of duration 5 and 7 days. Here, $T(24)=1$,
and $T(28)=5$, triggering the condition for $D=5$. The flow remains blocked
through $T(30)=7$, such that day 24 would also count as the onset of a $D=7$
day blocking event.

With this formulation, our central question becomes: given a $T(t)=1$ state at
time $t$ (the flow has just become blocked), will it stay blocked for $D$
days, $T(t+D-1)=D$, or not? We address this question by estimating the
conditional probability:

$q(\boldsymbol{x}(t))=\mathbb{P}[T(t+D-1)=D\,|\,\boldsymbol{X}(t)=\boldsymbol{x}(t),T(t)=1].$
(9)

Unless otherwise specified, we adopt $D=5$ to maintain consistency with the
common blocking indices [Tibaldi Molteni (1990), Dole Gordon (1983), Pelly
Hoskins (2003)]. We also consider more extreme events with $D=7$ and $D=9$.

##  5 Convolutional Neural Network Training and Performance

Convolutional Neural Networks (CNN) have gained widespread application in
probabilistic forecasting problems [Miloshevich . (2023), Ham . (2019), Liu .
(2016)] for their outstanding performance on multidimensional data sets with
spatial structure. A CNN differs from a dense neural network in the use of
convolutional layers with shared weights and biases across layers within the
network, designed to extract features that exhibit translation invariance
across the input space [Goodfellow . (2016)]. Originally developed in the
context of image processing, CNN excels in scenarios where target features,
such as the face of a cat, may appear at different places within the training
image. Convolutional layers allow the network to efficiently learn these
features, combining information across multiple images. In our context,
atmospheric eddies and Rossby waves share similar dynamics across all
longitudes. A CNN can potentially more effectively extract these dynamics,
while still learning how they vary with longitude and zonal asymmtries induced
by topography, etc.

The structure of the CNN in this investigation follows Bouchet2023 and is
shown in Fig. 3. It consists of a three-layer architecture, combining
convolutional filters followed by ReLu activations. Specifically, we use 32
and 64 filters ($3\times 3$) for the first and last two convolutional layers.
Between each pair of convolutional layers is a max-pooling layer. The output
is then flattened and passed to a dense layer with 64 neurons that produces 2
outputs. The output is then passed through a softmax function to form two
normalized probabilities that sum to 1.

We performed experiments with alternative CNN structures and found that
reducing the widths of layers can mitigate overfitting, but this also reduces
the performance at the best epoch (not shown). Therefore we adopt the
architecture in Fig. 3 and use early-stopping to avoid overfitting, as
detailed below.

Figure 3:  Convolutional Neural Network structure. The three convolutional
layers (yellow) respectively use 32, 64 and 64 filters ($3\times 3$), followed
by ReLu activations. Between each pair of convolutional layers is a max-
pooling layer (red) with window size $2\times 2$. Then the output is flattened
(green) and passed to a dense layer with 64 neurons that produces 2 outputs
(blue). The output is then passed through a softmax function (blue).

###  5.1 Training and Test Datasets

To study whether a nascent blocked state will persist, we create a training
and test set of all states where the flow has just become blocked:
$\\{(\boldsymbol{X},T)|T=1\\}$, where $\boldsymbol{X}$ are $18\times 90\times
3$ (latitudes $\times$ longitudes $\times$ pressure at levels of 200 hPa, 500
hPa, 800 hPa) grid maps of geopotential height from 20∘N to 87∘N. Our goal is
to classify which of these cases persist into blocking events ($Y=1$) versus
states that do not ($Y=0$). Fig. 2 shows a sample time series with 4 instances
of a nascent blocked state, $t=16,24,38$ and 47, only the second of which
evolves into a persistent blocking event, $Y=0$, 1, 0, and 0, respectively.
For each case, the model must classify $Y=0$ or $Y=1$ given only
$\boldsymbol{X}$ at the onset time.

We examined the sensitivity of CNN model performance with respect to different
amounts of training data. To prepare the dataset, we integrate the MM model
for 1250k days in total. The computational cost is low, requiring 1 CPU core
and approximately 11 hours. We select the first $n$ days (with $n$ ranging
from 1k to 1000k) to create the training data set, and always take the last
250k days for the test dataset. Thus all models can be fairly compared. The
trajectory length and the corresponding number of nascent blocked state states
are shown in Tab. 1. The likelihood $q$ of forming a blocking event varies
depending on different persistence thresholds $D$. This dependence
relationship is illustrated in Tab. 2.

Training data
---
Days | Nascent blocked states
1k | 63
10k | 699
100k | 7024
500k | 35078
1000k | 70635

Test data
---
Days | Nascent blocked states
250k | 17755

Table 1: Length of trajectory (in thousands of days) vs. number of nascent
blocking states ($T=1$) in training set and test sets of varying size.

###  5.2 Learning procedure

For simplicity, we use binary cross entropy as a loss function, a common
choice for classification [Miloshevich . (2023)]. Alternative loss functions
have been studied by rudy2023output. The loss function $L(q)$ is defined as as
follows:

$\displaystyle L(q)=-\frac{1}{N}\sum_{i=1}^{N}$ $\displaystyle\Big{[}Y_{i}\log
q(Y_{i}=1)+(1-Y_{i})\log(1-q(Y_{i}=1))\Big{]}$

where $q(Y_{i}=1)\in(0,1)$ is the probability of the event $Y_{i}=1$ as
predicted by the CNN. $L(q)$ is small when the CNN predicts high probability
for positive events, and low probability for negative events.

Given the rarity of blocking events, the data exhibit a pronounced class-
imbalance, which becomes increasingly severe for longer block durations. As
shown in Tab. 2, for $D=5$, only about 1 in 5 nascent blocked states persist
into an event, but $D=9$, less than 1 in 20 evolve into persistent events.
With this extreme imbalance, a model that never predicts an event will be
correct over 80% or 95% of the time, respectively. However, such a model would
clearly underperform in terms of precision and recall, which would both be
zero.

Threshold | $Y=1$ | $Y=0$ | Positive rate
---|---|---|---
$\geq 5$ days | 18748 | 69642 | 0.212
$\geq 7$ days | 8522 | 79868 | 0.096
$\geq 9$ days | 3891 | 84499 | 0.044
Table 2: The statistics of blocking events in our MM 1250k day simulation. The
full dataset exhibits 88390 nascent blocking states ($T=1$ states). $Y=1$
marks the number of these nascent blocks that persist for 5, 7, or 9 days,
thus evolving into a blocking event under these respective thresholds, while
$Y=0$ denotes the number that don’t make it to the threshold.

To address the class imbalance, for our results in this section we employ
over-sampling [Johnson Khoshgoftaar (2019)] techniques during training. In
each epoch, we sample an equal number of nascent blocks from both classes
until we complete an iteration over all the nascent blocks in the
overrepresented class. As a result, the nascent blocks that persist have been
sampled multiple times during each epoch.

###  5.3 Performance metrics

Throughout this study, we evaluate model performance using two key metrics:
_precision_ and _recall_. We monitor the values of these metrics on the test
dataset throughout the training process to determine the stopping point in
order to avoid overfitting. The precision and recall are respectively defined
as

Precision $\displaystyle=\frac{\text{True positives}}{\text{True positives +
False positives}},$ (10) Recall $\displaystyle=\frac{\text{True
positives}}{\text{True positives + False negatives }},$ (11)

where “True positives” is the number of data points with $Y=1$ for which our
CNN predicts a persistent blocking event, “False positives” the number of data
points with $Y=0$ for which our CNN predicts a persistent blocking event, and
“False negatives” the number of data points with $Y=1$ for which our CNN
predicts a blocked state that does not persist.

More informally, if the method forecasts that an event will occur, the
precision measures the fraction of times this forecast is correct. The recall,
on the other hand, is the fraction of the successfully forecasted events of
all the positive events. If, regardless of the system state, one randomly
predicts events with the climatological mean rate, in which an overall
fraction $p$ of the data labels are True, then the precision and recall are
both given by $\frac{p^{2}N}{p^{2}N+(1-p)pN}=p$. This sets the floor for a
useful predictor: both the precision and recall must be better than
climatological rate.

There can be tradeoffs between improving the precision and recall. Predicting
the event all the time will give you a perfect recall, but climatological
precision $p$. A low recall implies missing a substantial number of positive
events, leading to inadequate preparation and increased risk of damage.
Conversely, a low precision suggests over-predicting events, “crying wolf” too
often. In the context of extreme weather forecasting, this can lead to over-
preparation, consequently reducing the efficiency of regular societal
operations, as well as trust.

A reasonably high value of both recall and precision is crucial for an
effective and resource-efficient forecasting model. We use a simplistic
definition of ‘best’ performance, expressed as

$\displaystyle\text{Overall performance}=\text{Precision}+\text{Recall}\,.$
(12)

However, it is crucial to note that in practical scenarios, designing overall
performance metrics requires careful consideration of the cost of preparing
vs. risk of damage associated without preparation. This naive criteria only
works when the precision and recall are both reasonably high, since
forecasting the event all the time will yield a performance score of 1+p
(recall of 1 and precision of $p$). We used caution in ERA5 based forecasts,
requiring our trained models exhibit nontrivial precision above the
climatological rate.

###  5.4 Performance and early stopping technique

The top row of Fig. 4 shows the precision and recall evaluated on the test
data for varying training data sets for $D=5$. Both the precision and recall
metrics are plotted starting from the end of Epoch 1 (the leftmost point on
the horizontal axis of Fig. 4); From Epoch 2 to Epoch 10, the precision
increases, chiefly reflecting a decrease in the false positive rate, as the
CNN becomes better at discriminating between persistent and non-persistent
flow configurations. At the same time, the recall slowly decays: the false
negative rate rises slightly as the network becomes more conservative and less
likely to over-predicting persistent cases. Except for the low data regime (1k
days), the performance of the CNN asymptotes after approximately 10 epochs
where the precision and recall are approximately equal, but this is not
necessarily the ideal stopping time [Miloshevich . (2023)].

Applying the definition of best performance in Eq. (12), the “best” CNN is
obtained by training on the full data set of 1000k days for 4 epochs,
indicated by the star in Fig. 4. It achieves precision of 0.70 and recall of
0.87, exhibiting significant predictive power over the climatological mean
prediction (the black dashed line with value 0.21). Therefore, we use it for
further analysis in Sec. 6.

Figure 4: Top: Precision and recall results for models trained on data sets of
varying sizes with $D=5$. The dashed black line is the predicted recall and
precision from the climatology computed using the largest data set. The blue
stars indicate precision=0.70 and recall=0.87. Middle: Same results for $D=7$.
Bottom: Same results for $D=9$. Fewer curves are displayed for $D=7$ and $D=9$
for the sake of clarity.

All of our CNNs significantly outperformed the climatological mean prediction
for any amount of data or training length. Interestingly, although the best
performance is always realized with the longest trajectory of 1000k days, the
sensitivity of precision and recall to the training data size is different.
For $D=5$ events, the precision improves with more data up to 100k days
(equivalent to approximately 1000 winters), after which additional data does
not lead to much improvement. The recall, however, is more data-hungry; its
performance continues to improve until data reaching 500k days, equivalent to
5 millennia of winter data. This reflects the fact that more data continues to
help the CNN avoid missing events after its ability to limit false positive
forecasts has saturated.

Fig. 4 also shows the results for higher persistence thresholds, $D=7$ and 9.
These thresholds correspond to rarer evnts, and even with the longest
trajectory of 1000k days, the precision and recall curves suffer for two
reasons. First, as seen from Tab. 2, the number of positive events drops,
effectively limiting the data set almost by a factor of 5 for the most extreme
$D=9$ cases. More importantly, however, it simply becomes harder to
discriminate rare events as the data set becomes more imbalanced: less than 1
in 10 nascent blocking states will evolve into a 7 days block, and less than 1
in 20 into a 9 day blocking event. Without our efforts to overcome this
imbalance, a network can classify almost all events correctly by never
predicting a persistent case.

Despite the difficulties, the CNNs still show some skill in rare event
forecasting. Given the full 1000k dataset, for $D=9$ the precision and recall
converges to about 0.35. While this is only half the values achieved by the
CNN in the $D=5$ case, this is almost 10 times the climatological values of
precision and recall in that case. As with the D=5 cases, we found that the
recall for $D=7$ and 9 suffers more than the precision when the data set
shrinks: with less events to learn from, the CNNs become more conservative and
less likely to call an event. The recall depends on the false negative rate,
thus appears more sensitive to class imbalance. More data gives the network
more true positive cases to learn from, appearing to help overcome this
challenge.

The low precision and recall values for smaller data sets (1k and 10k) does
not bode well for training our CNN on ERA5 data, which will be discussed in
detail in Section 8. For $D=5$, there are 273 nascent blocked states in the
ERA5 record, 84 of which persist into blocking events (see Table 3). This data
amount falls between our 1k and 10k cases where data clearly limit
performance. Consistent with our experience with the MM model, achieving a
high recall is the most difficult with limited data, and it is with this
metric that transfer learning will have the largest impact.

##  6 Feature analysis: What is our CNN using to predict blocking events?

Before turning to forecasting in the realistic data regime, we ask what our
best CNNs have learned to make these forecasts. Explainable Artificial
Intelligence (XAI) is an array of techniques used to try to gain some
understanding of the basis on which neural networks make predictions
[Linardatos . (2020)]. In this section, we use SHapley Additive exPlanation
(SHAP) value analysis to dissect the contributions of different atmospheric
pressure levels and geographic areas that our CNN is using to make its
predictions. We further construct a sparse model using the identified
important features as inputs to quantitatively justify their relative
importance in the prediction process.

###  6.1 Method

SHapley Additive exPlanation (SHAP) values, introduced by lundberg2017unified
and shrikumar2017learning, draw inspiration from Shapley values in game theory
[Lipovetsky Conklin (2001)]. In the domain of weather and climate science,
SHAP values have found broad use, with applications ranging from Earth System
model error characterization [Silva . (2022)] to drought forecasting [Dikshit
Pradhan (2021)].

Intuitively, given a function $f:\mathbb{R}^{d}\to\mathbb{R}$ (such as the
conditional probability function $q$ in Eq. 9), SHAP assigns an importance
value $\phi_{i}$ to each feature $x_{i}$ of the argument
$\boldsymbol{x}\in\mathbb{R}^{d}$, which combine additively:

$\displaystyle
f(\boldsymbol{x})=\mathbb{E}[f(\boldsymbol{x})]+\sum_{i=1}^{d}\phi_{i}(f,\boldsymbol{x}).$
(13)

With no knowledge of $\boldsymbol{x}$, the optimal prediction of $f$ (in a
mean-square sense) is the climatological average over the distribution of
$\boldsymbol{x}$: $\mathbb{E}[f(\boldsymbol{x})]$ . SHAP values quantify how
much is gained beyond this baseline by incorporating information from each
component $i$ of $\boldsymbol{x}$. The SHAP values
$\phi_{i}(f,\boldsymbol{x})$ are unique for each sample of $\boldsymbol{x}$,
but features $i$ for which $|\phi_{i}(f,\boldsymbol{x})|$ are large for most
$\boldsymbol{x}$ (that is, a large SHAP value on average) can be singled out
as important, or useful, for the prediction of $f(\boldsymbol{x})$. SHAP
values possess advantageous theoretical properties as well, and we refer the
reader to lundberg2017unified for a detailed theoretical analysis. In this
study, SHAP values are computed using the Python package Deep SHAP. The
function $f(\boldsymbol{x})$ is taken as the estimated conditional probability
$\hat{q}(\boldsymbol{x})$ computed by the CNN, i.e., the probability,
according to the CNN, that the blocked state will extend $\geq D$ days,
leading to a blocking event.

###  6.2 Results

Fig. 5 shows the composite of SHAP values for true positive data. Because few
nascent blocks persist for $D=5$, 7, or 9, the climalogical probability of a
persistent event $\mathbb{E}[\hat{q}(\boldsymbol{x})]=0.21$, $0.096$, and
$0.044$, respectively. For our CCN to call a positive event, we require the
conditional forecast probability $\hat{q}(\boldsymbol{x})$ to be larger than
0.5. Hence a positive (negative) value of $\phi_{i}(\hat{q},\boldsymbol{x})$
indicates that knowing the geopotential height anomaly at this level and
location increases (decreases) the likelihood of a positive event. Therefore,
the shading in Fig. 5 can be interpreted as the average influence of each grid
point for the CNN to successfully predict a long-lasting blocking event.

The SHAP composite is approximately uniformly non-negative because it is based
only on true positive events: additional information should always increase
the forecast probability. This indicates that the CNN has been well-trained to
only use geopotential height information that improves the blocking event
probability, and suggests it has identified robust features that herald a
persistent block. A composite based on true negative cases (not show), reveals
similar patterns, but of the opposite sign.

Figure 5: Composite maps of SHAP values, $\overline{\phi}$, of geopotential
height at 200, 500, and 800 hPa, for true positive cases, i.e., when the CNN
accurately forecasts a persistent blocking event. The unit is the probability
of a positive forecast per feature (see equation 13), indicating the average
incremental increase in the CNN’s confidence that the nascent blocked state
will evolve into a persistent blocking event, given knowledge of $Z$ at at a
given location and pressure. The boundaries of the most important regions
learned by the CNN are marked by solid lines and denoted region 1 (Florida,
black), region 2 (north Atlantic, blue), region 3 (northeastern North America,
green) and region 4 (Iceland, red) .

The first thing to notice is that anomalies upstream from the blocking region
(to the west) are more valuable for predicting the persistence of the blocked
state. Moreover, the commonality among different pressure levels reflects the
relatively barotropic nature of the MM model. In general, however, the CNN
prediction relies most on the upper level flow (200 hPa).

The SHAP values emphasize four distinct regions in a quadrupole arrangement to
the west of the Atlantic blocking region, as marked in Fig. 5. We chose these
regions to encapsulate high SHAP values using the following algorithm: after
objectively identifying regions where SHAP values exceeded a set threshold, we
defined boundaries by hand with the goal of enclosing these regions across all
three levels within the smallest encompassing rectangle. While part of the
goal of choosing these regions was to build a sparse predictor in the next
section, they give us physical insight on their own.

The meaning of the SHAP values can be more easily interpreted with the aid of
composites of the true positive events (Fig. 6), which show us the sign of
anomalies that favor persistence. Positive geopotential anomalies in region 1
(black, centered over Florida) and 4 (red, over Iceland, just east of the
blocking region itself) at the onset of blocking indicate to the CNN that a
block will persist, while negative anomalies over Regions 2 (blue, North
Atlantic Ocean) and 3 (green, northeast US) also favor persistence.

Regions 2 and 4 project onto opposing centers of action of the North Atlantic
Oscillation (NAO). They indicate that a more negative NAO state at the onset
of blocking increases the likelihood of a persistent block. Previous studies
have also found that blocks tend to be more persistent when the NAO is
negative [Barnes Hartmann (2010)]. While a blocking pattern off Europe
projects weakly onto the NAO itself, SHAP analysis indicates that the wider
structure of the pattern is important. Regions 1, 3, and 4, on the other hand,
appear to be part of a wave train arching southwest from the blocking region.
Their importance suggests that downstream development of a wave packet
propagating along the jet stream helps drive persistent blocking events in the
North Atlantic.

Figure 6: Average states of nascent blocking states that evolve into
persistent blocking events ($T=1,y=1$) of (top row) MM dataset and (bottom
row) ERA5. The colorbar represents values of geopotential height anomalies
normalized by the standard deviation at each location and height.

##  7 Building a sparse model: Logistic regression

To substantiate the importance of the regions highlighted by the CNN in
prediction, we constructed a sparse model. The success of this model reveals
that a small set of well-chosen variables and a model with a simple structure
can recover a sizeable portion of the predictability. We computed the local
mean of $Z200$, $Z500$, $Z800$ for each of the four rectangles shown in Fig.
5, resulting in 12 time series. We then applied logistic regression with
different combinations of these 12 features. The results for the sparse models
with the best predictive skill within models of 1 to 5 dimensions on the test
set are illustrated in Fig. 7(a). The horizontal axis denotes the variable
combinations that achieve the predictive scores shown in the figure.

We draw three key conclusions from Fig. 7(a). First, to predict the
persistence of a blocked state, the best one-dimensional feature is Z200 in
region 1, upstream over Florida and the Gulf, not Z500 in region 4, the
$Z-$field nearest to the blocking region we focus on. Second, the combination
of Z200 in region 1, Z500 in region 4 forms a two-dimension model (shown in
Fig. 7(b)) that already recovers a recall value of 0.75 – it captures three
quarters of all blocking events – with a precision of 0.44, twice the
climatological rate. The precision and recall of the full CNN, however, are
0.87 and 0.70. This leads us to the third key message: the large discrepancy
in precision between CNN and logistic regression. Even with 5 predictors, the
precision of our sparse model is only 0.5.

The poor precision indicates that the sparse model makes too many false
positive predictions. This could suggest that the decay of the Atlantic
blocked state is a more nonlinear dynamical phenomenon, which cannot be
modeled as a simple linear statistical model. A CNN can capture these
nonlinearities more effectively than sparse regression, which is consistent
with previous research which found North Atlantic blocks are associated with
nonlinear processes [Evans Black (2003)]. It could also indicate that more
subtle features outside these 4 centers (and variation within these regions)
are important. Fig. 5 indicates that the CNN uses information across all of
the North Atlantic, eastern North America, and even off the west coast of the
US, to make skillful predictions.

Figure 7: (a): Sparse model predictive skill on the test data set. The
horizontal axis represents the dimension $d$ of the sparse model from 1 to 5,
with labels showing the combination of variables (“R1” = “region 1”) that
achieves the best predictive skill among all combinations of $d$ variables.
The $(+)$ and $(-)$ indicate the sign of the coefficient before the variable
in the logistic regression. (b) Conditional probability of a persistent block,
$q$, as a function of mean normalized geopotential height anomaly at 200 mb
over region 1 and at 500 mb over region 4 (the second column of (a)). (c) The
marginal density (likelihood of observing these anomalies) as a function of
the same variables. Densities below $10^{-5}$ are cut off.

To explore the effectiveness of the two-dimensional sparse model, we
visualized the conditional probability of a block persisting, $q$, projected
onto this simple subspace (shown in Fig. 7(b)). Introduced in Eq. (9), $q$
quantifies the probability that the system will evolve into a persistent
blocking event before the flow becomes unblocked. For example, the lightest
pink region, corresponding to $q\approx 0.5$ indicates that if, at the onset
of blocking, $Z$ at 200 hPa over region 1 (Florida) is particularly high or
$Z$ at 500 hPa in region 4 (Iceland) is abnormally high, the system has a
roughly 50% chance of evolving into a persistent block, more than double the
climatological rate of 21%. In the red region at the top right, where both of
these regions exhibit abnormally high pressure, the odds of a persistent block
increase to near 100%.

Fig. 7(c) shows the likelihood of observing these Z200 and 500 anomalies. Most
often, the system exists in the middle of the diagram, where the probability
of a blocking event hovers around the climatological value or below. The most
likely state that exhibits a high chance of a block lies along the diagonal
from the upper left to the lower right, with moderately high Z200 and 500
anomalies. The states in the top right corner, for which a persistent block is
nearly certain, are very rare.

The sparse models suggest physical links between blocking events and the
upstream flow. The Atlantic blocking region lies at the end of the Atlantic
storm track [Michelangeli Vautard (1998)]. Persistent blocks, at least in the
MM model, are favored when there is enhanced wind off the east coast of the US
(high pressure over Florida, region 1) and low pressure over regions 2 and 3
(which are highlighted in the higher dimensional sparse models). This
displaces the climatological winds upstream of the blocking region
equatorward. This will modify the input of storm activity into the blocking
region, consistent with prior studies that have highlighted the relation
between the storm track and blocking events [Zappa . (20142), Yang . (2021)].

##  8 Extending to ERA5 Using Transfer Learning

Given sufficient data, it was possible to construct a CNN that skillfully
forecasts the maintenance of blocking events in the MM model. ERA5 December,
January and February (DJF) data from 1940-2022 exhibit only 273 nascent
blocked states in our Atlantic region of focus. A significant degradation in
performance was evident in Fig. 4 when we restricted the amount of training
data from the MM model, the drop unfortunately occuring in the data regime
available in reanalysis. The curve associated with the trajectory of 10k days
(699 nascent blocked states) plateaued at lower values for both the precision
and recall. With only 1k days (63 nascent blocked states) performance was
poor, and the learning unstable, oscillating significantly across epochs.

The class imbalance between $Y=0$ and $Y=1$ adds to the difficulty (see Tab.
3), particularly when longer blocks are considered. An extreme example is the
set of blocking events that last $\geq$9 days: there are only 18 such events
($Y=1$) in the reanalysis record out of 273 data points. Such a small sample
of positive data can hardly support any meaningful training, and makes it
impractical to get meaningful uncertainty bounds on performance. In a standard
training-test data split with a ratio of 90:10, only around 2 positive events
typically fall in the test set, making it challenging to robustly assess the
skill.

When training on the limited number of events in the reanalysis, a CNN can
more easily suffer from overfitting, where the network uses ‘noise’ (unrelated
features) to classify blocking events. Overfitting can be diagnosed when the
performance on the test set diverges from the training set. Yang and Gerber
(submitted) found that the oversampling strategy used so far in this study was
more prone to overfitting than a weighted loss function strategy [Johnson
Khoshgoftaar (2019)]. With this latter strategy, one emphasizes the rare class
(in our case, positive events) by increasing its weight in the loss function.
In our remaining experiments, we weighted positive and negative events inverse
to their occurance rate.

Threshold | $Y=1$ | $Y=0$
---|---|---
$\geq 5$ days | 84 | 189
$\geq 7$ days | 36 | 237
$\geq 9$ days | 18 | 255
Table 3: The statistics of ERA5 dataset in 1940-2022 DJF with $T=1$.

###  8.1 Direct training

The scarcity of events makes direct training (DT) on ERA5 blocks challenging.
In our study of the MM model data, we had the luxury of a large test data set
(which we intentionally kept the same for fair comparison of the different
CNNs), even for the case with only 1k training days. For ERA5 data, we use
cross validation [Goodfellow . (2016)] to make the best use of the smaller
dataset. The limited number of states were partitioned into training and test
sets in ratios of 90:10; we also tried 80:20, and the results were similar
(not shown). These splits were chosen to balance two difficulties: a small
training set can prevent robust learning, while a small test data set limits
accurate evaluation, even for a well-trained model.

To proceed, we first reduced the resolution of the ERA5 data to a comparable
size of the MM output, considering geopotential height on the same three
levels at the same coarse resolution. Reducing the resolution allowed us to
use the same CNN architecture, and made transfer learning possible (as
discussed below). It also helped avoid overfitting, reducing the number of
input variables relative to the number of events. Then we created the test-
train splits, yielding 10 cross validation sets with distinct test events.
Finally, for each test-train split, we trained and evaluated 10 CNNs, where
variations were confined to random weight initialization and shuffling of
training data.

Providing meaningful uncertainty on the precision and recall statistics from
direct training, shown in the left column of Fig. 8, is challenging. As the 10
CNNs trained on each train-test split are not independent and identically
distributed (IID), we first average the skill scores within each split. The 10
test sets, however, can be viewed as IID samples. The solid lines and shades
respectively represent the mean and two-standard deviation bounds of the
precision and recall, as a function of epoch, across the 10 splits.

For 5 day blocks, a CNN trained by DT can beat the climatological forecast,
albeit only modestly. Given the small testing data set (27 nascent blocks, of
which roughly 8 persist into events), it is important not to put too much
stock in the best possible performing network, for CNN can get lucky on a
small size of samples. The average performance more reliably quantifies the
potential skill. On average, a CNN can achieve a precision of approximately
0.45: when it calls a persistent blocking event, 4-5 out of 10 times it is
correct, as compared to about 3 of 10 in the climatology. The recall was
modestly better, the network only missing 4 of 10 actual events, while a
climatological forecast would miss 7 of 10.

We also explore 7 day events, where only 13% of nascent blocks evolve into 7+
day events. Again, the average CNN modestly beat the climatological forecast
in terms of precision: 1/5 of the cases it calls evolve into persistent
events, roughly double the success rate by a guess with a Bernouli random
variable. The recall was initially deceptively high (the network captured 5 of
10 blocks), but this skill rapidly decreased with training. This was due to
the fact that CNNs at early stages of DT call too many events. As it trains
further, it reduces the forecast rate, declaring fewer false positives at the
expense of missing more events.

###  8.2 Transfer learning

Transfer learning (TL) has found broad application in atmospheric science,
such as detecting gravity waves [González . (2022)], improving extreme
heatwave forecast in climate model [Jacques-Dumas . (2022)], subgrid-scale
(SGS) models [Subel . (2021)], image restoration [Guo . (2022)] and parameter
retrieval from raw dew point temperature profiles [Malmgren-Hansen . (2018)].

TL involves pre-training a model on a larger dataset that is similar to the
dataset of interest (source domain), then fine-tuning the model on the smaller
target dataset (target domain). This approach is particularly beneficial when
labeled data for the target task is limited, as it allows the model to exploit
learned features and representations from the larger dataset to enhance its
performance on the smaller dataset. With this strength, TL has shown its power
in forecasting, combining the data from a climate model [Rasp Thuerey (2021)]
or a dynamical model [Mu . (2020)] with the observational record to improve
medium-range weather forecasting and ENSO prediction.

In this section, we applied TL to leverage our MM dataset to predict events in
the reanalysis data. As a quasi-geostrophic model, MM has complexity between
full climate [Rasp Thuerey (2021)] and low order [Mu . (2020)] models used in
previous transfer learning studies. The overall process is to first ‘pre-
train’ a CNN the MM model dataset, learning to capture the characteristic
features of blocking. While significantly simplified, the MM model is skillful
in representing atmospheric variability [Lucarini Gritsun (2020)], but more
importantly provides extensive positive and negative cases to learn from,
supporting optimal CNN training, as demonstrated in Sec. 5. After pre-
training, our CNN is then fine-tuned on the ERA5 dataset, where the weights
are modified to account for biases in the MM model, and the parameter scales
are calibrated.

In most applications of TL, only the weights in the last few layers of a
neural network are fine-tuned on the target domain [Yosinski . (2014), Hussain
. (2019), Talo . (2019)]. Following this convention, we only retrain the last
layer of the CNN on ERA5 while keeping the other layers frozen. This allows
the CNN to correct biases it inherits from MM, but not to fall back into the
poorly constrained limit we reached with direct training. We also tried
retraining other single layers too, but retraining the last layer performed
the best. To avoid overfitting, we set the learning rate to $1/10$ the
learning rate of pre-training.

We tested different lengths of pre-training and then evaluated the performance
of the resulting models with the peak precision and recall in the transfer-
learning phase. The results show that CNN parameters taken at earlier pre-
training epochs show better peak performance after transfer learning (results
not shown). This suggests that overfitting on the source domain cannot be
fully corrected by fine-tuning on the target domain. For the displayed results
in Figs. 8, 9 and 10, we use a pre-training of 2 epochs for $D=5$, and 1 epoch
for $D=7$. Given the 1000k days of MM integration we had at our disposal, this
means that the neural network has explored more than unique 70,000 nascent
blocking states (all of them twice, for $D=5$) before seeing any of the 273
events in ERA5.

We follow a similar procedure as with DT to assess the ensemble-average
performance. We pre-train 10 CNNs with the 1000k-day MM dataset; the only
differences are due to randomness in the initialization and training data
shuffling. We then carry out a 10-fold cross-validation procedure with 90:10
splits: for each split, we perform TL fine-tuning on the 10 pre-trained CNNs.
We compute the mean precision and recall for each split. The results in the
right column of Fig. 8 show the mean and 2-standard deviation bounds across
all the splits.

Compared to DT, TL begins with a higher precision but lower recall due to pre-
training. With additional fine-tuning, the precision stays almost unchanged,
while the recall grows markedly. The network is able to increase the number of
events that it can capture (lowering the number of false negatives) with
minimal degradation in reliability of its forecast (that is, only slighly
increasing the false positive rate).

Uncertainty in the precision is dominated by differences in the true positive
events between the splits; consequently, the 2-standard deviation error bounds
are comparable for DT and TL. The recall is less sensitive to differences
among the splits, however, and at least for the $D=5$ case, there is
noticeably less spread across the splits with transfer learning. This is
understandable because recall, by definition, doesn’t depend on the positive
rate of the test dataset, which varies a lot for small data sets (around 27
states in each test set after splitting). On the other hand, precision relies
on the positive rate of the test dataset, so it has more intrinsic
variability.

We still evaluate the overall performance by Eq. (12). Focusing first on $D=5$
events, the best mean performance with DT is a precision of 0.45 and recall of
0.61, which is realized at Epoch 3. With TL, we achieve an average performance
with a similar precision of 0.45 and higher recall 0.82 (at Epoch 4). A
noticable advantage of TL is the significantly reduced variance in recall
compared to DT, indicating TL’s superior robustness in prediction, attributed
to its enhanced capacity for capturing predictive features. For $D=7$ day
events, the best mean performance with DT is a precision of 0.21 and recall of
0.48, achieved after 3 epochs. TL, however, achieves a precision of 0.22 and
recall of 0.76 at Epoch 6.

To ensure that these gains in recall are statistically significant, we
conducted a Wilcoxon signed-rank test [Conover (1999)]. Fig. 9 shows
histograms of the difference in precision and recall between direct training
and transfer learning. For example, each of the 10 values in histogram of
$D=5$ is defined for a specific train-test split, evaluated by subtracting the
mean precision (recall) of 10 randomly initialized TL models taken at Epoch 4
from the mean precision (recall) of 10 randomly initialized DT models taken at
Epoch 3. The spread here stems primarily from the fluctuation in 10 small-size
test sets, not uncertainty in the networks due to randomness in training. The
values for small-size test sets are taken at the same epoch of the best mean
performance.

The average recall with TL surpasses that of DT by 34% ($p=0.001$) for 5 day
events and by over 50% ($p=0.002$) for 7 day events. While there is not a
significant difference between the TL and DT precision, it is critical that
transfer learning was able to improve the recall without sacrificing
precision. One could easily inflate the recall by declaring more positive
cases, but without any skill, the precision would suffer and approach the
climatological rate.

Figure 8: Comparison of CNN forecast skill between direct training (DT, blue)
and transfer learning (TL, red). The top row shows the precision and the
bottom row, the recall, as a function of training epoch of DT (columns 1 and
3) and fine-tuning epoch of TL (columns 2 and 4). The black dashed line
indicates the climatological event rate $p$. The left two columns show the
results for $D=5$ (standard blocking events) and the right two columns show
the results for $D=7$ (longer blocking events). The shading shows a two-
standard deviation uncertainty bound, as detailed in the text. Figure 9:
Histograms of the performance gap between the best performing CNNs obtained
with transfer learning versus the best performing CNNs obtained with direct
training, for (left) precision and (right) recall. The top panels are for 5
day events and the lower panels are for 7 day events. “Best performing” was
determined by stopping the training procedure at the epoch when the best
overall balance between high precision and recall was achieved in the mean
(solid lines in Fig. 8). The 90:10 split yields 10 different CNN scores, and
the differences between pairs of TL and DT based CNNs, scored on the same test
split, are shown.

###  8.3 What has transfer learning learned?

When we show ERA5 events to CNNs first trained on the MM dataset, what exactly
is the CNN learning to improve the recall? For example, do the key
geographical regions and levels (Fig. 5) retain the same level of
significance? It is reasonable to expect that this might not be the case. In
the MM dataset, the duration of the Atlantic blockings could be related to
upstream flow, specifically to the structure of the wave train at the blocking
onset. The mechanism for blocking in the real world is more complicated, and
the correlated pattern may shift, intensify, and/or weaken. To address these
questions, we compare the SHAP values of the pre-trained CNNs when directly
applied to ERA5 (i.e., without fine-tuning step) to the SHAP values of the CNN
after 4 epochs of fine-tuning, as shown in row $a$ and row $b$ of Fig. 10. The
most evident difference after fine-tuning is a decrease in the amplitude of
the SHAP values. This is because the climatological rate of positive blocking
events in ERA5 is higher: almost 1/3 of nascent blocked states persist for 5
days in ERA5, compared to about 1/5 in MM. As the expected fraction of events
is larger, $\hat{q}(\mathbf{x})-\mathbb{E}[\hat{q}(\mathbf{x})]$ from equation
(13) will be smaller, and the SHAP value increments
$\phi_{i}(\hat{q},\mathbf{x})$ will tend to be smaller. It is the sum of the
SHAP values that build up the probability for a $Y=1$ prediction; for a more
likely event, one does not need to build up the probability as much, so fine-
tuning quickly adjusts the weights.

Figure 10: Rows 1 through 4 are composite maps of SHAP values,
$\overline{\phi}$, for geopotential height (200, 500, and 800 hPa), averaged
over true positive predictions of blocking events in ERA5 by the CNNs listed
below. This is the same quantity shown in Fig.5, but now applied to ERA5
events. Row $a$ shows $\overline{\phi}^{\text{MM}}$ for the pre-trained CNNs
before transfer learning (i.e., networks that have only learned from MM, but
applied to ERA5). Row $b$: $\overline{\phi}^{\text{TL}}$ of these pre-trained
CNNs after fine-tuning. Row $c$: $\overline{\phi}^{\text{DT}}$ of CNNs
directly trained on ERA5 dataset (i.e., networks that never saw the MM
events). Row $d$ shows the change in the SHAP values, $\Delta\phi$, between
the first two rows, after normalization as detailed in the text. This
quantifies the effect of transfer learning: positive values indicate that
information from the region became more important for the prediction, while
negative values indicate that anomalies in the region became less important
for prediction.

To assess the more subtle change in the relative contribution of each feature
on the predicted result after transfer learning, we show the difference in the
normalized composite map $\Delta\phi$ in row $d$ of Fig. 10. $\Delta\phi$ is
defined for each input $i$ (i.e., geopotential height $Z$ at a particular
latitude, longitude, and pressure level) by
$\Delta\phi_{i}\equiv\max\left(\frac{\overline{\phi}^{\text{TL}}_{i}}{\frac{1}{d}\sum_{j=1}^{d}\overline{\phi}^{\text{TL}}_{j}},0\right)-\max\left(\frac{\overline{\phi}_{i}}{\frac{1}{d}\sum_{j=1}^{d}\overline{\phi}_{j}},0\right)$.
The maximum function is used to avoid spurious negative SHAP values, which
should not arise in a composite of true positive events, as discussed in the
context of Fig. 5. The normalization makes the total integral of the SHAP
values the same for both cases, so that one can focus on where the CNN is
using information, as opposed to the overall reduction of the SHAP values
driven by the difference in rates.

The “normalized” SHAP values increase mainly in region 4 (the region right
around the block), and additionally over Quebec and Atlantic Canada, a region
less used for predictions with the MM model. The SHAP values decrease in a
relative sense over regions 1 (Florida and the Gulf), 2 (North Atlantic
Ocean), 3 (northeastern North America), and central North America. This change
in relative importance reveals a general de-emphasis of the regions farther
upstream and an increased emphasis on regions more immediately upstream. This
indicates that while it is still upstream information that is most important
for predicting a persistent blocking state in ERA5, the structure and westward
extension of the wave train has changed.

For further insight, we compare the SHAP value patterns with a more
traditional metric for understanding predictability: composite analysis. Fig.
6 shows composite maps of nascent blocks that evolve into persistent events in
the MM model and ERA5. Persistent blocks are associated with wave activity
south and west of the blocking region in both the model and reanalysis, but
the pattern shifts. The wave train in MM initially arcs westward before
turning southward, with a strong center of high pressure east of Florida,
while the wave train in ERA5 arcs more to southwest at first, then further
westward.

The SHAP values change over Quebec, capturing this shift in the wave train,
but overall the CNN seems to shift to more local information with transfer
learning. We speculated that the dry, quasi-geostrophic MM model
overemphasizes long range teleconnections. It only captures deformation scale
dynamics, and this only at low resolution, and so lacks smaller, local modes
of instability, e.g., instability associated with latent heat release due to
precipitation, present in our atmosphere. The CNN makes more use of these
local features when predicting the persistence of blocks, but still focuses on
the upstream flow, consistent with our intuition.

Finally, we contrast the feature importance analysis of the CNN with transfer
learning (Fig. 10 row $b$) to that of the CNNs trained only directly on the
ERA5 output (Fig. 10 row $c$). DT struggles to develop nuanced features with
limited data. The SHAP values with DT are also more barotropic than those with
TL. Moreover, in general, the SHAP values with TL capture finer details across
a wider spatial range, while the SHAP values with DT are more localized.
Geopotential height anomalies over Iceland, especially in the Z500 map, are
more emphasized for TL than DT. The same applies to upstream anomalies over
Florida and the Gulf of Mexico in the Z200 map. Additionally, the importance
of geopotential height anomalies over the Atlantic, immediately upstream of
the target region west of north Africa, is neglected in DT, though it appears
in TL. This is closely correlated to the blocking event prediction from the
ERA5 composite in Fig. 6, which does not show as strong composite Atlantic
anomaly as in the MM model.

In summary, the superiority of CNNs trained with transfer learning, as
compared to direct training, appears to lie in their ability to leverage
learned features from the pre-trained dataset, helping the network to take
advantage of information further upstream of the blocking region. In either
case the precision is modest: when the networks call an event, the rate of
success is at best 50% higher than a naïve climatological forecast. Pre-
training the network, however, has a significant impact on the recall,
increasing the forecast rate to capture more events without decreasing the
precision.

##  9 Conclusion

The impact of data-driven science on weather and climate science has grown
substantially in recent years. In this paper, we suggest two data-driven
approaches to help predict and understand atmospheric blocking events. First,
given sufficient data, convolutional neural networks (CNNs) are capable of
identifying subtle features that differentiate short-lived blocked states from
those that persist for an extended period. Moreover, XAI methods can provide
insight into what features matter most to this differentiation. Second,
transfer learning has the potential to make data-driven forecasts possible for
our atmosphere, making the most of the limited extreme events in the
observational record by leveraging insight from longer, albeit imperfect,
numerical simulations.

We began in a data-rich regime with the idealized Marshall-Molteni model,
showing that a CNN can accurately predict the persistence of North Atlantic
blocks in terms of both precision and recall. Leveraging XAI (SHAP feature
importance analysis), we identified crucial regions for the prediction of
persistent blocked states, given a nascent high-pressure anomaly. Our results
suggest that incorporation of both local and non-local features is important
for prediction skill.

To validate our discovery, we constructed a two-dimensional model that used
only upstream anomalies over Florida and the Gulf of Mexico, and anomalies
immediately upstream of the blocking region. The sparse model exhibited
precision significantly above the climatological rate and recall nearly as
good as the full CNN. It struggled, however, with false positives (and hence
exhibited low precision relative to the CNN) which could not be improved
within the log linear logistic regression framework. This suggests the CNN
learns non-trivial relations in the upstream flow, extending all the way to
the Pacific, to better discriminate between short-lived and long-lived blocks.

The challenge of conducting direct training on ERA5 data stems from the
paucity of available events. Small training and test datasets make training
and evaluation difficult. With the MM model, we observed a systematic
degradation in forecast skill when the training data was limited, particularly
for the recall statistic. Through transfer learning, we leverage the abundance
of data generated by simplified dynamical models to enhance real-world
forecasting. By pre-training a CNN on the MM model dataset and retraining the
deepest layer on the ERA5 dataset, the recall was improved by 34% compared to
a CNN developed with direct training alone for 5 day events, and over 50% for
more extreme 7 day events, without any loss of precision.

In addition to advancing predictive skill, transfer learning in combination
with SHAP analysis allowed us to compare the predictive features between
weather systems in ERA5 and the idealized quasigeostrophic model. The bottom
row of Fig. 6 reveals biases in the MM model, which appears overly dependent
on upstream features over Florida and the Gulf of Mexico relative to blocks in
ERA5. This approach provides a new angle of how a machine learning approach
could guide the diagnosis and quantification of model biases. This said, the
success of transfer learning results underscore the MM model’s ability,
despite its simplicity, to capture features that are important for predicting
the persistence of blocked states in the real world. We believe that greater
strides could be made by pre-training on a more advanced climate model, or
even hindcasts in the subseasonal-to-seasonal (S2S) data set [Vitart . (2017),
Finkel . (2023)].

The methods presented here are not limited to the context of blocking events,
and can be generalized to the study of other challenging natural phenomena,
especially in scenarios where data may be limited, and the potential
influencing factors are complex. An immediate future goal is to push further
on the physical and dynamical mechanisms that causes the differences in
prediction mechanisms for ERA5 and MM model. Another goal is to adapt the
present approach to investigate the statistical behavior and mechanisms for
the onset of the blocking events.

## Open Research Section

The code for computing SHAP values, transfer learning and producing plots is
publicly available in the Github repository at https://github.com/hzhang-
math/Blocking_SHAP_TL.

###### Acknowledgements.

We thank Valerio Lucarini and Andrey Gritsun for sharing their Marshall-
Molteni Fortran code. This work was supported by the Army Research Office,
grant number W911NF-22-2-0124. EPG acknowledges support from the National
Science Foundation through award OAC-2004572. J. F. is supported through the
MIT Climate Grand Challenge on Weather and Climate Extremes, and the Virtual
Earth Systems Research Institute (VESRI) at Schmidt Sciences.

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