# Inferring the past: a combined CNN–LSTM deep learning framework to fuse
satellites for historical inundation mapping

Jonathan Giezendanner
University of Arizona
87521 Tucson, USA
https://jgiezendanner.com Rohit Mukherjee
University of Arizona
87521 Tucson, USA
http://rohitmukherjee.space Matthew Purri
University of Arizona
87521 Tucson, USA
<EMAIL_ADDRESS>Mitchell Thomas
Columbia University
10964 Palisades, USA
<EMAIL_ADDRESS>Max Mauerman
Columbia University
10964 Palisades, USA
<EMAIL_ADDRESS>A.K.M. Saiful Islam
Bangladesh University of Engineering and Technology
1000 Dhaka, Bangladesh
<EMAIL_ADDRESS>Beth Tellman
University of Arizona
87521 Tucson, USA
<EMAIL_ADDRESS>
###### Abstract

Mapping floods using satellite data is crucial for managing and mitigating
flood risks. Satellite imagery enables rapid and accurate analysis of large
areas, providing critical information for emergency response and disaster
management. Historical flood data derived from satellite imagery can inform
long-term planning, risk management strategies, and insurance-related
decisions. The Sentinel-1 satellite is effective for flood detection, but for
longer time series, other satellites such as MODIS can be used in combination
with deep learning models to accurately identify and map past flood events. We
here develop a combined CNN–LSTM deep learning framework to fuse Sentinel-1
derived fractional flooded area with MODIS data in order to infer historical
floods over Bangladesh. The results show how our framework outperforms a CNN-
only approach and takes advantage of not only space, but also time in order to
predict the fractional inundated area. The model is applied to historical
MODIS data to infer the past 20 years of inundation extents over Bangladesh
and compared to a thresholding algorithm and a physical model. Our fusion
model outperforms both models in consistency and capacity to predict peak
inundation extents.

##  1 Introduction

Mapping floods supports building resilience and the adoption of mitigation
strategies to minimize losses in current and future events. Timely and
accurately identifying regions affected by flooding is crucial for emergency
responses and disaster management efforts. Satellite imagery provides a unique
vantage point for near real time mapping, especially for remote and
inaccessible regions.

Satellite data is an effective tool for mapping floods as it allows for rapid
and accurate analysis of large areas, even in remote or inaccessible regions.
With satellite imagery, flood-affected areas can be monitored and assessed in
near-real-time, providing critical information for emergency response and
disaster management efforts.

Additionally, satellite data can provide historical information that can be
used to analyze trends and patterns in flooding over time, helping to inform
long-term resiliency planning and risk management strategies. Historical flood
data derived from satellite imagery can be an essential resource for
understanding flood risk over time, particularly for insurance-related cases
[42]. By analysing historical data, insurers can better assess the likelihood
and potential severity of future flood events, which can inform risk
management decisions, pricing, and coverage options.

Traditional flood insurance for financial protection from flood damages
involves contracts where adjusters visit individual homes to assess damages
and determine corresponding payouts. However, traditional insurance is often
unavailable in regions of the world most vulnerable to floods, such as
Bangladesh. Flood index insurance is an affordable and scaleable alternative
whereby a data source, often from weather stations or satellites, generates a
“flood index” (_e.g_., inundated area) that should correlate with the damage
insured [41]. This makes insurance scalable in remote areas where traditional
insurance penetration is low. Public satellite sensors such as MODIS are
currently being piloted for flood index insurance in Southeast Asia [28] and
Colombia [16]. MODIS is used because its 20+ year history makes it possible to
estimate probability of exceedance needed to both price an insurance
instrument and determine an appropriate trigger for payout [42]. Additionally,
MODIS is available on a daily basis, and its derived 8-days composite image
provides a relatively cloud free image. Other products like Landsat, although
available for a longer time, do not provide such a consistent image series
(low revisit frequency, smaller extent, harmonisation issues between missions
and instrument failure amongst other challenges).

MODIS is an optical sensor with two images acquired daily, and is commonly
used for global and near real time flood mapping [43]. However, clouds often
obscure the view in optical sensors, making it difficult to map floods in many
cases. Furthermore, the coarse spatial resolution of 250m, 500m, and 1km of
MODIS makes it unsuitable for accurately extracting flooded features in
complex and heterogeneous landscapes such as urban regions. Together, the
coarse spatial resolution and cloud cover can lead to basis risk, where the
data for index insurance is not correlated with damage on the ground. Recent
evaluations of using MODIS for inundation maps in Bangladesh show low to no
correlation with reported damage, raising concerns that insurance product
based on this sensor could cause basis risk [45]. To overcome the limitations
of optical sensors, radar sensors [21, 19, 18] detecting water signals through
clouds consistently are preferable for insurance applications [46, 9].

Sentinel-1 is an ideal satellite for flood detection due to its radar sensor
that provides high-resolution (10 meters) images of the Earth’s surface, even
in cloudy conditions, making it suitable for accurate and consistent flood
extent mapping. However, it is only reliably available since 2017 and is
therefore unsuitable for historical flood mapping. For applications where
longer time series are needed, such as return period estimates that require at
least 20 years of data [5], other satellites have to be considered.

Fusing together Sentinel-1 with a longer MODIS time series is a potential
solution to overcome the disadvantages of either sensor for index-based
insurance and leverage the advantages of high resolution modern flood
detection and historical data availability.

We propose a deep learning model trained on Sentinel-1 derived fractional
inundated area to predict historical MODIS satellite data. We utilize a
Convolutional Neural – Long Short-Term Memory Network (CNN–LSTM) to take
advantage of both the spatial and temporal feature representation of
inundation and compare it to a traditional Convolutional Neural Network (CNN).
We apply the CNN–LSTM model from 2000 - 2021 over all of Bangladesh at 500m
resolution, to demonstrate the potential of our fusion model to be used in
index insurance pricing applications.

##  2 Methods

We train a deep learning fusion model (Fig. 1) to regress fractional inundated
area based on an 8-day MODIS composite of satellite imagery and static
landscape features (elevation, slope and height above nearest drainage
(HAND)). The inundated labels are derived from 10 meters resolution Sentinel-1
imagery. The fractional inundated area is the percentage of area covered with
water.

Figure 1:  Modelling framework: The target flood maps are generated from
Sentinel-1 data. Water is detected with a dynamic thresholding algorithm
tailored to Bangladesh and then upscaled (averaged) to MODIS resolution. The
fusion model uses hydrologically relevant indices (elevation, slope, and
height above nearest drainage (HAND)) and a time series of 10 MODIS Terra
8-day composite images to regress the target fractional inundated area.

Our proposed deep learning framework combines the spatial encoding ability of
CNNs with the temporal encoding of LSTMs to form a CNN–LSTM (not to be
confused with Convolutional LSTMs [7]). CNNs are commonly used in computer
vision applications [26, 49, 2], and numerous applications show how they can
be used in the context of satellite based flood detection [31, 36, 3, 6].
LSTMs on the other hand have proven particularly powerful in sequential time
series applications [1]. In hydrology in particular, LSTMs have been applied
in the context of rainfall-runoff predictions [14, 23, 15, 13, 30, 24] and
flood forecasting [25, 11, 37]. Our motivation for using LSTMs over other ways
of capturing time (_e.g_., transformers, stacked image time series as CNN
features), is physically informed. LSTMs conserve the sequence of the time
series, whereas transformers reshuffle the pixel stack [22], and CNNs
arbitrarily link the features together. LSTMs should thus capture the physical
dynamic of the inundation in time. Additionally, transformers are very data
hungry [50], especially when training them on a problem that is very different
from the problem they were designed for [34], which makes their use
unreasonable for this proof of concept paper. Future work should explore these
alternatives. CNN-LSTMs have been applied in various remote sensing contexts
[20, 40, 47], but, although applications of CNN–LSTMs exist in hydrology [48,
27], to the best of our knowledge, CNN–LSTMs have not been used to regress
satellite based inundation estimates, where the time series of satellite
imagery is the main driver of the model.

###  2.1 Data

Figure 2: Example of data collected and used for this modelling framework: a)
example of target fractional inundated area for a chosen week (week of August
9, 2021) upscaled from binary Sentinel-1 water maps, along a False Color
Composite (FCC: near-infrared (NIR), red, blue) image of MODIS for the same
date. The black chips indicate a chip where at least one pixel is missing for
this week at this place (if more than one pixel is missing, the chip is
discarded, due to missing sentinel overpass during the 8 days). Note that the
shown data overlap between Sentinel-1 and MODIS is specific for the chosen
week, but varies in time. b) zoom in on a single 32 x 32 chip with MODIS FCC
on the left, and Fractional Inundated Area on the right. The resolution for
both a) and b) is 500 meters.

####  2.1.1 Target Fractional Inundated Area Maps

We generate fractional flooded area at a 500 meters resolution. The target
resolution is chosen to match MODIS Terra’s 500 meter resolution. We first
generate binary flood maps at the Sentinel-1 native resolution (10 m). We
apply a dynamic thresholding method derived from DeVries _et al_. [10]
specifically developed and verified over Bangladesh [45].

This Sentinel-1 algorithm identifies a dry-season baseline from a set of VV
and VH backscatter images taken during periods of low soil moisture. With
these dry season baseline images, the pixel-wise mean and standard deviation
images are calculated. From these, a z-score image (number of standard
deviations from the mean, unitless) is computed for each Sentinel-1 image in
the historical record. Surface water is defined as any pixel with a Z-score
below -2 in either VV or VH bands, or by VH backscatter below -27 dB. The map
is then smoothed by a majority filter at 30 meters to remove SAR noise. The
final result of this algorithm is a binary surface water map at 10 meters
resolution available every $\sim$5 days on average.

Based on this binary inundation map, we calculate the fractional inundated
area by upsampling the 10 meter resolution map to 500 meters. Each 500 m x 500
m grid cell thus represents the average number of pixels where water is
detected over 2,500 pixels. We extract images at a chip size of 32x32 to allow
the network to learn spatial relationships, and permit a reasonable overlap
between the Sentinel-1 extent and MODIS resolution. Larger chips would benefit
the CNN and allow for a better spatial context, but matching large MODIS and
Sentinel-1 extents appears very difficult. Each 32x32 image patch contains
information from 1,600 x 1,600 Sentinel-1 pixels or covers an area of 256 km2.

We extract all 32 x 32, 500 meters resolution, chips for the years 2017 to
2021 (5 years) in a regular grid over Bangladesh (_cf_. Fig. 2). With this
method, we generate a total of 372,263 target chips covering all of
Bangladesh, over 5 years, with a 2 - 10 days interval between images
(Sentinel-1 revisitation time). Out of those, 150,946 had no missing
Sentinel-1 data (complete chip) and were used in the context of this study.

####  2.1.2 Features

We use a combination of dynamic (satellite remote sensing data) and static
(derived from digital elevation models) features to regress the target
fractional inundated area, for a total of 10 features. Each feature stack is
stored in a 32 x 32, 500 meters resolution raster chip.

Dynamic Features: For each chip, we select the 8-Day Global 500 meters MODIS
Terra Surface Reflectance image (MOD09A1.061) that overlaps the Sentinel-1
collection date. We use the 7 first bands (red, near-infrared (NIR), blue,
green, SWIR 1-3), which are min-max normalised between 0 and 1 based on the
sensor’s data range (-100, 16,000). We also extract the 9 previous MODIS
images, to be used in the LSTM (see below), for a total of 10 images for each
Sentinel-1 date. This period of approximately 2.5 months covers a majority of
the inundation peak in Bangladesh. Future work should consider extending this
time series to a full year of data to capture yearly dynamics, although this
would come to the cost of high computational resources.

Static Features: Hydrologically relevant features and indexes have been shown
to improve model performance in flood related applications [35]. We here
complement the MODIS data with elevation, slope and height above nearest
drainage (HAND [32]). Elevation is given as the re-sampled
(downsampled/upscsaled using the mean from 30 meters to 500 meters) elevation
in each 500m pixel derived from the Digital Elevation Model (DEM) product
FABDEM [17]. The output value is min-max normalised between 0 and 1 with a
reasonable elevation range for Bangladesh (0, 100). Slope is first calculate
based on the FABDEM product, and then re-sampled (mean) and expressed as its
hyper tangent to bound the values between 0 and 1. HAND is extracted from
Yamazaki _et al_. [51], the logarithm is applied to reduce the data spread,
and after re-sampling (mean) a min-max normalisation is applied to bound the
values between 0 and 1.

###  2.2 Model

As mentioned above, we propose a deep learning framework that combines CNNs
and LSTMs sequentially (_cf_. Fig. 3). For the particular implementation of
CNN–LSTM we propose, for each time step, the features are initially passed
through the first CNN (CNN A in Fig. 3) to summarize the multi-band images
into a single band. This output generates, for each pixel in the 32 x 32 chip,
a time series of spatially contextualised data. Note that, although each group
of features (time step) is passed through CNN A independently, CNN A is the
same network across groups and the weights are shared. CNN A is comprised of 4
convolution layers (Conv2D $\rightarrow$ BatchNorm2D $\rightarrow$ ReLU) that
inflates the number of layers from 10 to 128, and one final Conv2D layer which
brings it down to 1 layer just before the LSTM input. The LSTM is then fed,
for each pixel, a time series consisting of the 9 encoded values.

Figure 3:  Schematic representation of the deep learning architecture: A CNN
(A) is run on each time step of a 10 input MODIS images and hydrologically
relevant indices, for a total of 10 input bands per time step. The CNN outputs
of the 9 time steps before time $t$ is passed through a LSTM network, where
each of the 1’024 pixels is considered as an independent time series. The
output of the LSTM is passed through an inverse convolution layer, and
,concatenated with the output of CNN A at time step $t$, passed through a
second CNN (B). Finally, a sigmoid is applied to bound the prediction between
0 and 1. Note that CNN A is the same network for all time steps and that the
weights are shared, whereas CNN B is different. The CNN’s architecture is
described in further detail in the text. Also note that the light gray numbers
in the figure denote the different layer’s dimensions.

The output of the LSTM is converted from a one-dimensional (1 x 1024) layer to
a square image (32 x 32) with a single inverse convolutional layer, and then
merged with the CNN output of the target time, to then pass through one last
CNN (CNN B in Fig. 3). CNN B is composed of a single Conv2D layer. Finally,
the output is passed through a sigmoid activation layer to bound the
prediction between 0 and 1.

Both CNNs use a convolution scheme with a kernel size of 3, reflected padding,
and a stride of 1. The input image size is retained throughout the convolution
layers (32 x 32). This choice was motivated by the size of the image.

####  2.2.1 Training and Cross-Validation

The model is trained with a leave-one-out cross-validation scheme, where each
of the 5 years is withheld for validation one at a time, _i.e_., the model is
trained 5 times with varying number of chips (_cf_. Tab. 1).

Leave-out Year | Number of chips
---|---
2017 | 24,845
2018 | 21,558
2019 | 17,279
2020 | 43,815
2021 | 43,449
Total | 150,946
Table 1:  Number of complete chips collected over Bangladesh for each year.
Only complete chips are considered here, chips where parts of the Sentinel-1
data was missing are not considered.

For each cross-validated year, we train the model on 20 epochs with a learning
rate of 10${}^{\text{-3}}$, then 5 epochs with a learning rate of
10${}^{\text{-4}}$ and finally 5 epochs with a learning rate of
10${}^{\text{-5}}$. We use root mean square error (RMSE) for the loss function
and the Ranger optimiser. All models reached convergence of loss after the 30
epochs. We use rotation, flip and dihedral image transformation to augment the
data.

####  2.2.2 Model Baseline

In order to assess the performance of the proposed model, we compare it to a
CNN-only model without temporal data. We train a CNN with the same
architecture as CNN A (_cf_. Fig. 3) with the same cross-validation and
training scheme described in Sec. 2.2.1 and compare it to the combined
CNN–LSTM network.

##  3 Results

###  3.1 Capturing the Temporal Evolution in the Deep Learning Model

It is important to assess whether including temporal information improves the
prediction of fractional inundated area or if the spatial context is
sufficient. The CNN model is trained with the same set of hyperparameters and
cross-validation scheme as the CNN–LSTM, and the R${}^{\text{2}}$ values are
reported in Tab. 2.

Year | CNN | CNN–LSTM
---|---|---
2017 | 0.62 | 0.66
2018 | 0.55 | 0.62
2019 | 0.55 | 0.67
2020 | 0.63 | 0.72
2021 | 0.60 | 0.67
Table 2: Comparison between R${}^{\text{2}}$ values calculated on leave-out
year for a pure CNN model and the deep learning CNN–LSTM model presented here.
The R${}^{\text{2}}$ values are calculated over all Bangladesh. The CNN–LSTM
model consistently outperforms the pure CNN model on all years.

The CNN–LSTM consistently outperforms the pure CNN network on all leave-out
years. Fig. 4 compares, for a selected chip, the target, baseline and proposed
model’s temporal evolution of the fractional inundated area, and, at peak
monsoon season, the inundation extent.

Figure 4:  Example for a single chip (red square on the map insert (f)) of the
different performance of validation data of pure CNN (chip b and black line in
e) compared to the CNN–LSTM model(chip c and blue line in e), with the
Sentinel-1 derived target (chip d and red line in e) as comparison. The model
with the 2020 leave-out year is used in this example. The three chips (a) show
the fractional inundated area at the flood peak for the monsoon season in year
2020 (at the black dashed line in the time series (e)). The CNN–LSTM model
outperforms the pure CNN model both in temporal and spatial patterns.

The spatial patterns and inundation intensity at peak are closer matched by
the CNN–LSTM than the pure CNN model. The temporal trend as well is closely
matched by the CNN–LSTM, whereas the pure CNN misses peaks in spring, and
grossly underestimates the peak inundation during the monsoon.

Figure 6:  Time series of median (with confidence interval) fractional
inundated area over Bangladesh for the “worst” (2018) and “best” (2020) cross-
validation models (“best” and “worst” in terms of R${}^{\text{2}}$). The
hatched area represents the data used for training, and the blank are for the
validation on the cross-validation scheme.  Figure 5:  Statistical summary of
the cross-validation scheme for training (white) and testing(grey). The years
represent the year left out in the cross-validation scheme (_i.e_. all other
years are used for training). The figure shows four statistics:
R${}^{\text{2}}$, slope, Spearman rank-order correlation coefficient
($\rho_{s}$) and the root mean square error (RMSE). The training id displayed
along side the testing to root out over-fitting. The average testing scores
over all years are 0.66 (R${}^{\text{2}}$), 0.68 (slope), 0.69 ($\rho_{s}$)
and 0.1 (RMSE).

Overall, the CNN-LSTM model seems to capture the temporal trends, whereas the
pure CNN misses the peaks and fails to reproduce the lows during the dry
season.

Figure 7:  Cumulative per pixel error (difference between model and target
output) averaged over the cross-validated years for the whole country. The
blue tint indicates the model under-estimating the inundation area, and the
red tint an over-estimation. Note that chips that appeared less than 5 times
over the 5 years were removed from this figure to avoid visual clutter.

###  3.2 CNN-LSTM Model Performance

Fig. 5 shows performance metrics of the CNN–LSTM aggregated over Bangladesh
for the training and testing sets, for all the leave-out years. The statistics
are consistent amongst cross-validated years. The R${}^{\text{2}}$ (on average
0.66), slope (on average 0.68) and Spearman’s $\rho_{s}$ (on average 0.69) all
depict the model as explaining a large portion of the observed variance, and
follow a positive, linear and monotonic relation between the observed and
predicted inundation values. The model deviates from the mean prediction with
a RMSE of 0.1, _i.e_. on average, the model is within 10% of the observed
inundation value.

The training is displayed along the testing to assess whether the model is
overfitting. For most years this does not seem to be the case, but 2018 could
display signs of overfitting. To assess the difference between model fitting
and possible overfitting, the “worst” and “best” year (in terms of testing
R${}^{\text{2}}$), aggregated over the country, are shown in Fig. 6. In this
figure, comparing 2018 when the year is excluded (“worst”) and included
(“best”) from the training dataset, the model seems to slightly overfit when
the year is excluded. In the “worst” year, the model appears to overestimate
the flood peaks, and underestimate the dry season. The same does not occur
when 2020 is excluded (the “best” year).

Generally, the model reproduces the peaks during the monsoon season, and
valleys during the dry season with a high fidelity.

Fig. 7 shows the error between the model output and the target (model -
target) fractional inundated area. A majority of the error seems to be located
around Bangladesh’s major rivers leading to the ocean, where the delineation
of water and land is rendered more difficult by the river’s meanders. The
model does not seem to be systematically over or underestimating the
inundation extent, but rather simply unable to accurately model the intricate
and evolving river extents. Additionally, close to the ocean, in the southern
part of the country, as well in the more hilly regions (west side), the model
underestimates the fractional inundation area. A majority of the territory
shows a negligible average error.

###  3.3 Inference for Historical Inundation

The goal of the model is to infer the historical inundation extents over
Bangladesh. Fig. 8 shows the inference over 20 years of MODIS data of the
ensemble of cross-validated models.

Figure 8:  Inference run on 20 years of MODIS data (ensemble of the 5 cross-
validated models). Example of inundation extent for the monsoon flood peak in
2003 over Bangladesh and historical time series, with the monsoon season
highlighted in red. Note that spring of year 2002 is removed from the time
series due to an error in the MODIS data that is propagating through the model
with the LSTM.

The raster image shows an example of inundation at the peak for the year 2003
(July 7${}^{\text{th}}$). The irrigation inundation peaks occurring during
spring are separated from the monsoon inundation peaks.

Fig. 9 shows the annual maximum inundation extent extracted from the fusion
model, and the output of two other algorithms for comparison: the global flood
database algorithm (GFD [44]) and the Bangladesh Flood Forecast Warning Center
(FFWC) prediction (based on the MIKE11 Model from the Danish Hydraulic
Institute (DHI)).

Figure 9:  Annual monsoon maximal inundation extent extracted from the
inferred ensemble CNN–LSTM model (red line, _cf_. Fig. 8 for the full time
series), Global Flood Database (GFD [44]) algorithm and the Bangladesh Flood
Forecast Warning Center (FFWC) predictions (based on the MIKE11 Model from the
Danish Hydraulic Institute (DHI)).

GFD is based on MODIS daily 250 meters resolution images, where each pixel
gets a binary inundation prediction. The model output is aggregated to 500
meters, the fractional inundated area is thus the average over 4 pixels, then
aggregated over all of Bangladesh (mean). Mike11’s model is a physically based
model, the accuracy of which, like all flood models, is affected by large
uncertainties, mainly related to the DEM [4]. The DEM in Bangladesh is
particularly inaccurate in part because of poor interpolation techniques [33],
as well as dynamic seasonal changes to the elevation due to the monsoon
filling the water table [38] and subsidence due to sediment dynamics and
climate change [39]. The flood map from such a model is not ground truth, but
provides a useful reference to compare to remotely sensed flood maps as an
independent data source when no ground truth is available, an approach used in
other remote sensing flood studies [8]. Comparisons with MIKE11 and Sentinel-1
flood maps, as well as conversations with the Bangladesh Flood Forecasting and
Warning Center that calibrates the model and provided the data, indicate the
Mike11 model likely over predicts flood peaks. Future work should validate the
extracted inundation maps with recent high resolution remotely sensed data,
_e.g_., Planetscope.

For all models, the inundation peaks are extracted during the monsoon season.
The monsoon season’s inundation peak are uncontrolled (as opposed to
controlled during the irrigation season), and of interest for insurance
purposes, but for model optimisation it is of interest to use all available
inundation observations, _i.e_., include the irrigation season as well.

The fusion model replicates all major flood peaks shown in MIKE11 (2004, 2007,
2011 and 2017), whereas the GFD seems to miss some of them.

In terms of baseline, the fusion algorithm and GFD have, on average, $\sim$0.2
fractional inundated area difference. The GFD algorithm is very conservative
in terms of water detection, whereas the fusion model here aims at replicating
the 10 meters observed fractional flooded area.

##  4 Discussion

The temporal features in the deep learning network appear to strongly improve
identifying inundation and filling in gaps under clouds, whereas the pure CNN
approach seems to struggle in some cases. The CNN–LSTM likely picks up the
“hydrograph” signature of the inundation, _i.e_. comprehend the rising and
falling of the inundation level and predict whether the water is receding or
not in the next time step. Most importantly, in 2020, which is the highest
inundation year in the time series available with Sentinel-1, the LSTM makes
the largest contribution, increasing accuracy from 0.63 to 0.72, _i.e_. an
additional 10 percent. Large flood years are especially critical to predict
correctly as, for example, in index based insurance applications, large flood
years are the years where payouts have to be made by insurers.

Model performance is highest in places with seasonal inundation, where a
temporal feature can be learned. For example in Sylhet, a region in the north
east of the country, the seasonality of the lakes filling and recessing are
relatively predictable. Here is where the LSTM shines. In contrast, coastal
areas, areas with steep topography (_i.e_. mountains) and areas prone to storm
surges appear to be the least well represented. This could either be because
in these areas, a slow onset of the event, required to generate a temporal
feature to fill in the flood gaps, is missing, or because Sentinel-1 (which
has a 2 and 10 revisiting period) does not capture storm surge events well
along the coast, and fast moving floods in mountainous regions.

Around rivers, where the meanders have changed multiple times over the years,
and the river course isn’t predictable from one year to the other, the LSTM
struggles. In these cases, a more complicated CNN could learn from more
convoluted patterns, and could be coupled with the LSTM framework.

Overall, the CNN–LSTM model proposed here shows an impressive improvement over
the CNN-only approach, and proves essential to capture inundation dynamics in
time, in regions where the CNN’s pattern recognition isn’t enough and learning
from the past is required.

##  5 Conclusion

The recent failure of Sentinel-1B [12], as well as the planned decommission of
MODIS in mid-February 2023 [29], have made it apparent that products derived
from the fusion between different satellite sources are highly important to
reduce the dependency of applications to a single product. Although this work
is based on Sentinel-1 and MODIS, future work will bridge the gap between
MODIS and VIIRS, MODIS’s successor, as 11 years of overlapping data exists
between the two. Additionally, the proposed framework can be operationalised
to predict the inundation area when and where Sentinel-1 isn’t available,
reducing the dependency on Sentinel-1 for, _e.g_., insurance payouts.

While impressive, the historical MODIS record of 20 years is still relatively
short in terms of hydrological return period estimates. Future work could take
advantage of the impressive Advanced Very High Resolution Radiometer (AVHRR)
record available since 1979 and combine it with historical Landsat data to
create a harmonised dataset and provide longer inundation return period
estimates, although capturing the peak inundation extents would remain a
challenge with these products. Additionally, other SAR data, such as, _e.g_.,
ENVISAT or PALSAR, could be considered to extend the observed inundation time
series.

The proposed model architecture performs reasonably well given the complexity
of the task, but could certainly be improved. The advantage of the proposed
approach, treating CNNs and LSTMs sequentially, means that any CNN
architecture could replace the one proposed. Future work should compare the
model to a single LSTM to understand the benefits of combining LSTMs with
CNNs. Additionally, future work should explore other architectures, and
compare them to alternative solutions to incorporate time (transformers, stack
time stamps as features for a CNN).

Overall, the parametric insurance, along with green climate funds, is
expanding in countries like Bangladesh. To support these activities, research
like the one proposed here is not only highly relevant, but crucial to allow
for realistic and fair deals for all parties involved.

## Code and Data Availability

The code and data for this paper can both be found on GitHub:
https://github.com/GieziJo/cvpr23-earthvision-CNN-LSTM-Inundation

## Acknowledgements

This work is undertaken as part of the NASA New (Early Career) Investigators
(NIP) Program (80NSSC21K1044). Additionally, funding was provided by the
Syngenta Foundation for Sustainable Agriculture InsuResilience project and was
supported as part of the Columbia World Project, ACToday, Columbia University
in the City of New York.

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††thanks: Corresponding authors:
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<EMAIL_ADDRESS>Corresponding authors:
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<EMAIL_ADDRESS>
# Observation of Giant Band Splitting in Altermagnetic MnTe

T. Osumi  Department of Physics, Graduate School of Science, Tohoku
University, Sendai 980-8578, Japan  S. Souma  Center for Science and
Innovation in Spintronics (CSIS), Tohoku University, Sendai 980-8577, Japan
Advanced Institute for Materials Research (WPI-AIMR), Tohoku University,
Sendai 980-8577, Japan  T. Aoyama  Department of Physics, Graduate School of
Science, Tohoku University, Sendai 980-8578, Japan  K. Yamauchi  Center for
Spintronics Research Network (CSRN), Osaka University, Toyonaka, Osaka
560–8531, Japan  A. Honma  Department of Physics, Graduate School of Science,
Tohoku University, Sendai 980-8578, Japan  K. Nakayama  Department of Physics,
Graduate School of Science, Tohoku University, Sendai 980-8578, Japan  T.
Takahashi  Department of Physics, Graduate School of Science, Tohoku
University, Sendai 980-8578, Japan  K. Ohgushi  Department of Physics,
Graduate School of Science, Tohoku University, Sendai 980-8578, Japan  T. Sato
Department of Physics, Graduate School of Science, Tohoku University, Sendai
980-8578, Japan  Center for Science and Innovation in Spintronics (CSIS),
Tohoku University, Sendai 980-8577, Japan  Advanced Institute for Materials
Research (WPI-AIMR), Tohoku University, Sendai 980-8577, Japan  International
Center for Synchrotron Radiation Innovation Smart (SRIS), Tohoku University,
Sendai 980-8577, Japan  Mathematical Science Center for Co-creative Society
(MathCCS), Tohoku University, Sendai 980-8577, Japan

###### Abstract

We performed angle-resolved photoemission spectroscopy (ARPES) on hexagonal
MnTe, a candidate for an altermagnet with a high critical temperature
($T_{\mathrm{N}}=307\textrm{ K}$). By utilizing photon-energy-tunable ARPES in
combination with first-principles calculations, we found that the band
structure in the antiferromagnetic phase exhibits a strongly anisotropic band-
splitting associated with the time-reversal-symmetry breaking, providing the
first direct experimental evidence for the altermagnetic band-splitting. The
magnitude of the splitting reaches 0.8 eV at non-high-symmetry momentum
points, which is much larger than the spin-orbit gap of $\sim$0.3 eV along the
$\Gamma\textrm{K}$ high-symmetry cut. The present result paves the pathway
toward realizing exotic physical properties associated with the altermagnetic
spin-splitting.

A new type of magnetism called “altermagnetism” is attracting a great deal of
attention because it is regarded as the third type of magnetism neither
categorized into conventional ferromagnetism nor antiferromagnetism Šmejkal
_et al._ (2022a, b). In ferromagnets, the energy bands are spin split due to
the breaking of time-reversal symmetry (TRS). In colinear antiferromagnets,
the energy bands are generally spin degenerate because of the antiparallel
configuration of magnetic moment and the zero net magnetization. Intriguingly,
altermagnets are different from these two magnets; despite the zero
magnetization as in antiferromagnets, the bands in altermagnets are spin split
due to the TRS breaking as in ferromagnets. This band splitting is guaranteed
by the existence of opposite-spin sublattices connected by the crystal-
rotation symmetries Šmejkal  _et al._ (2022a, b); Hayami  _et al._ (2019);
Yuan  _et al._ (2020); Ma  _et al._ (2021). In contrast to ferromagnets where
the band splitting is typically isotropic in the momentum (k) space,
altermagnets exhibit an anitotropic band-splitting. This is because the sign
of band splitting must be reversed between the positive and negative k’s to
meet the requirement from the zero net magnetization, leading to band
splitting with nodes on high-symmetry lines in the bulk Brillouin zone (BZ)
under the absence of the spin-orbit coupling (SOC). Although an anisotropic
band-splitting appears in nonmagnets as a consequence of space-inversion
symmetry breaking and SOC Rashba (1960), the altermagnetic band-splitting is
distinct from it, because the splitting appears even without SOC Šmejkal  _et
al._ (2022a, b); Hayami  _et al._ (2019); Yuan  _et al._ (2020); Ma  _et al._
(2021); Hayami  _et al._ (2020); Yuan  _et al._ (2021). Besides the
fundamental interest in the unconventional band splitting, the altermagnetic
band-splitting is predicted to host a variety of exotic quantum phenomena such
as the anomalous Hall effect, spin current generation, spin splitter torque,
magnetic circular dichroism, crystal Nernst effect, and piezomagnetic effect
Šmejkal  _et al._ (2022a, b); Ma  _et al._ (2021); Naka  _et al._ (2019);
Šmejkal  _et al._ (2020); Naka  _et al._ (2021); Mazin  _et al._ (2021);
Nguyen and Yamauchi (2023); Shao  _et al._ (2023); Rooj  _et al._ ; Mazin
(2023); Zhou  _et al._ . Some of them were recently verified experimentally
Chen  _et al._ (2021); Bai  _et al._ (2022); Karube  _et al._ (2022); Bose
_et al._ (2022); Bai  _et al._ (2022); Gonzalez Betancourt  _et al._ (2023);
Bai  _et al._ (2023); Hariki  _et al._ ; Aoyama and Ohgushi ; Fedchenko  _et
al._ . However, the direct experimental evidence for the altermagnetic band-
splitting, which would serve as a base to understand the exotic phenomena of
altermagnets, is still missing. It is thus urgently required to experimentally
verify the altermagnetic band-splitting to further advance this exciting
field.

Here we focus on hexagonal MnTe which is suited to observe the possible large
spin splitting Šmejkal  _et al._ (2022a); Rooj  _et al._ ; Mazin (2023) due to
its high Néel temperature ($T_{\mathrm{N}}=307\textrm{ K}$) Squire (1939);
Komatsubara  _et al._ (1963); Szuszkiewicz  _et al._ (2005); Efrem D’Sa  _et
al._ (2005). MnTe crystallizes in the NiAs-type structure (space group:
$P6_{3}/mmc$) consisting of alternately stacked Mn and Te planes characterized
by a simple hexagonal Mn lattice and a hexagonal closed packed Te lattice with
the AB stacking [for crystal structure and bulk BZ, see Figs. 1(a) and 1(b),
respectively] Juza  _et al._ (1956). In the antiferromagnetic (AF) phase, the
Mn2+ ions placed at the center of the Te octahedra are ferromagnetically
aligned within the same Mn plane with the magnetic moment along the (0001)
plane, whereas the Mn ions between adjacent Mn planes are
antiferromagnetically coupled Komatsubara  _et al._ (1963); Szuszkiewicz  _et
al._ (2005); Kriegner  _et al._ (2017). Intriguingly, opposite-spin
sublattices essential for the altermagnetism are realized in the Mn sites
because the octahedral coordination is inequivalent between the AB- and BA-
stacked Mn sites. This situation does not allow spin reversal with a single
symmetry operation and leads to the breaking of global TRS. The DFT (density-
functional-theory) calculations predict a giant spin splitting Šmejkal  _et
al._ (2022a); Rooj  _et al._ ; Mazin (2023); Faria Junior  _et al._ (2023),
whereas such a splitting has yet to be clarified.

In this article, we report an ARPES study of MnTe bulk single crystal and show
a direct evidence for the altermagnetic band-splitting. This was enabled by
the utilization of photon-energy tunable micro-focused ARPES in combination
with DFT calculations. We found that the magnitude of altermagnetic band-
splitting strongly depends on the $k$ location in the 3D BZ, and is also
affected by the SOC. We discuss implications of the present results in
relation to other experiments on altermagnets.

First, we discuss the overall band structure of MnTe in the AF phase. Details
of experiments and calculations are provided in Note 1 of Supplemental
Material (SM). We carried out $h\nu$-dependent ARPES measurements and
estimated the $k_{z}$ value (see Fig. S2 of SM). The contour map of ARPES
intensity at the Fermi level ($E_{\textrm{F}}$) for the AHL plane obtained
with $h\nu=117\textrm{ eV}$ in Fig. 1(c) signifies a bright spot centered at
each A point that follows the periodicity of hexagonal BZ. The band dispersion
along the AH high-symmetry cut shown in Fig. 1(d) signifies a dispersive hole
band (white arrow) at the A point, together with a shallower hole band outside
this band (red arrow). These bands produce a spot centered at the A point in
Fig. 1(c). One can also recognize in Fig. 1(d) a broad feature at the binding
energy ($E_{\textrm{B}}$) of $\sim$2 eV around the A point (yellow arrow) and
another broad feature which weakly disperses upward on approaching the H point
(blue arrow). We carried out the ARPES measurements also for the
$\Gamma\textrm{KM}$ plane with $h\nu=80\textrm{ eV}$ and found that the band
dispersion [Fig. 1(e)] and the energy contour [Fig. 1(f)] are markedly
different. As shown in Fig. 1(e), instead of a simple hole band, the top of
the valence band along the $\Gamma\textrm{K}$ cut shows an M-shaped dispersion
with its maxima slightly away from the $\Gamma$ point (green arrow) producing
six petals surrounding the $\Gamma$ point in Fig. 1(f).

To discuss the band character in more detail, we show in Fig. 1(g) the energy
distribution curves (EDCs) around $E_{\textrm{F}}$ along the
$\Gamma\textrm{K}$ cut. The result signifies the band splitting with a maximum
value of $\sim$0.3 eV at slightly away from the $\Gamma$ point. This band
splitting is also seen in Fig. 1(e) (green and purple arrows) and associated
with SOC. We found that the DFT calculations with a spin configuration and
experimental geometry of either ($S_{a{\parallel}},\Gamma\textrm{K}_{2}$) or
($S_{a{\perp}},\Gamma\textrm{K}_{1}$) setting reproduce well the experimental
results (for details, see Supplemental Note 3). Since these two cases show
almost identical band structure, we hereafter adopt the former case just for
our convenience.

As shown in Fig. 1(h), the experimental band structure along the AH cut is
well reproduced by the band calculation. In particular, the steeper inner hole
band (white arrow) and the outer hole band (red arrow), as well as the feature
at $E_{\textrm{B}}\sim$ 1.3–2.0 eV (yellow and blue arrows) seen by ARPES are
well reproduced by the calculation. These bands are attributed to the
hybridized Te-5$p$ and Mn-3$d$ $t_{2g}$ orbitals. We also found an overall
agreement of the band dispersion between the experiment and calculation along
the $\Gamma\textrm{K}$ cut [Fig. 1(i)]. The valence-band top is experimentally
located at the A point around $E_{\textrm{F}}$, in line with the calculation,
signifying the semiconducting nature of MnTe. Taking into account the total
band-gap size of $\sim$1.3 eV Allen  _et al._ (1977); Ferrer-Roca  _et al._
(2000), it is suggested that MnTe is a $p$-type semiconductor, consistent with
the transport measurement Wasscher ; Kriegner  _et al._ (2016).

Next we focus on the observation of the altermagnetic band-splitting. Since
the altermagnetic band-splitting was predicted to have nodes along high-
symmetry lines under negligible SOC, it is necessary to carry out ARPES
measurements along non-high-symmetry cuts to observe the altermagnetic band-
splitting. Non-high-symmetry $\Gamma\textrm{L}$ cut [see Fig. 2(a)] is suited
for this sake because the DFT calculations identified a sizable splitting
along this cut Šmejkal  _et al._ (2022a); Rooj  _et al._ . On the other hand,
the ARPES measurement needs a special care because $h\nu$ must be changed at
every step of momentum to correctly sweep k along the $\Gamma\textrm{L}$ cut.
Since the altermagnetic band-splitting can show up even without SOC, we first
explain the calculated band structure without SOC shown in Fig. 2(b). One can
recognize the spin-split bands in a whole ($E,k$) region. Although many bands
are observed in this region, one can easily identify a spin-split partner of
each band, thanks to the band degeneracy along high-symmetry lines. For
example, bands labelled 1 and 2 (sequentially labeled from the highest
occupied band around $\Gamma$) are spin-split partners to each other because
they degenerate at the $\Gamma$ and L points. The altermagnetic band-splitting
associated with these bands are strongly anisotropic; it reaches $\sim$0.8 eV
at around $k_{z}\sim 0.5\pi$, whereas it is zero at the $\Gamma$ ($k_{z}=0$)
and L ($k_{z}=\pi$) points. Similarly, the altermagnetic band-splitting for
bands 3 and 4 has nodes at the $\Gamma$ and L points and takes a maximum at
$\sim$1.0 eV around $k_{z}\sim 0.7\pi$. The band 2 intersects the band 3 due
to the large spin splitting. Such a large splitting is associated with the
$\mathcal{PT}$ (space- and time-reversal) symmetry breaking of the Te site and
strong hybridization between Te-5$p$ and Mn-3$d$ orbitals. When the SOC is
included in the calculation, these bands hybridize each other to produce a
small spin-orbit gap at the intersection, but the overall spin-splitting
feature is essentially preserved [Fig. 2(c)]. Although the inclusion of SOC
makes the assignment of spin-spilt partners not so straightforward, one can
discuss the characteristics of the spin splitting by referring to the
calculations without SOC. As shown in Fig. 2(d), the experimental band
structure along the $\Gamma\textrm{L}$ cut shows a rough agreement with the
calculation [Fig. 2(b) or 2(c)].

To investigate the altermagnetic band-splitting in more detail, we performed
ARPES measurements along representative k cuts which cross the k points in the
$\Gamma\textrm{KHL}$ plane including the $\Gamma\textrm{L}$ cut [cuts 1–5 in
Fig. 2(a)]. Cut 1 with ($k_{x},k_{z})=(0,0)$ and cut 5 with
$(k_{x},k_{z})=(\pi,\pi)$ correspond to the $\Gamma\textrm{K}$ and LH high-
symmetry cuts, respectively, whereas cuts 2–4 trace non-high-symmetry k
points. Figures 2(g), 2(e), and 2(f) show the obtained ARPES intensity, the
corresponding calculated band structures without SOC, and those with SOC,
respectively. The calculated bands along the $\Gamma\textrm{K}$ cut (cut 1) in
Fig. 2(e1) are spin degenerate irrespective of k when the SOC is neglected, as
highlighted by the case of bands 1 and 2 (thick black curves). When the SOC is
included, the degeneracy is lifted in most of the k points except for the
$\Gamma$ point. The spin polarization remains zero along the entire
$\Gamma\textrm{K}$ line due to the two-fold rotation and mirror symmetries of
the crystal [Fig. 2(f1)]. As the k cut moves away from the high-symmetry k
points, the bands start to spin split [cut 2; Fig. 2(e2)]. In particular,
bands 1 and 2 indicated by thick red and blue curves, respectively, show a
strongly k-dependent exchange splitting accompanied by the sign reversal upon
varying $k_{y}$, in contrast to the case of a simple itinerant ferromagnet
showing a k-independent exchange splitting. We found that the k points in
which the sign reversal occurs are exactly on the
$\overline{\Gamma\textrm{K}}$ high-symmetry lines of the surface BZ. A similar
anisotropic band-splitting is also recognized along cuts 3 and 4 [Figs. 2(e3)
and 2(e4)] whereas the overall magnitude is gradually suppressed and
eventually vanished along the high-symmetry LH cut [cut 5; Fig. 2(e5)].
Intriguingly, the magnitude of calculated altermagnetic band-splitting reaches
+0.8 eV at $|k_{y}|\sim 0.25\pi$ and -0.8 eV at $|k_{y}|\sim 0.5\pi$ along cut
3 (we define a positive splitting when the energy level of band 1 is higher
than that of band 2, and vice versa for the negative splitting).

The experimental band structure [Fig. 2(g)] shows a reasonable agreement with
the calculated band dispersion including SOC [Fig. 2(f)]. For example, one can
recognize a common trend that the topmost band gradually sinks downward and
becomes flatter on moving from cut 1 to cut 5 in both the experiment and
calculation. Along cut 1 ($\Gamma\textrm{K}$ cut), the SOC-induced band
splitting of the topmost valence band (green and purple arrows) is reproduced
by the calculation as already shown in Fig. 1. Along cut 3, the calculation
signifies two local maxima in the band dispersion at the $\Gamma$ point for
band 1 (red arrow) and slightly away from the $\Gamma$ point for band 2
($|k_{y}|\sim 0.4\pi$; blue arrow). Since these bands are the spin-split
partners, the experimental observation of a double-peaked dispersion along cut
3 supports the existence of altermagnetic band-splitting. We also find a
signature of the band bottom (yellow arrow) which may correspond to the local
minimum of calculated band 1. This suggests that the magnitude of
altermagnetic band-splitting in the experiment is comparable to that of
calculation, i.e. as large as $\sim$0.8 eV. These results provide a
spectroscopic evidence for the existence of altermagnetic band-splitting in
the colinear AF phase of MnTe.

To highlight our key findings, we draw in Fig. 3(a) the schematic band
dispersion for the spin-split partners, bands 1 and 2, in the
$\Gamma\textrm{KHL}$ plane based on the DFT calculation supported by the ARPES
observation. When the SOC is neglected, these bands are degenerate along the
$\Gamma\textrm{K}$ and LH high-symmetry lines, producing dispersive nodal
lines (orange curves) across which the sign of altermagnetic band-splitting is
reversed. There exist additional nodal lines running along the
$\overline{\Gamma\textrm{K}}$ cut. All these nodal lines are regarded as a
slice of nodal surfaces in the $\Gamma\textrm{KHA}$ and MKHL planes. The
existence of multiple nodal lines is also visualized by the intensity plot of
calculated altermagnetic band-splitting in hexagonal BZ in which one can
recognize the nodal lines running along the $\overline{\Gamma\textrm{K}}$ and
$\overline{\textrm{KM}}$ high-symmetry lines of the surface BZ as well as the
sign reversal of altermagnetic band-splitting across the nodal lines. Such
sign-reversal band splitting is a unique characteristic of altermagnets,
distinct from conventional sign-preserving ferromagnetic exchange splitting.

When the SOC is included, the band degeneracy of the nodal surfaces is lifted
by the spin-orbit gap in entire BZ, whereas the bands are still spin
degenerate along some particular high-symmetry lines such as the
$\Gamma\textrm{A}$ line, protected by the $C_{2z}$ symmetry of the crystal.
Low-energy excitations of such nodal line are characterized by the
altermagnetic quasiparticle with a quadratic band dispersion Šmejkal  _et al._
(2022a). Importantly, since the overall energy scale of the spin-orbit gap
($<0.3\textrm{ eV}$) is smaller than that of the altermagnetic band-splitting
($<0.8\textrm{ eV}$), the anisotropic sign-reversal exchange splitting, a key
characteristic of altermagnets, is still maintained even under the presence of
SOC. The present result thus establishes that the theoretically predicted
altermagnetic band-splitting is indeed realized in MnTe. It is emphasized that
the altermagnetic band-splitting verified in this study would be responsible
for the piezomagnetic effect and the x-ray magnetic circular dichroism
recently reported for MnTe Aoyama and Ohgushi ; Hariki  _et al._ , both of
which are sensitive to the global TRS breaking. Also, the multiple band
crossing and sign reversal of the altermagnetic band-splitting inherent to
MnTe would give rise to a high Berry curvature region in k space and may be
responsible for the spontaneous anomalous Hall effect as experimentally
verified recently Gonzalez Betancourt  _et al._ (2023).

The micro-ARPES measurements and the DFT calculations have established the
presence of altermagnetic band-splitting in the colinear AF phase of MnTe. The
altermagnetic band-splitting is strongly k dependent and reaches $\sim$0.8 eV
at non-high-symmetry k points in the $\Gamma\textrm{KHL}$ plane, whereas the
splitting at the high-symmetry k points is suppressed, producing multiple
nodal lines. We also observed a small ($\sim$0.3 eV) band splitting along the
high-symmetry $\Gamma\textrm{K}$ cut, suggestive of an additional contribution
from the SOC. The present results lay a foundation for exploring unique
physical properties inherent to the altemagnetic MnTe, paving a pathway toward
investigating the altermagnetic band-splitting in other altermagnets.

###### Acknowledgements.

We thank K. Ozawa, M. Kitamura, K. Horiba, and H. Kumigashira for their
assistance in the ARPES experiments. This work was supported by JST-CREST (No.
JPMJCR18T1 and JP19198318), JST-PRESTO (No. JPMJPR18L7), and Grant-in-Aid for
Scientific Research (JSPS KAKENHI Grant Numbers JP21H04435, JP19H01845,
JP22H00102, JP19H05823, and JP19H05822), Grant-in-Aid for JSPS Research Fellow
(No: JP18J20058), and KEK-PF (Proposal number: 2021S2-001). A.H. thanks GP-
Spin and JSPS for financial support.

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Figure 1: (a),(b) Crystal structure and bulk hexagonal BZ of MnTe,
respectively. Spin configuration for the $S_{a{\parallel}}$ case is indicated
by red arrows in (a). (c) ARPES intensity maps at $T=40\textrm{ K}$ as a
function of $k_{x}$ and $k_{y}$ at $E_{\textrm{B}}=E_{\textrm{F}}$ for
$k_{z}\sim\pi$ (AHL) plane. (d),(e) ARPES intensity as a function of wave
vector and $E_{\textrm{B}}$ measured along the AH cut of bulk BZ obtained at
$h\nu=117\textrm{ eV}$ and the $\Gamma\textrm{K}$ cut at $h\nu=80\textrm{
eV}$, respectively. (f) Same as (c), but at $E_{\textrm{B}}=0.1\textrm{ eV}$
for the $k_{z}\sim 0$ ($\Gamma\textrm{KM}$) plane. (g) EDCs near
$E_{\textrm{F}}$ along the $\Gamma\textrm{K}$ cut. (h),(i) Calculated band
structures along the AH and $\Gamma\textrm{K}$ cuts, respectively, for the AF
phase with the ($S_{a{\parallel}},\Gamma\textrm{K}_{2}$) configuration with
including the spin-orbit coupling (SOC), overlaid with the ARPES intensity
with gray scale.

Figure 2: (a) Bulk BZ (top) and surface BZ (bottom) of MnTe, together with the
$k$ cuts (blue solid lines) where calculations and ARPES measurements shown in
(e)–(g) were carried out. (b),(c) Calculated band structure along the
$\Gamma\textrm{L}$ cut in the AF phase without and with SOC, respectively. Red
and blue curves correspond to up and down spin components respectively, in
which the quantization axis for spins is defined along the $k_{y}$ axis. Gray
color corresponds to the zero spin polarization. (d) Second-derivative plot of
ARPES intensity at $T=40\textrm{ K}$ along the $\Gamma\textrm{L}$ cut. (e),(f)
Calculated band structures in the AF phase without and with SOC, respectively,
along the $k_{y}$ cut passing through the $\Gamma\textrm{L}$ line (cuts 1-5)
shown by blue lines in (a). Thick red and blue curves highlight bands 1 and 2,
respectively. Calculations for cuts 1–5 were carried out at
$(k_{x},k_{z})=(0,0),(0.2\pi,0.2\pi),(0.4\pi,0.4\pi),(0.75\pi,0.75\pi),\textrm{and
}(\pi,\pi)$, respectively. $|k_{y}|=\pi$ corresponds to the $\Gamma\textrm{K}$
length. Green and purple dashed lines correspond to the $k$ points on the
$\overline{\Gamma\textrm{K}}$ high-symmetry line of the surface BZ, as also
indicated in the bottom panel of (a). (g) Corresponding second-derivative
plots of ARPES intensity as a function of $|k_{y}|$ measured along cuts 1–5.

Figure 3: (a) Schematic 3D band dispersion of MnTe in the $\Gamma\textrm{KHL}$
plane without SOC, signifying the dispersive Dirac-nodal lines running along
the $\overline{\Gamma\textrm{KM}}$ high-symmetry lines of the surface BZ
(orange curves). (b) $\Gamma\textrm{KHL}$ plane (shaded area) in 3D BZ in
which the band dispersion in (a) is shown. Intensity plot of calculated
altermagnetic band-splitting in the hexagonal surface BZ is shown in the upper
panel.

Supplemental Material for “Observation of Giant Band Splitting in
Altermagnetic MnTe” by T. Osumi et al.

Supplemental Note 1. Experiments and calculations

Single crystals of MnTe were grown by the chemical vapor transport method with
I2 as an agent gas. Obtained single crystals [photograph of a crystal is shown
in the Fig. S1(a)] were characterized by x-ray diffraction, magnetization, and
electrical resistivity measurements. Details of the sample preparation and
characterization were described elsewhere Aoyama and Ohgushi .

Fig. S 1: (a) Photograph of a MnTe single crystal. (b) Photoemission spectrum
of MnTe in a wide energy region obtained at $h\nu$ = 120 eV. (c)
Representative Laue x-ray diffraction pattern of the (0001) surface.

ARPES measurements were performed with micro-focused vacuum-ultraviolet (VUV)
synchrotron light at BL-28A in KEK-PF Kitamura  _et al._ (2022). We used
linearly polarized light of 60–170 eV. The energy resolution was set to be
10–20 meV. Samples were cleaved in situ along the (0001) plane of hexagonal
crystal in an ultrahigh vacuum of $1\times 10^{-10}\textrm{ Torr}$, and the
clean surface nature was confirmed by observing sharp core-level peaks
originating from the Mn 3$p$ and Te 4$d$ orbitals with no inclusion of
contaminant peaks, as shown in Fig. S1(b). The crystal orientation was
determined by the x-ray Laue backscattering measurement which signifies clear
six-fold symmetric diffraction spots [Fig. S1(c)] consistent with the (0001)
cleaved plane. Since MnTe single crystal is hard to cleave, we tried to cleave
several times in an ultrahigh vacuum (UHV), occasionally obtained a small flat
area of the crystal with a few tens $\mu\mathrm{m}$ square and then focused
the micro photon beam on it. Temperature during ARPES measurements was kept at
$T=40\textrm{ K}$, well below $T_{\mathrm{N}}$ (= 307 K). The Fermi level
($E_{\mathrm{F}}$) of samples was referenced to that of a gold film
electrically in contact with the sample holder. First-principles band-
structure calculations were carried out by using a projector augmented wave
method implemented in Vienna Ab initio Simulation Package (VASP) code Kresse
and Furthmüller (1996) with generalized gradient approximation (GGA)
pseudopotential. The lattice constants were fixed to the experimental values
($a=b=4.158\textrm{ \AA}$ and $c=6.726\textrm{ \AA}$) Li  _et al._ (2022). The
total energy was calculated self-consistently with the tetrahedron sampling of
$12\times 12\times 8$ $k$-point mesh taking into account SOC.

Supplemental Note 2. Normal-emission ARPES spectra of MnTe

Figures S2(a) and S2(b) show the energy distribution curves (EDCs) and
corresponding ARPES intensity, respectively, along the wave vector
perpendicular to the sample surface ($k_{z}$), measured with the normal-
emission setup by varying $h\nu$ in the vacuum ultraviolet (VUV) region
($h\nu=$ 60–170 eV). One can find some energy bands displaying a finite
$k_{z}$ dispersion, e.g., in the binding energy ($E_{\textrm{B}}$) range of
$E_{\textrm{F}}$–0.7, and 1.2–2.6 eV. The observed band dispersions well
follow the periodicity of the bulk Brillouin zone as well as the calculated
band dispersion including SOC (green curves). The periodic dispersion along
the $k_{z}$ direction indicate the bulk origin of the observed bands.

Fig. S 2: (a), (b) Plots of the normal-emission EDCs and corresponding ARPES
intensity, respectively, as a function of $h\nu$ or $k_{z}$ (corresponding to
the out-of-plane $\Gamma\textrm{A}$ cut). Inner potential was set to be
$V_{0}=8.0\textrm{ eV}$ from the periodicity of the band dispersion. Green
curves represent the calculated band structure including SOC along the
$\Gamma\textrm{A}$ cut, for the ($S_{a{\parallel}},\Gamma\textrm{K}_{2}$)
setting.

Supplemental Note 3. Band-structure calculations with different in-plane spin
configurations

In the presence of spin-orbit coupling, the energy eigenvalue in AF phase
generally differs depending on spin direction Faria Junior  _et al._ (2023).
Although the magnetic torque and neutron diffraction experiments Komatsubara
_et al._ (1963); Szuszkiewicz  _et al._ (2005); Kriegner  _et al._ (2017) have
suggested the in-plane nature of the magnetic moment in the AF phase, it is
experimentally unclear whether the magnetic moment is directed along the
$a$-axis corresponding to the nearest neighbor Mn-Mn bonding direction
($k_{y}$ axis in the $k$ space) or perpendicular to it ($k_{x}$ axis in $k$
space). To examine to what extent the choice of possible spin configurations
and selection of inequivalent k cuts influence the calculated band structure,
we carried out DFT calculations for bulk MnTe with spin configurations
parallel and perpendicular to the $a$ axis, called here, $S_{a{\parallel}}$
and $S_{a{\perp}}$, respectively. Here we focus on the band dispersion along
the $\Gamma\textrm{K}$ cuts, because the band splitting of the topmost valence
band is useful to pin down a possible spin configuration and actual k cuts in
the experiment. For the $S_{a{\parallel}}$ configuration, there exist two
inequivalent $\Gamma\textrm{K}$ cuts, one directed along the spin axis,
$\Gamma\textrm{K}_{1}$, and the other rotated by $60^{\circ}$ from it,
$\Gamma\textrm{K}_{2}$ [Fig. S3(b)]. For the calculation with $S_{a{\perp}}$
configuration, we fix these k cuts but just rotate the spin direction by
$90^{\circ}$. As shown in Fig. S3(c) for the $S_{a{\parallel}}$ configuration,
the band dispersion for the topmost valence band shows no band splitting along
the $\Gamma\textrm{K}_{1}$ cut, whereas it apparently splits into two bands
(bands 1 and 2) along the $\Gamma\textrm{K}_{2}$ cut [Fig. S3(d)] with a
maximum splitting size of 0.25 eV. Such a difference is due to the symmetry
difference of these two k cuts with respect to the magnetic moment. For the
$S_{a{\perp}}$ configuration, the $\Gamma\textrm{K}_{1}$ cut exhibits a band
splitting comparable to that of the $\Gamma\textrm{K}_{2}$ cut for the
$S_{a{\parallel}}$ configuration, whereas the band splitting along the
$\Gamma\textrm{K}_{2}$ cut is overall reduced.

We show in Fig. S3(k) the experimental band dispersion determined by tracing
the numerical fittings to the EDCs in Fig. 1(g), in comparison with the
calculated band structure for four sets of possible spin configurations and k
cut. Taking into account that the experimental band splitting is as large as
$\sim$0.3 eV along the $\Gamma\textrm{K}$ cut, it is likely that the
experimental spin and measurement geometry is either
($S_{a{\parallel}},\Gamma\textrm{K}_{2}$) or
($S_{a{\perp}},\Gamma\textrm{K}_{1}$). As shown in Figs. S3(g)-S3(j), there
are no much differences in the band dispersion along cut 3. This suggests that
the key band argument based on cut 3 in the main text is not much affected by
the choice of k cuts and spin configurations.

Fig. S 3: (a) Crystal structure with in-plane spin configuration, called here
$S_{a{\parallel}}$. (b) k cuts in surface BZ along which DFT calculations were
carried out (green and blue lines). Direction of spin vectors,
$S_{a{\parallel}}$ and $S_{a{\perp}}$, is also indicated. We specify two
inequivalent K points as K1 and K2, and also for the M points (M1 and M2). As
a result, the $\Gamma\textrm{K}$ cut [cut 1; $k_{z}$ = 0] has two
configurations, namely, $\Gamma\textrm{K}_{1}$ and $\Gamma\textrm{K}_{2}$.
(c),(d) Calculated band structure with SOC for the $S_{a{\parallel}}$
configuration along $\Gamma\textrm{K}_{1}$ and $\Gamma\textrm{K}_{2}$ cuts,
respectively. (e),(f), Same as (c) and (d) but for the $S_{a{\perp}}$
configuration. (g)–(j) Same as (c)–(f) but for cut 3 ($k_{z}=0.4\pi$).
Definition of cuts 1 and 3 is the same as that in the main text. (k)
Experimental band dispersion obtained by fitting to the EDCs in Fig. 1(g),
compared with the calculated band structure along the $\Gamma\textrm{K}$ cut
in the AF phase for the four sets of spin configurations.