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# Improving the Thermal Infrared Monitoring of Volcanoes: A Deep Learning
Approach for Intermittent Image Series

Jeremy Diaz, Guido Cervone, and Christelle Wauthier J. Diaz is currently with
the United States Geological Survey Water Mission Area. At the time of this
work, he was with the Department of Geography and the Institute of
Computational and Data Sciences, The Pennsylvania State University, University
Park, PA, 16801 USA (e-mail: jad6655@psu.edu)G. Cervone is with the Department
of Geography and the Institute of Computational and Data Sciences, The
Pennsylvania State University, University Park, PA, 16801 USAC. Wauthier is
with the Department of Geosciences and the Institute of Computational and Data
Sciences, The Pennsylvania State University, University Park, PA, 16801
USAManuscript received April 19, 2005; revised August 26, 2015.

###### Abstract

Active volcanoes are globally distributed and pose societal risks at multiple
geographic scales, ranging from local hazards to regional/international
disruptions. Many volcanoes do not have continuous ground monitoring networks;
meaning that satellite observations provide the only record of volcanic
behavior and unrest. Among these remote sensing observations, thermal imagery
is inspected daily by volcanic observatories for examining the early signs,
onset, and evolution of eruptive activity. However, thermal scenes are often
obstructed by clouds, meaning that forecasts must be made off image sequences
whose scenes are only usable intermittently through time. Here, we explore
forecasting this thermal data stream from a deep learning perspective using
existing architectures that model sequences with varying spatiotemporal
considerations. Additionally, we propose and evaluate new architectures that
explicitly model intermittent image sequences. Using ASTER Kinetic Surface
Temperature data for $9$ volcanoes between $1999$ and $2020$, we found that a
proposed architecture (ConvLSTM + Time-LSTM + U-Net) forecasts volcanic
temperature imagery with the lowest RMSE ($4.164^{\circ}$C, other methods:
$4.217-5.291^{\circ}$C). Additionally, we examined performance on multiple
time series derived from the thermal imagery and the effect of training with
data from singular volcanoes. Ultimately, we found that models with the lowest
RMSE on forecasting imagery did not possess the lowest RMSE on recreating time
series derived from that imagery and that training with individual volcanoes
generally worsened performance relative to a multi-volcano data set. This work
highlights the potential of data-driven deep learning models for volcanic
unrest forecasting while revealing the need for carefully constructed
optimization targets.

###### Index Terms:

Global volcano monitoring, eruption forecasting, deep learning (DL), remote
sensing, thermal infrared, ASTER, neural networks, LSTM, convolutional neural
network (CNN), data-driven architecture, irregular time series.

##  I Introduction

Forecasting volcanic unrest has been a major scientific goal for at least 100
years [1]. This is important because over 800 million people live within 100
kilometers of a Holocene volcano [2]. Beyond direct volcanic hazards,
volcanoes can perturb agriculture, infrastructure, the environment, as well as
(inter)national business and travel [3] during the course of an eruption or
during inter-eruptive periods. Short-term (e.g., days to months) volcanic
unrest forecasts are made by identifying patterns in monitoring observational
time series (e.g. seismic, deformation, thermal, emissions). Improving these
forecasts has been identified as a grand challenge within the field of
volcanology [4]. Ultimately, this task is very challenging because there are
various methods, expertise, and/or instrumentation to monitor volcanoes on the
ground, as well as remotely from the air and space (see [4] section 1.4 for a
light overview, see [5] for a more detailed remote sensing perspective). As a
result, reliable and timely monitoring and, thus, the ability to forecast
unrest accurately is costly and absent from many active volcanoes [3].

Short-term forecasting of volcanic unrest is conducted through a range of
domain and statistical approaches, both quantitative and qualitative. At the
highest level, these forecasts are generated by expert committees at
observatories [6] and described through event trees which communicate all
possible outcomes with the associated probabilities at each stage of the
outcome [7]. Likewise, the very few examples of quantitative methods which
directly output broadly prescriptive forecasts (i.e., many metrics or
probabilities) rely upon event trees as input data [8, 9]. These event tree
approaches are very promising but also very demanding, requiring a plethora of
expert knowledge, monitoring streams, and volcano-specific knowledge. On the
other hand, quantitative methods are often more limited in scope, instead
being applied to individual time series of monitoring data to aid in the
identification of early warnings of change (e.g., values above background,
changes in pattern), and this exercise is often considered forecasting, see
[10, 11]. Examples of this approach that are heavily influenced by domain
knowledge include modeling seismic signals from a material failure perspective
[12, 13] and monitoring gas ratios [14] and lava discharge rates [15] from an
equilibrium/disequilibrium perspective. On the other hand, more purely
quantitative approaches include the use of simple thresholds [1], examination
of current events among their broader distributions [16], and various modeling
projects that have made use of Poisson point processes [17, 18], variograms
[19, 20], Markov chain Monte Carlo methods [21], and ARIMA models [22, 23].

With global volcanic monitoring in mind, there are many difficulties and
opportunities with this existing toolkit of forecasting options. Particularly,
individual forecasting efforts generally differ in the variety and historical
reach of their available data. Committee-based and event tree approaches are
particularly ambitious, but their reliance on many experts makes them
inaccessible due to global volcanic monitoring already being financially
restrictive. Likewise, quantitative methods that employ domain knowledge often
make use of specialized ground equipment (e.g., seismometers, infrared
spectrometers) that is lacking at many active volcanoes. Thus, financially
accessible solutions which are globally homogeneous are highly appealing.
Because of this, satellite remote sensing has been thoroughly studied [24] and
widely adopted to both establish and supplement monitoring efforts [25, 26,
27, 28]. Currently, multiple online tools exist to aid global volcanic
monitoring via remote sensing; examples of these include MODVOLC [28, 29],
MIROVA (Middle InfraRed Observation of Volcanic Activity) [30], and MOUNTS
(Monitoring Unrest From Space) [31]. Thermal monitoring has seen particular
operational success due to the relative ease of access, processing, and
interpretation. This imagery is often provided for viewing while accompanied
by time series of higher-level information, such as the maximum temperature
above the background (i.e., “excess temperature”) [28], the number of hotspots
[31], the maximum distance of a hotspot from the summit [30], and/or measures
of radiance [30].

Thermal remote sensing imagery is therefore a promising data stream for global
volcanic monitoring and forecasting. However, for a predictive image task, the
routine statistical models mentioned earlier have multiple weaknesses:
notably, they make rigid assumptions about the functional relationships of
data and the distributions underlying those data, do not necessarily consider
spatially correlated data, and, perhaps most importantly, are relatively
simple—exploring a parameter space much smaller than current computational
technology allows. Meanwhile, since 2012 [32], neural networks have seen an
explosive growth in popularity (including within the field of remote sensing
[33]) due to their incredible performance in making image-based predictions by
utilizing millions of optimized parameters and lots of data. Likewise,
machine/deep learning has been applied in the field of volcanology for various
detection and classification tasks (particularly with seismic signals and
Interferometric Synthetic Aperture - InSAR - geodetic data) [34, 35, 36, 37,
31, 38], but the task of forecasting remains poorly explored [33, 1]. However,
in the existing examples of short-term volcanic forecasting with machine
learning [39, 40, 41], neural networks have more commonly excelled [39, 40]
and among neural networks, more sophisticated architectures have proven
beneficial [40]. Lastly, while it is undesirable that neural networks are
infamous for having uninterpretable mappings of input data to output targets,
they are additionally favorable for this cloud-susceptible task because they
have a documented history for performing well with noisy and missing data
[42].

In summary, improving short-term volcanic forecasts are valuable for society
worldwide. There are many ways to produce these forecasts, but most cannot be
implemented globally due to the costs of experts and specialized ground
equipment. There are, however, many globally accessible remote sensing data
streams; particularly, satellite thermal imagery has been widely adopted.
Forecasting remotely sensed imagery—especially imagery that can be inhibited
by atmospheric conditions—is not well explored. Here, we draw on existing and
proposed models (with an emphasis on neural networks) to explore and evaluate
the task of forecasting volcanic temperature imagery. Specific experiments
include:

  * •

Which model (among naive methods, autoregression, and numerous neural
networks) can best forecast volcanic AST_08 image sequences?

  * •

How do derived time series from these models compare to real observations?

  * •

How does performance on individual volcanoes change when the training data set
is limited to individual volcanoes instead of all 9 in the data set?

##  II Data and Methods

###  II-A Data Set

Nine active volcanoes were targeted in this project: Mount Erebus
(Antarctica), Erta Ale (Ethiopia), Mount Etna (Italy), Kīlauea (Hawaii),
Masaya (Nicaragua), Nyamulagira (Democratic Republic of Congo), Nyiragongo
(Democratic Republic of Congo), Pacaya (Guatemala), and the distant Pu’u ’Ō’ō
cone of Kīlauea (Hawai’i, US). These volcanoes were chosen for their active
nature and/or because they have displayed lava lakes (a superficial and
persistent thermal phenomenon) for at least some percentage of the data
record. Among these, the Hawaiian volcanoes and Mount Etna are among the most
studied and monitored in the world; Erebus, Masaya, and Pacaya have some
degree of ground monitoring; Nyamulagira and Nyiragongo have decommissioned
ground monitoring efforts due to civil unrest in the region; and, Erta Ale
does not have regular ground monitoring [2].

Among thermal monitoring options, nighttime imagery is preferred in the field
of volcanology because it is generally less cloudy and decreases the effects
of topography and solar heating, “allowing better delineation of low-
temperature geothermal anomalies” [43, p. 18]. In general, many platforms do
not make nighttime thermal imagery easily accessible (e.g., Landsat has
special request targets [44]); however, the $>20$-year NASA Terra platform
which hosts the MODIS and ASTER instruments are an exception to this. MODIS is
very popular because it provides a very high temporal sampling rate (four
images per day [45]) which can aid in the rapid detection of changes [29].
ASTER, on the other hand, provides a 10-fold increase in spatial resolution
which allows the less frequent detection of subtler changes months in advance
[46, 47, 5]. Nighttime thermal data from ASTER (L1T) is available for bulk
download in the form of 5 bands of radiance values (8-12$\mu$m) from the
online DAAC2Disk Utility, courtesy of the NASA Land Processes Distributed
Active Archive Center (LP DAAC), USGS Earth Resources Observation and Science
(EROS) Center, Sioux Falls, South Dakota,
https://lpdaac.usgs.gov/tools/daac2diskscripts/. The level-2 Surface Kinetic
Temperature product (AST_08) [48] which is more thoroughly corrected for the
atmosphere and with radiance converted to temperature is available for order
from the NASA LP DAAC via NASA’s Earthdata Search
(https://search.earthdata.nasa.gov/search?q=ast_08).

First, all L1T data for each volcano between the dates December 18, 1999, and
May 1, 2020, were downloaded. Initial inspection revealed a poor correlation
between viable scenes and cloud cover metadata, so all scenes were manually
inspected for viability in the data set. Upon manual inspection, each scene
was denoted viable, uncertain, or nonviable (Figure 1). Due to the large
number of nonviable scenes, uncertain scenes were ultimately included in our
data set; these were scenes with partial cloud cover or scenes with large
extents of cloud cover, through which, thermal features were still observable.
The final set of viable and uncertain scenes were then ordered as level-2
AST_08 data for the remainder of the project. The distribution of time between
these scenes is presented in Figure 2.

Figure 1: Manual scene inspection results by volcano. Scenes are from the
beginning of ASTER record to May 1, 2020. Figure 2: Distribution of the amount
of time between scenes in this data set. Mean = $37.138$. Standard deviation =
$73.850$.

###  II-B Data Processing

The AST_08 data contained missing values representing areas outside the
sensor’s field of view as well as pixels amid very volcanically active areas
(see Figure 3, top row) – with sources referencing the latter as recovery
pixels [49]. In the level-2 data, the latter appears as counter-intuitive dark
spots within a bright hotspot, while the level-1 data does not contain these
missing pixel observations. Since locations outside the field of view will
have valid observations in at most two cardinal directions, pixels with valid
values in at least three cardinal directions were filled with nearest-neighbor
interpolation; qualitatively, this better resembled level-1 data and did not
assign improbable ”cold” pixels islands in the middle of large lava
lakes/flows.

To narrow the focus of the image forecasting task, a $96\times 96$ pixel
subset of the scene was used in the data set; with a 90-meter spatial
resolution, this translates to an $8.64\times 8.64$ kilometer subset. This
precise size is somewhat arbitrary, but, qualitatively, it was large enough to
observe significant portions of large lava flows while not being too large to
neglect small lava lakes. The subset was centered on the latitude and
longitude of the volcano summit. Converting the scenes’ coordinate reference
systems to latitude and longitude for summit locating and subsetting
introduced geometric distortions that were visually apparent for Mount Erebus
(located in Antarctica); however, these relics were not problematic for
thermal interpretation.

In volcanology studies, temperature measurements are often examined relative
to the environmental background (i.e., the excessive temperature of the
volcanic feature relative to the surrounding environment); as such, the AST_08
scenes were processed this way (see Figure 3 for steps up to this point).
Volcano studies often manually identify a cloud-free, representative part of
the environment to determine the environmental background temperature. To
automate this process in a robust way, the following process was used to
compute the average environmental background temperature: first, a $10\times
10$ pixel subset was extracted from each corner of the reduced image; these
corners were then considered appropriate if less than 10% of their pixel
values were outside the sensor’s field of view (i.e., missing values). The
mean was then calculated for each subset, and the mean of those means was used
as the environmental background temperature. Mount Erebus’s location along the
orbital path led to unique viewing geometries where this method would fail to
identify any valid corners; in those cases, the $10\times 10$ pixel subsets
were rotated around the perimeter of the image to identify suitable areas (see
Figure 4). Once the environmental background temperature was computed, it was
subtracted from every pixel value. Notably, in addition to being consistent
with volcanology studies, this removes the task of forecasting the seasonality
of the background environment.

Figure 3: Visualization of three steps of the data processing for a five-scene
sequence of Pu’u ’Ō’ō. Each row is a stage of data processing, and each column
is the same scene. The green shade denotes formally acknowledged missing
values. Figure 4: The two ways that background areas were defined to calculate
the average background temperature of the scene. Red boxes denote the
background area, and the green shade denotes formally acknowledged missing
values.

After the above, there remained missing values from the $96\times 96$ scene
subsets that represented areas outside the sensor’s field of view. These
pixels were then filled with the values from the previous scene and the date
of those previous scenes was noted. When this occurred for the first scene
acquired for a volcano (i.e., no previous scenes), the missing values were
filled with nearest-neighbor interpolation (Figure 5). Lastly, prior to
training the data was scaled between 0 and 1 (i.e.,
$x_{new}=(x_{original}-x_{min})/(x_{max}-x_{min})$) to facilitate training
convergence.

Figure 5: The two methods for filling in missing values. Left: when no scenes
exist prior to the scene of interest, missing values were filled with nearest
neighbor interpolation. Right: when scenes exist prior to the scene of
interest, missing values were filled with prior observations while tracking
the difference in time.

###  II-C Forecasting Configuration

The forecasting task here uses the previous $n$ scenes to forecast the next
scene. For most models, this is done by treating each pixel as an independent
observation, but for convolution-based neural networks, this is done at the
image-level. The exact choice of $n$ is not straightforward, but
realistically, it must be large enough to make use of past information and
small enough to not magnify computational and storage burdens. A popular
method in volcanology studies is to examine the maximum excess temperature
through time, so these time series were extracted from the L1T data (prior to
AST_08 retrieval) and their autocorrelation structure was examined for each
volcano. The average of the largest significant autocorrelation lag was
$6.43$, so $n$ was set to $6$.

In the case of volcano-specific training sets (see Results and Discussion,
section C), the largest significant autocorrelation lag for each volcano was
used with a minimum accepted value of $3$ and a maximum accepted value of $10$
previous scenes.

###  II-D Models

####  II-D1 Baseline predictive methods

ARIMA models are common time series models (see [50] for a comprehensive
coverage) with published applications in volcanic forecasting [22, 23]. The
ARIMA acronym stands for “autoregressive integrated moving average” because it
combines the simpler autoregressive (AR) model that predicts $x_{t}$ based on
the linear combination of its previous $\\{x_{t-1},\dots,x_{t-p}\\}$ values
and the moving average (MA) model that defines $x_{t}$ based on the linear
combination of the past $q$ random perturbations to the process mean. The
“I/integrated” part of the name refers to the process of $d$-order
differencing, a method for approximating constant mean and, to a lesser
extent, autocovariance, through time; the simplest and most common example of
this is first-order differencing which is concerned with modeling the change
in $x_{t}$ ($\Delta x_{t}=x_{t}-x_{t-1}$) instead of $x_{t}$ directly. Thus,
an ARIMA model is defined by 3 parameters: $p$, $d$, and $q$.

This work examined two baseline models related to the commonly used ARIMA
model—more precisely, with both being variants of the AR model. All neural
network architectures examined in this work utilize $6$ previous scenes to
forecast the next scene (unless stated otherwise), so the AR($p=6$) model is
an intuitive baseline analog. This can be expressed mathematically as:

$x_{t}=\sum_{i=1}^{6}\phi_{i}x_{t-i}$ (1)

Likewise, scientifically, the move naive forecast is to assume no change from
the most recent scenes, so this will also be examined by deploying an
AR($p=1$) model with $\phi_{1}=1$.

Lastly, a final baseline model is inspired from the domain’s data
processing—predicting all $0$ values as the forecast. In light of the data
processing, this forecast can be interpreted as predicting a scene of
homogeneous temperature with no pixels exceeding the background average.

####  II-D2 Existing neural network architectures

Among neural network architectures, the long short-term memory (LSTM) model is
very popular. Like its preceding recurrent neural network (RNN), this model is
designed for sequential (rather than independent) data, and it expanded upon
its predecessor by introducing gate units that are trained to learn the
optimal flow of new data and retention of prior information. The equations of
this model are given below:

$i_{t}=\sigma(W_{xi}x_{t}+W_{hi}h_{t-1}+b_{i})$ (2)
$f_{t}=\sigma(W_{xf}x_{t}+W_{hf}h_{t-1}+b_{f})$ (3) $c_{t}=f_{t}\odot
c_{t-1}+i_{t}\odot tanh(W_{xc}x_{t}+W_{hc}h_{t-1}+b_{c})$ (4)
$o_{t}=\sigma(W_{xo}x_{t}+W_{ho}h_{t-1}+b_{o})$ (5) $h_{t}=o_{t}\odot
tanh(c_{t})$ (6)

Here, $\circ$ denotes the Hadamard product for element-wise multiplication of
two matrices, $\sigma$ represents the sigmoid activation function, lower case
variables indicate scalars or vectors, and upper case variables indicate
matrices. Likewise, these notation rules are adapted throughout the section,
modifying original equations as needed.

The traditional LSTM was modified by Shi et al. [51] to perform precipitation
nowcasting on radar maps by changing both the inputs and the hidden states
from scalars/vectors to matrices and applying the learned weights via the
convolution ($\ast$) or “sliding window” operator in Equations 2-5:

$I_{t}=\sigma(W_{xi}\ast X_{t}+W_{hi}\ast H_{t-1}+b_{I})$ (7)
$F_{t}=\sigma(W_{xf}\ast X_{t}+W_{hf}\ast H_{t-1}+b_{F})$ (8)
$C_{t}=F_{t}\odot C_{t-1}+I_{t}\odot tanh(W_{xc}\ast X_{t}+W_{hc}\ast
H_{t-1}+b_{C})$ (9) $O_{t}=\sigma(W_{xo}\ast X_{t}+W_{ho}\ast H_{t-1}+b_{O})$
(10) $H_{t}=O_{t}\odot tanh(C_{t})$ (11)

The result is the now-popular convolutional LSTM with proven results on
scientific raster time series, and the architecture has since been used to
predict future temperature [52], precipitation [53], and land cover [54].

Separately, Zhu et al. [55] incorporated two time gates to allow the “notion
of interval” or time between observations to improve content recommendation
services (e.g., music and scientific literature). These time gates serve as
further learned modulators to the flow of short-term and long-term memory,
yielding the Time-LSTM:

$t1_{t}=\sigma(W_{x1}x_{t}+\sigma(W_{t1}\Delta t_{t})+b_{1})$ (12) $\mbox{such
that }W_{t1}\leq 0$ $t2_{t}=\sigma(W_{x2}x_{t}+\sigma(W_{t2}\Delta
t_{t})+b_{2})$ (13) $\widetilde{c_{t}}=(1-i_{t}\odot t1_{t})\odot
c_{t-1}+i_{t}\odot t1_{t}\odot\sigma(W_{xc}x_{t}+W_{hc}h_{t-1}+b_{c})$ (14)
$c_{t}=(1-i_{t})\odot c_{t-1}+i_{t}\odot
t2_{t}\odot\sigma(W_{xc}x_{t}+W_{hc}h_{t-1}+b_{c})$ (15) $h_{t}=o_{t}\odot
tanh(\widetilde{c_{t}})$ (16)

Similarly, Baytas et al. [56] developed the Time-Aware LSTM that analyzed
irregularly timed patient records to predict patient groups with similar
disease progression pathways. Instead of using further nonlinear and learned
gates, however, the Time-Aware LSTM simply reallocates the memory
contributions of long-elapsed inputs from short-term to long-term memory
proportional to the time elapsed via a monotonically non-increasing function
$g$:

$c^{S}_{t-1}=tanh(W_{d}c_{t-1}+b_{d})$ (17)
$\hat{c}^{S}_{t-1}=c^{S}_{t-1}\odot g(\Delta_{t})$ (18)
$c^{T}_{t-1}=c_{t-1}-c^{S}_{t-1}$ (19)
$c^{\ast}_{t-1}=c^{T}_{t-1}+\hat{c}^{S}_{t-1}$ (20)
$\widetilde{c}=tanh(W_{xc}x_{t}+W_{hc}h_{t-1}+b_{c})$ (21) $c_{t}=f_{t}\odot
c^{\ast}_{t-1}+i_{t}\odot\widetilde{c}$ (22)

Thus, the Time-LSTM and Time-Aware LSTM largely modify and expand Equation (4)
of the original LSTM 111It is worth noting that the Time-LSTM defines
$\Delta_{t}$ as the time following an observation ($t_{i+1}-t_{i}$), while the
Time-Aware LSTM defines $\Delta_{t}$ as the time preceding an observation
($t_{i}-t_{i-1}$)..

####  II-D3 Proposed neural network architectures

There does, however, remain the opportunity to modify these equations to
include the composite spatial handling of the convolutional LSTM and the
irregular temporal handling of the Time-LSTM or Time-Aware LSTM, and this is
somewhat straightforward due to their shared origin and minimal modifications
of the traditional LSTM. I express these modifications below for (1) Time-LSTM
+ ConvLSTM:

$I_{t}=\sigma(W_{XI}*X_{t}+W_{HI}*H_{t-1}+b_{I})$ (23)
$T1_{t}=\sigma(W_{XT1}*X_{t}+\sigma(W_{TT1}*\Delta T_{t})+b_{T1})$ (24)
$\mbox{such that }W_{TT1}\leq 0$
$T2_{t}=\sigma(W_{XT2}*X_{t}+\sigma(W_{TT2}*\Delta T_{t})+b_{T2})$ (25)
$\begin{split}\widetilde{C_{t}}=(1-I_{t}\circ T1_{t})\circ C_{t-1}+I_{t}\circ
T1_{t}\circ\\\ \sigma(W_{XC}*X_{t}+W_{HC}*H_{t-1}+b_{C})\end{split}$ (26)
$\begin{split}C_{t}=(1-I_{t})\circ C_{t-1}+I_{t}\circ T2_{t}\circ\\\
\sigma(W_{XC}*X_{t}+W_{HC}*H_{t-1}+b_{C})\end{split}$ (27)
$O_{t}=\sigma(W_{XO}*X_{t}+W_{TO}*\Delta T_{t}+W_{HO}*H_{t-1}+b_{O})$ (28)
$H_{t}=O_{t}\circ\sigma(\widetilde{C_{t}})$ (29)

and (2) Time-Aware LSTM $+$ ConvLSTM:

$I_{t}=\sigma(W_{XI}*X_{t}+W_{HI}*H_{t-1}+b_{I})$ (30)
$F_{t}=\sigma(W_{XF}*X_{t}+W_{HF}*H_{t-1}+b_{F})$ (31)
$O_{t}=\sigma(W_{XO}*X_{t}+W_{HO}*H_{t-1}+b_{O})$ (32)
$C^{S}_{t-1}=tanh(W_{D}*C_{t-1}+b_{D})$ (33)
$\hat{C}^{S}_{t-1}=C^{S}_{t-1}\circ g(\Delta T_{t})$ (34)
$C^{T}_{t-1}=C_{t-1}-C^{S}_{t-1}$ (35)
$C^{*}_{t-1}=C^{T}_{t-1}+\hat{C}^{S}_{t-1}$ (36)
$\widetilde{C}=tanh(W_{XC}*X_{t}+W_{HC}*H_{t-1}+b_{C})$ (37) $C_{t}=F_{t}\circ
C^{*}_{t-1}+I_{t}\circ\widetilde{C}$ (38) $H_{t}=O_{t}\circ tanh(C_{t})$ (39)

The above equations are minimally modified from their original forms, but,
notably, the use of peephole connections in the original Time-LSTM were not
included because (i) the exclusion yields a simpler model, and (ii) a seminal
paper [57] found that removing peephole connections from LSTMs did not
significantly reduce performance. In both cases, the convolutional LSTM’s C
gate has been modified to incorporate the potentially nonlinear effect of the
time interval ($\Delta T_{t}$) between the current observation ($X_{t}$) and
the previous state ($H_{t-1}$) on the current state ($H_{t}$). Here, the time
interval ($\Delta T_{t}$) is also a spatial structure that allows temporal
specification per pixel; this enables the use of partially obstructed images
which may be caused by different orbital paths and/or incomplete cloud
obstruction. The current state ($H_{t}$) is naively a spatial structure, but
it may be further processed to some metric of interest (e.g. maximum pixel
value, the number of anomalous pixels) or used as inputs to further neural
network layers.

####  II-D4 U-Net

Manual inspection of early results for the convolution-based models suggested
that these models tended to produce overly smoothed predictions. This can be
seen as a problem with precise localization, which the U-Net architecture [58]
provides a solution for. The U-Net architecture was designed for image
segmentation with no intention for sequential tasks. In the original U-Net
implementation, a deep convolutional neural network is improved by creating
symmetrical hidden dimensions which start relatively low then increase and
return to the original specifications (e.g., $64$, $128$, $64$); as this
happens, the spatial size shrinks then expands while propagating the outputs
of earlier layers into the later but symmetrical layers (i.e., $64_{1}$’s
output propagates to $64_{2}$’s input) to promote localization and improved
resolution. Here, we implement the symmetrically sized hidden dimensions and
the propagation of earlier symmetrical layers’ output to the convolution-based
model with the lowest validation set RMSE.

###  II-E Optimization Details

The data set was separated with a $70/15/15$ split chronologically by each
volcano into a training, validation, and test set. The training set was used
to fit model parameters, the validation set was used to compare different
models as well as compare the same model fit with different regularization
strengths, and the test set was aside for evaluation of the final, best model.
Every model was fit with $5$ different regularization strengths ranging from
$0.0001$ to $1$ by a factor of $10$; higher values of this parameter help
reduce overfitting to the training set by penalizing large model parameters.

Choosing the exact number of hidden layers as well as the number of
units/neurons in each layer is not straightforward, with the only general
rules being that more parameters increase performance, computational burden,
and the need to combat overfitting. The exact hidden layer choices used in
this project were taken from the published best results of the existing neural
networks used here [51, 55, 56]. All models were trained for 100 epochs with
the Adam optimizer [59] to minimize mean squared error. Optimization occurred
in varying mini-batch sizes that were determined to maximize GPU utilization.

All models were implemented in the Python programming language via the PyTorch
machine learning framework. The hardware used for this project was an HPC
cluster with 10 NVIDIA Tesla V100 GPUs with 32 gigabytes of RAM each; all
neural network computations used this hardware in parallel.

##  III Results and Discussion

###  III-A Experiment 1: Best Model for Forecasting Images

Table 1 presents the validation set RMSE along with important training details
for all the models. Among the naive methods, using the most recent scene as
the forecast for the next scene led to an RMSE of $5.291^{\circ}$C while,
alternatively, assuming a homogenous temperature landscape (“All Zeros”) led
to an RMSE of $4.738^{\circ}$C. Representing the only learned statistical
model, the AR(6) returned an RMSE of $4.322^{\circ}$C. Among the non-
convolutional neural networks, the standard LSTM models returned RMSEs ranging
from $4.217$ to $4.265^{\circ}$C, the Time-LSTM models returned RMSEs ranging
from $4.694$ to $4.870^{\circ}$C, and the Time-Aware LSTM models returned
RMSEs ranging from $4.660$ to $4.726^{\circ}$C. The latter two converged very
closely to the All Zeros model, returning low magnitude predictions in the
best results. Convolution-based neural networks returned RMSEs between $4.164$
and $4.428^{\circ}$C with less variance than the non-convolutional neural
networks. The lowest RMSE was achieved by the proposed ConvLSTM + Time-LSTM +
U-Net architecture. It was approximately $10$ times faster than the lowest
RMSE LSTM and required approximately $40$ times fewer parameters.

TABLE I: Validation set performance and training details for all models Model Type | Model | Hidden Dim.s | RMSE (∘C) | Number of Parameters | Training Time (hours)
---|---|---|---|---|---
Naive | Last Scene | – | 5.291 | – | –
| All Zeros | – | 4.738 | – | –
Statistical | AR(6) | – | 4.322 | 6 | 0.10
Non-spatial NN | LSTM | 128 | 4.265 | 67080 | 0.09
|  | 1028, 512 | 4.229 | 7393384 | 2.05
|  | 2000, 2000 | 4.217 | 48032008 | 4.91
| Time-LSTM | 128 | 4.870 | 51213 | 0.29
|  | 1028, 512 | 4.706 | 5555825 | 4.00
|  | 2000, 2000 | 4.694 | 36052013 | 10.10
| Time-Aware LSTM | 128 | 4.660 | 83594 | 0.25
|  | 1028, 512 | 4.719 | 8713854 | 4.15
|  | 2000, 2000 | 4.726 | 56036010 | 11.37
Spatial NN | ConvLSTM | 64, 64 | 4.230 | 447528 | 0.26
|  | 128, 64, 64 | 4.428 | 1335080 | 0.41
| ConvLSTM + Time-LSTM | 64, 64 | 4.255 | 341967 | 0.31
|  | 128, 64, 64 | 4.250 | 1013903 | 0.46
| ConvLSTM + Time-Aware LSTM | 64, 64 | 4.256 | 521394 | 0.31
|  | 128, 64, 64 | 4.394 | 1556530 | 0.45
| ConvLSTM + Time-LSTM + U-Net | 64, 128, 64 | 4.164 | 1233359 | 0.53

For the best model of each type (i.e., statistical, non-convolutional, and
convolutional), the models were separately trained for $1000$ epochs to
determine if further training resulted in better performance, but, instead, it
was found that validation set RMSE increased for all models with this extended
training. For the AR(6), validation set RMSE increased from $4.322$ to
$4.357^{\circ}$C; for the LSTM, $4.217$ to $4.293^{\circ}$C; and, for the
ConvLSTM + Time-LSTM + U-Net, $4.164$ to $4.182^{\circ}$C. The models
originally trained for $100$ epochs are thus used for subsequent discussion,
and the three mentioned in this paragraph will be referred to as the “selected
models”.

Predictions for the lowest RMSE model are presented in Figure 6 and Figure 7.
Both of these demonstrations compare observations with predictions during
large effusive events. In Figure 6 the predictions are well aligned with the
observations. It can be noted that the predicted sequence failed to predict
new spatial features until after they emerged in observations. Interestingly,
as the lava flow continues, the model emphasized particular hotspot areas that
were present in prior scenes while broadening the width of more linear
features. While Figure 7 is also a large lava flow, its data are much less
ideal—note the persistent presence of partial cloud cover as well as missing
edge values which required filling with previous scenes. Despite these
obstructing conditions, the model continued to predict the presence of a large
lava flow in the correct general direction and shape.

Figure 6: Demonstration of validation set predictions for the best model. Top
row: observations. Bottom row: predictions. Figure 7: Demonstration of
validation set predictions for the best model. Top row: observations. Bottom
row: predictions. Note the cloudy conditions and missing values that have been
filled (as apparent by the diagonal boundaries in the observations for
2015-12-01 and 2016-01-02).

The distributions of predicted pixel values for the selected models as well as
a poor-performing model (Time-LSTM) were examined and are presented in Figure
8. Barring the poor-performing model, it was found that the models with lower
RMSE tended to predict lower-valued extremes than higher RMSE models and all
models predicted lower-valued extremes than the true observations.

Figure 8: Cumulative histograms of pixel values in the validation set for the
observations and selected models. Vertical lines indicate maximums.

All in all, the explicit modeling/optimization task was to predict image
sequences that are irregularly spaced in time with minimal RMSE relative to
the true observations. Each model had different characteristics which
influenced its ability to accomplish this task. For example, the AR(6) had
very few parameters while some of the neural networks had millions of
parameters; the Time-LSTM considered the time between scenes while the
ConvLSTM considered a pixel’s spatial neighbors. Ultimately, the proposed
model, specifically the ConvLSTM + Time-LSTM + U-net, considered more
characteristics of the data (e.g., spatial, irregular in time, and high
resolution) and accomplished this explicit goal the best. It is worthwhile to
note that while all models have low error relative to the full range of the
data that these error differences are spread across $2211840$ pixels—$240$
images in the withheld validation set, each of which contains $9216$ pixels.

###  III-B Experiment 2: Derived Time Series

With infrared-based imagery, there are many derived time series of interest in
volcanic monitoring. Specifically, with temperature data, radiance-based
measurements cannot be examined, so here we focus on (1) maximum temperature
above the background, (2) the number of hotspots, and (3) distance of furthest
hotspot from the summit. All derived time series are concerned with anomalous
or extreme temperatures, so it was expected that the models’ inability to
predict the true scale of extreme temperatures would inhibit derived time
series. To alleviate this, predicted time series were derived both before and
after histogram matching the validation set predictions to the training set
observations. A demonstration of this can be seen in Figure 9.

Figure 9: A demonstration of histogram matching a validation set prediction to
the training set. Left: observation. Middle: prediction. Right: prediction
with histogram matching. The same color scale is used for all three images.
Note the increased abundance of observation-scale extremes after histogram
matching to the training set.

Table 2 contains the error of the selected models as well as the poor-
performing Time-LSTM on the derived time series. See Figure 10 for a
demonstration of these time series using the validation set, histogram
matching, and the ConvLSTM + Time-LSTM + U-Net model.

TABLE II: Derived Time Series Metrics for Selected Models Model | Histogram | Maximum | Number of | Maximum Distance of a Hotspot (m)
---|---|---|---|---
| Matching | Temperature (∘C) | Hotspots | from the Summit
AR(6) | No | 33.304 | 20.553 | 1190.338
AR(6) | Yes | 32.255 | 18.180 | 1067.456
LSTM | No | 40.604 | 25.249 | 1421.756
LSTM | Yes | 32.120 | 18.303 | 1105.383
Time-LSTM | No | 57.930 | 25.347 | 1513.082
Time-LSTM | Yes | 32.361 | 17.468 | 825.800
ConvLSTM + Time-LSTM + U-net | No | 46.262 | 25.347 | 1513.082
ConvLSTM + Time-LSTM + U-net | Yes | 36.642 | 17.884 | 1090.002
Figure 10: A demonstration of derived time series using validation set data
from the best model. Included here are time series for maximum temperature
above the background, the number of hotspots, and distance of furthest hotspot
from the summit. “Hot” is defined as $10^{\circ}$C above the background, and
volcanoes plotted here include Kīlauea, Nyiragongo, Pu’u ’Ō’ō, Pacaya, and
Erta Ale.

Before histogram matching, the AR(6) achieves the lowest RMSE on all three
derived time series. This is attributable to the fact that it predicts the
largest maximum values of the available models. After histogram matching,
which accounts for the apparent tendency of neural networks to produce low-
scale extremes, the AR(6) model is no longer the best at any derived time
series. Likewise, the proposed model (ConvLSTM + Time-LSTM + U-net), which
achieved the lowest RMSE on predicting the image sequences was unable to
reproduce any derived time series with the lowest RMSE. Instead, after
histogram matching, the traditional LSTM achieved the lowest RMSE recreating
the maximum temperature above the background ($32.120^{\circ}$C), while the
Time-LSTM achieved the lowest RMSE on the number of hotspots ($17.468$ pixels)
and the maximum distance of a hotspot from the summit ($825.800$ meters).

This discrepancy is interesting because the Time-LSTM was included in further
comparisons since its best optimization resulted in the worst validation set
image sequences (RMSE = $4.694^{\circ}$C). Further examination revealed that
this is because it tended to converge to near-zero image predictions (minimum
= $-0.907^{\circ}$C, maximum = $0.958^{\circ}$C). Together, this suggests that
despite struggling immensely with scale, the Time-LSTM learned superior
pattern recognition within a very narrow comfort zone for this task that was
indirectly optimized for.

Scientifically, this suggests an important implication—when forecasting
multipurpose data with many potential uses, directly minimizing error on the
data does not necessarily minimize error on its potential uses or derivations.
Moving forward, it would be best to explicitly account for these other metrics
directly in the loss function (i.e., optimization problem). Compared to highly
engineered, domain-based algorithms, deep learning models are relatively
simple and rely on heavy parameterization along with large amounts of data to
improve performance. With immense interest in the value of deep learning for
scientific tasks, it will likely be a very promising pursuit to engineer loss
functions with the multiple scientific priorities and constraints in mind—an
example of this is can be seen when forecasting water temperature constrained
by physical laws [60].

###  III-C Experiment 3: Training with Individual Volcanoes

Broadly, RMSE decreased when the training set was limited to individual
volcanoes (Table 3). In general, volcanoes with fewer data produced worse
results, but performance did not increase exclusively with training set
quantity. For example, training exclusively with Erta Ale or Kīlauea data led
to lower RMSE than training exclusively with Erebus data ($4.356^{\circ}$C, n
= 178 and $4.390^{\circ}$C, n = 150 versus $4.684^{\circ}$C, n = 336), and
training exclusively with Nyiragongo data led to higher RMSE than training
exclusively with Masaya data ($5.365^{\circ}$C, n = 74 versus
$5.343^{\circ}$C, n = 44).

TABLE III: Validation set performance for different training sets: all
volcanoes or specific volcanoes

|  | Validation Set Performance (∘C) |
---|---|---|---
Training Volcano | Number of Sequences | All | Erebus | Erta Ale | Etna | Kīlauea | Masaya | Nyamulagira | Nyiragongo | Pacaya | Pu’u ’Ō’ō
All | 1127 | 4.164 | 3.311 | 5.875 | 2.631 | 2.583 | 3.458 | 5.019 | 7.423 | 2.611 | 3.890
Erebus | 336 | 4.684 | 3.959 | 6.160 | 4.629 | 2.880 | 5.532 | 5.369 | 7.100 | 3.052 | 3.884
Erta Ale | 178 | 4.356 | 3.897 | 5.534 | 4.451 | 2.742 | 3.141 | 4.868 | 6.691 | 3.085 | 3.890
Etna | 132 | 5.696 | 5.629 | 7.341 | 3.667 | 4.533 | 6.369 | 5.898 | 7.652 | 3.347 | 5.206
Kīlauea | 150 | 4.390 | 3.863 | 5.486 | 4.737 | 2.729 | 3.229 | 4.955 | 6.745 | 3.350 | 3.882
Masaya | 44 | 5.343 | 4.772 | 6.671 | 5.362 | 3.739 | 6.073 | 5.966 | 7.527 | 3.992 | 4.585
Nyamulagira | 53 | 5.345 | 4.846 | 6.727 | 5.122 | 3.785 | 6.074 | 5.920 | 7.508 | 3.838 | 4.620
Nyiragongo | 74 | 5.365 | 4.788 | 6.678 | 5.422 | 3.765 | 6.103 | 6.001 | 7.552 | 3.970 | 4.611
Pacaya | 50 | 5.453 | 5.103 | 6.919 | 4.717 | 4.011 | 6.166 | 5.915 | 7.540 | 3.736 | 4.802
Pu’u ’Ō’ō | 120 | 4.946 | 4.340 | 6.356 | 4.967 | 3.119 | 5.636 | 5.469 | 7.241 | 3.543 | 4.109

It is commonly noted that every volcano is different and, as such, patterns
learned from one volcano may not generalize to another. It could have been
possible that a volcano’s validation set performance was minimized by training
only with data from that volcano, but that was not found to be the case; in
general, training with a specific volcano did not reduce RMSE on that volcano
itself. The only exception to this was Erta Ale (all volcanoes RMSE =
$5.875^{\circ}$C, Erta Ale only RMSE = $5.534^{\circ}$C). The reasons for this
are not known; it may be that patterns from the other volcanoes do not apply
well to Erta Ale, and/or the fact that, unlike most volcanoes in this data
set, Erta Ale has relatively flat, cloud-free terrain.

In $8/9$ cases, a training set consisting of one volcano did not benefit
validation set performance on that volcano itself, but in $4/9$ cases, this
did reduce validation set RMSE on at least one other volcano. This was the
case when the training set consisted of only Erebus, Erta Ale, Kīlauea, and
Pu’u ’Ō’ō resulting in validation set RMSE reductions in $2/8$, $3/8$, $5/8$,
and $1/8$ other volcanoes, respectively. Among these, the model fit using only
Kīlauea data improved validation set performance on at least the same
volcanoes that the others did (i.e., training sets of Erebus, Erta Ale, and
Pu’u ’Ō’ō). Kīlauea is one of the most well-studied volcanoes in the world
and, fortunately, this appears to suggest that thermal patterns learned from
it may generalize better than other well-known basaltic volcanoes.

###  III-D Opportunities

A forefront weakness in this forecasting project is that high-resolution
thermal imagery is the only data considered. Even within thermal data sources,
there are additional data sources that could be integrated—most obviously
MODIS. Additionally, regional weather satellites such as GOES could prove
valuable and would still provide nighttime imagery. Beyond nighttime imagery,
there are also the options of Landsat and Sentinel-2. With additional
expertise it would also be beneficial to include other forms of remote
sensing, such as InSAR for ground deformation and IR/UV for gas emissions,
and, when available, performance would likely benefit from the inclusion of
ground monitoring efforts. Ultimately, volcanoes display a variety of behavior
that can be measured on the ground and from space; here, we focused on thermal
satellite imagery, one of the most globally available and widely adopted.

Moving forward, the task of integrating all of this data will likely not be
straightforward. Experts in the field have speculated that ”few developments
promise to revolutionize eruption forecasting — along with so many other
branches of science — as does machine learning.” [1, p. 23]. We argue here
that important developments moving forward will likely require the development
of machine/deep learning methods with scientific goals and data
characteristics in mind. Geoscience fields, for example, are generally data-
sparse but rich with theory-based models [61], and it has been found that
integrating scientific theory with machine learning helps alleviate the burden
of low data quantity [60]. Additionally, scientific data may not have the same
characteristics as the data that machine learning methods were developed for.
For example, Earth data are often irregular in time and space, and imagery may
be very different (e.g., gravity-based data, interferograms, hyperspectral)
from the typical optical situations that computer vision is concerned with. In
summary, there are many broad directions and considerations for machine
learning in science, and scientific discovery and application will likely
benefit from models that are creatively specified to not only the data but
also the governing scientific goals and principles.

##  IV Conclusion

Here we present one of the first projects forecasting volcanic activity with
machine learning. Specifically, we forecasted ASTER nighttime temperature
imagery for 9 volcanoes between $1999$ and $2020$. To better match the remote
sensing data characteristics (i.e., temporally irregular image sequences), we
proposed three new neural network architectures, and among them, one achieved
lower RMSE than all considered models (e.g., autoregression, LSTM, ConvLSTM).
All models struggled with scale, but it was found that histogram matching
alleviated this, producing derived time series (e.g., maximum temperature
above background, the number of hotspots, and distance of furthest hotspot
from the summit) that generally approximated reality. It was discovered,
however, that the lowest RMSE on forecasting future scenes did not ensure the
lowest RMSE on derived tasks. This suggests a need for science-driven
optimization targets, especially when forecasting multipurpose monitoring
streams. Lastly, it appears that patterns learned from multiple volcanoes
generally increase forecasting performance relative to single-volcano training
sets. The pattern of improved validation set performance was not exclusively
tied to data set size, suggesting that certain volcanoes provide generalizable
patterns despite small data sets. Interestingly, the models fit to Kīlauea,
one of the most well-studied volcanoes in the world, were found to generalize
better than any other volcano considered here—further reducing RMSE relative
to the entire data set in $5/9$ volcanoes.

This work demonstrates the value of implementing data-driven changes to
popular neural networks for remote sensing forecasting, and we expect that it
should generalize well to other Earth observation tasks with data that are
irregularly available through time. This also demonstrated a weakness in using
standard optimization targets (e.g., base data RMSE) for multipurpose
scientific data; performing best at predicting data does not necessarily
indicate the best performance on the final uses of that data. Machine learning
has the potential for transformative results on scientific tasks, but it is
paramount that these efforts make data-driven adjustments, consider the full
problem to be explicitly optimized, and integrate existing scientific
theory/principles.

##  Appendix A Test Set Performance

Many important findings in this paper revolved around training conditions
(specifically, comparing different types of models and models that utilized
different subsets of the available data). As such, the test set—sat aside for
final performance evaluation of the best model—could not play a role in these
narratives. Here, we present some test set findings when using the proposed
model (ConvLSTM + Time-LSTM + U-net) that achieved the lowest validation set
RMSE on the image forecasting task.

First, this model was fit on a combination of the training and validation sets
(i.e., the chronologically first 85% of scenes per volcano) to more fully
utilize the available data. The resulting model was then evaluated on the test
set. Overall, this model achieved a test set RMSE of $4.100^{\circ}$C—lower
than all previously found validation set RMSEs (see Table 1).

In line with the discussions found in Section 3, Subsections A and B, this
model refused to produce very anomalous temperatures—the maximum temperature
it predicted for the entire test set was $10.268^{\circ}$C above the
background, while the maximum observation for these scenes was
$104.789^{\circ}$C above the background. This was problematic for the derived
series where it achieved RMSEs of $46.718^{\circ}$C, $635.387$ pixels, and
$3420.979$ meters for forecasting the maximum temperature above the
background, the number of hotspots, and the maximum distance of a hotspot from
the summit, respectively. Histogram matching to the training and validation
set alleviated performance at forecasting the maximum temperature above the
background, reducing the RMSE to $30.202^{\circ}$C; however, performance
remained poor at forecasting the number of hotspots ($644.899$ pixels) and the
maximum distance of a hotspot from the summit ($3819.775$ meters). In these
tasks, stretching the small range of predicted values resulted in far too many
hot pixels whereas before there were too few. For reference these values can
be compared to Table 2, but it should be noted that those metrics were
computed for different temporal ranges of the data (i.e., mutually exclusive
dates between $2015$ and $2019$ versus dates between $2017$ and $2020$,
depending on the volcano).

One of the events of particular interest in the test set is the 2018 eruptive
activity experienced at Kīlauea and its East Rift Zone [62]—represented in
part by the Pu’u ’Ō’ō scenes in our data set. Unfortunately, no available
scenes collected between May and August were deemed viable or uncertain, but
in relation to our data set, this data would have been at the Pu’u ’Ō’ō and
Kīlauea subsets, not the highly active lower East Rift Zone. There is,
however, relatively dense data availability prior to May 2018.

Leading up the May 2018 lower East Rift Zone eruptions, the model does produce
two interesting patterns in its forecasts (that have been histogram matched to
the training and validation set to account for known issues in maximum
values). The first is a large increase then sharp decrease in maximum
temperature above the background at the Kīlauea summit (Figures 11 and 12);
while this did not greatly mimic observations at the time, it did shortly
proceed the May 2018 lava lake withdrawal that had consequences in local
seismic activity and was indicative of broader changes in the plumbing system.
It may be argued that this reduced temperature is an artifact of the preceding
cloud cover, but Figure 7 demonstrates that cloud cover alone is not
sufficient to prohibit active forecasts. The second interesting forecast
pattern is that the model predicted an increasing amount of hot spots at the
Pu’u ’Ō’ō subset (Figures 11 and 13) leading up the lower East Rift Zone
activity that stood in contrast to the fluctuating—if not declining—pattern
seen in observations. In reality, the Pu’u ’Ō’ō vent ultimately collapsed,
thus displaying no effusive hotspots; however, the ultimate result increased
eruptive activity in the lower East Rift Zone. In summary, derived time series
from the model forecasted sharply decreasing anomalous temperatures at
Kīlauea’s summit shortly before the lava lake withdrawal and increasingly wide
eastward lava flows from Pu’u ’Ō’ō prior to Pu’u ’Ō’ō’s vent collapse and the
large lava flows in the lower East Rift Zone.

Figure 11: Test set time series for maximum temperature above the background
for Kīlauea’s summit and number of hotspots for Pu’u ’Ō’ō. The dashed lines
indicate the segment of forecasted values prior to the May 2018 lower East
Rift Zone lava flows. Notably, the Pu’u ’Ō’ō vent collapsed on April 30, 2018,
the Kīlauea lava lake began withdrawing on May 1, 2018, and there were large
lava flows from multiple fissures in the lower East Rift Zone in May 2018.
Figure 12: Test set image forecasts for Kīlauea’s summit. Notably, the lava
lake began withdrawing on May 1, 2018. Figure 13: Test set image forecasts for
Pu’u ’Ō’ō. Notably, the Pu’u ’Ō’ō vent collapsed on April 30, 2018 and there
were large lava flows from multiple fissures in the lower East Rift Zone in
May, 2018.

##  Appendix B Varying the Time between Scenes with the Best Proposed Model

The existing neural networks used in this paper assume one of three data
scenarios: sequential data whose observations are evenly distributed in time,
sequential images whose observations are evenly distributed in time, or
sequential data whose observations are unevenly distributed in time. The merit
of the proposed neural networks is that they represent a combination of the
latter two scenarios—modeling images whose observations are unevenly
distributed in time. These proposed models achieved some of the lowest RMSEs
compared to existing models and did so efficiently with respect to computation
time and number of parameters (see Table 1). Here, we inspect how image
forecasts change when the time between scenes is varied for the proposed model
that achieved the lowest validation set RMSE on the image forecasting task
(ConvLSTM + Time-LSTM + U-net).

We consider six simple modifications to the time differences between scenes to
evaluate how these time differences affect the image forecast. First, we
consider multiplying and dividing the first time difference in the six-scene
input sequence by a factor of $10$; furthermore, we consider the same
procedure applied to the last time difference in the six-scene input sequence;
and, lastly, we consider adjusting every time difference in this manner. To
compare these effects, we consider the mean value of the difference between
the originally forecasted scene (with unaltered time differences) and the
newly forecasted scene. Additionally, we examine the RMSE of the altered
forecast relative to the original forecast. In summary, these two metrics will
provide a general direction and magnitude of change.

TABLE IV: Effect on forecast of changing time differences | Mean of Differenced | RMSE of Altered
---|---|---
Forecast | Forecast | Forecast Relative to
Adjustment | (Adjusted - Original) | Original Forecast
First $\Delta$T * 0.1 | 0.002 | 0.003
Last $\Delta$T * 0.1 | 0.005 | 0.010
All $\Delta$T * 0.1 | 0.006 | 0.017
First $\Delta$T * 10 | -0.019 | 0.034
Last $\Delta$T * 10 | -0.053 | 0.098
All $\Delta$T * 10 | -0.063 | 0.169

The results presented in Table 4 suggest that reducing the time differences by
a factor of $10$ has very little effect on image forecasts. The average time
difference in the data set is $37$ days, so this likely indicates that the
model expects similar changes on the scale of approximately one week to one
month. Alternatively, the metrics comparing a change in the forecast are
generally an order of magnitude higher when time differences were adjusted in
the same manner but by a factor of $10$ (instead of one-tenth). Additionally,
predicted pixel values (on average) decreased when time differences were
increased. The greatest changes in forecasts were caused by increasing all
time differences by a factor of $10$; qualitative inspection of some of the
most-changed forecasts showed that there was a tendency for background areas
to be forecasted as cooler while volcanically active areas often remained
unchanged or were forecasted as hotter. Figure 14 displays how some forecasts
were different when all of the time differences were increased by a factor of
$10$.

Figure 14: Selected examples of how test set forecasts changed when the time
between input scenes was increased by a factor of 10. Displayed data are the
altered forecasts minus the original forecast after both have been histogram
matched to the training set (to account for known issues with the full range
of values). Sharp linear features in the bottom images represent mosaicking
artifacts.

As is common with neural networks, these patterns are difficult to interpret.
The tendency to increase average predicted temperatures at short time scales
and decrease them at long time scales may be related to the effects of the
atmosphere. For example, the thermal signal from clouds does get considered by
the forecasts, and sometimes it is passed forward as image features in the
forecast. When the time between scenes was increased, the average temperature
in the scene decreased, reducing the effects of weather (e.g., relatively warm
nighttime clouds) on the long-term forecast. Additionally, one criterion when
considering scene viability was whether a volcanic phenomenon was observed
through partial cloud cover; for example, large lava flows (particularly at
Pu’u ’Ō’ō) were included with partially obstructing conditions. Perhaps the
model learned that shorter time between scenes was indicative of more volcanic
activity coupled with more nighttime clouds. The more pronounced patterns of
forecasting greatly increased or decreased patterns in volcanically active
areas are much harder to interpret. It is notable, however, that these
patterns are highly spatial, and the existence of differing forecast patterns
that emerge when changing the temporal structure is suggestive that the
temporal information is valuable and that machine learning algorithms are
inclined to find patterns within it.

## Acknowledgment

This material is based upon work supported by the National Science Foundation
Graduate Research Fellowship under Grant No. DGE1255832. Any opinions,
findings, and conclusions or recommendations expressed in this material are
those of the authors and do not necessarily reflect the views of the National
Science Foundation. Additionally, we thank Penn State’s Institute of
Computational and Data Sciences for both funding and computational resources.
Lastly, Helen Greatrex and Manzhu Yu for serving on the advising committee
which led to this paper.

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|  Jeremy Diaz received the B.A. degree in ecology and evolutionary biology
from the University of Colorado Boulder, Boulder, CO, and the M.S. degree in
geography from the Pennsylvania State University, University Park, PA, in
2021. From 2016 to 2019, he was a student assistant at the Cooperative
Institute for Research in Environmental Sciences, Boulder, CO. From 2017 to
2019, he worked with the Earth Lab Analytics Hub applying deep learning to
various natural hazards projects. During his time at the Pennsylvania State
University (from 2019 to 2021), he was affiliated with the Geoinformatics and
Earth Observation Laboratory (GEOlab), the Institute of Computational and Data
Sciences, and earned a minor in electrical engineering from the School of
Electrical Engineering and Computer Science. His main research interests
include applying quantitative methods and computing to natural hazards and
earth science research. He is now a Machine Learning Specialist for the United
States Geological Survey’s Water Mission Area. Mr. Diaz was a recipient of the
2020 National Science Foundation (NSF) Graduate Research Fellowship (GRF).
---|---
|  Guido Cervone received the B.S. degree from the Catholic University of
America, Washington, DC, USA, in 1998, and the master’s and Ph.D. degrees in
computer science from the George Mason University, Fairfax, VA, USA, in 2000
and 2005, respectively. He is currently Associate Director of the Institute
for Computational and Data Science and Professor in Geography, Meteorology and
Atmospheric Science at the Pennsylvania State University, State College, PA,
USA. He is also a member of the Earth and Environmental Systems Institute. His
research sits at the intersection of geospatial science, atmospheric science,
and computer science. His research focuses on the development and application
of computational algorithms for the analysis of remote sensing and numerical
modeling data related to environmental hazards and renewable energy.
---|---
|  Christelle Wauthier received a B.S. degree in Engineering, and a M.S. and
Ph.D. degrees in geological engineering from the University of Liege, Belgium,
in 2003, 2005 and 2011, respectively. She also received a M.S. degree in
Volcanology from the University Blaise-Pascal, France, in 2006. She is
currently an Associate Professor of Geosciences in the Department of
Geosciences and Institute for Computational and Data Sciences, Pennsylvania
State University, State College, PA, USA. Thanks to modeling the surface
deformation identified with satellite geodesy, Dr. Wauthier intends solving
for the subsurface dynamic characteristics of a volcanic system; i.e.,
characterize where and how is the magma transported and stored. A thorough
study and modeling of ground deformation, together with a stress change
analysis, can provide clues on when, where, and how a volcano will erupt,
provide insights on magma-tectonic processes and their causality. Her research
has deep impacts on society as the geophysical signals registered can be used
to assess and mitigate volcanic and seismic hazards.
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