# Structural Forecasting for Short-term Tropical Cyclone Intensity Guidance

###### Abstract

Because geostationary satellite (Geo) imagery provides a high temporal
resolution window into tropical cyclone (TC) behavior, we investigate the
viability of its application to short-term probabilistic forecasts of TC
convective structure to subsequently predict TC intensity. Here, we present a
prototype model which is trained solely on two inputs: Geo infrared imagery
leading up to the synoptic time of interest and intensity estimates up to 6
hours prior to that time. To estimate future TC structure, we compute cloud-
top temperature radial profiles from infrared imagery and then simulate the
evolution of an ensemble of those profiles over the subsequent 12 hours by
applying a Deep Autoregressive Generative Model (PixelSNAIL). To forecast TC
intensities at hours 6 and 12, we input operational intensity estimates up to
the current time (0 h) and simulated future radial profiles up to +12 h into a
“nowcasting” convolutional neural network. We limit our inputs to demonstrate
the viability of our approach and to enable quantification of value added by
the observed and simulated future radial profiles beyond operational intensity
estimates alone. Our prototype model achieves a marginally higher error than
the National Hurricane Center’s official forecasts despite excluding
environmental factors, such as vertical wind shear and sea surface
temperature. We also demonstrate that it is possible to reasonably predict
short-term evolution of TC convective structure via radial profiles from Geo
infrared imagery, resulting in interpretable structural forecasts that may be
valuable for TC operational guidance.

This work presents a new method of short-term probabilistic forecasting for
tropical cyclone (TC) convective structure and intensity using infrared
geostationary satellite observations. Our prototype model’s performance
indicates that there is some value in observed and simulated future cloud-top
temperature radial profiles for short-term intensity forecasting. The non-
linear nature of machine learning tools can pose an interpretation challenge,
but structural forecasts produced by our model can be directly evaluated and
thus may offer helpful guidance to forecasters regarding short-term TC
evolution. Since forecasters are time-limited in producing each advisory
package despite a growing wealth of satellite observations, a tool that
captures recent TC convective evolution and potential future changes may
support their assessment of TC behavior in crafting their forecasts.

##  1 Introduction

Tropical cyclones (TCs) are powerful, organized systems that pose a major risk
to coastal populations. Though many statistical models provide forecast
guidance on future TC intensity change (e.g., the Statistical Hurricane
Intensity Prediction Scheme [SHIPS]; DeMaria and Kaplan 1999), direct
measurement of most predictors such as relative humidity or vertical wind
shear used in such models is impossible due to the development of TCs over
open ocean far from land-based observing networks (Gray, 1979). Many
predictors must be inferred through a combination of remote observation and
dynamic models of ocean and atmospheric behavior.

Infrared (IR; 10.3-10.7$\mu$m) imagery from geostationary (Geo) satellites
such as the Geostationary Operational Environmental Satellites (GOES) provides
one of the few regular high-resolution observations of TC behavior over the
open ocean with a historical record spanning decades (Knapp and Wilkins, 2018;
Janowiak et al., 2020). Furthermore, modern Geo IR platforms such as GOES-16
provide observations at even greater spatial and temporal resolution (Schmit
et al., 2017). Since cloud-top temperature is related to cloud-top height, low
IR temperatures tend to indicate higher cloud tops and thus stronger
convection, and convective structures are known to be related to TC intensity
(Dvorak, 1975; Olander and Velden, 2007).

In light of this growing record of satellite observations, a broad array of
recent works have explored the wealth of information contained in the _spatio-
temporal_ structure of Geo IR imagery. The Dvorak technique and more recent
Advanced Dvorak Technique (ADT) have long related Geo IR imagery to TC
intensity (Dvorak, 1975; Olander and Velden, 2007), and more recent work has
leveraged neural networks to improve the nowcasting accuracy of the ADT (AI
enhanced Dvorak Technique; Olander et al. 2021).  Here, we define “nowcasting”
as estimating the current TC intensity based on intensity estimates up to 6 h
prior and IR features up to the current time (0 h). Spatial analyses of IR
imagery have been leveraged to improve forecasts of TC eye formation, a
process related to intensification (DeMaria, 2015; Knaff and DeMaria, 2017).
The deviation angle variance (DAV) technique, a measure of convective
organization in IR imagery, contains valuable information for short-term
($\leq$24 h) TC intensity guidance (Hu et al., 2020). The shape and evolution
of Geo IR radial profiles is known to relate to intensity and intensity change
respectively (Sanabia et al., 2014; McNeely et al., 2020). In this work, we
utilize the evolution over time of radial profiles (see Figure 1) to jointly
forecast short-term TC intensity and structure changes. We leverage deep auto-
regressive (AR) generative models to construct interpretable and high-
resolution structural probabilistic forecasts, which display entire functions
rather than time series of thresholded quantities, such as pixel counts
beneath a given temperature threshold.

Concurrent with the rise of high-resolution Geo IR imagery is the growing
application of convolutional neural networks (CNNs), powerful tools for
performing prediction tasks with images as input. Predicting TC intensity from
Geo IR data is an obvious candidate application; indeed, there are dozens of
such works in the machine learning literature applying CNNs to this problem,
including Pradhan et al. (2017); Combinido et al. (2018); Lee et al. (2019);
Tian et al. (2020); Wang et al. (2020); and Zhang et al. (2021). These models
achieve reasonable forecast accuracy via the traditional machine learning
framework with a CNN taking IR imagery as input to directly predict intensity
by, e.g., minimizing the average squared-error loss on independent test data.
Explainable AI approaches may then use methods such as layer-wise relevance
propagation, saliency maps, and activation maps to better understand _how_ the
model produced its point estimate (McGovern et al., 2019; Ebert-Uphoff and
Hilburn, 2020). For an example of explainable CNN-based TC intensity
forecasting in the meteorological literature, see Griffin et al. (2022).

Our proposed pipeline takes a different approach to explainability—one which
remains compatible with the above tools for insight into the relationships
leveraged by CNNs. Our approach (i) utilizes a dimensionality-reducing
functional transformation of IR imagery prior to analysis, and (ii) provides
12-hour ensemble forecasts of TC convective structure in addition to TC
intensity.

First, we extract scientifically-motivated functional features, reducing the
dimension of the problem (from 2-D images over time to 1-D functions over
time) in a directly interpretable summary, rather than directly relying on the
CNN to extract salient features from (high-dimensional and low-sample size)
raw Geo IR imagery. These rich summary functions are derived from the ORB
suite: Organization (e.g., DAV as a function of radius), Radial structure
(e.g., the radial profiles examined in this work), and Bulk morphology (e.g.,
pixel counts as a function of a temperature threshold). Temporal sequences of
radial profiles are highly relevant to both intensity and intensity change
(Sanabia et al., 2014; McNeely et al., 2020, 2022). Temporal changes in these
sequences of profiles  can be visualized via Hovmöller diagrams, which are
more readily digestible by users than inferring temporal patterns from
animations of satellite imagery.

Figure 1: Evolution of TC IR Imagery as Structural Trajectories. The raw data
at each time $t$ is a sequence of TC-centered cloud-top temperature images
from GOES. We quantify the image at time $t$ by its radial profile ($X_{t}$).
The sequence of consecutive radial profiles, sampled every 2 hours, defines a
_structural trajectory_ or Hovmöller diagram ($S_{<t}$).

Second, we provide a  probabilistic structural forecast, a prediction of  an
ensemble of possible TC convective evolution, rather than directly predicting
future intensity from past IR structure  and TC intensity. Our novel approach
to intensity guidance via Geo IR imagery results in interpretable intensity
forecasts such as “our model predicts  short-term intensification due to the
potential emergence of an eye-eyewall structure in the next 12 hours”. Though
methods such as layer-wise relevance propagation can provide further insight
into the CNN’s use of structural forecasts, the IR structural forecasts
themselves are the core of our proposed intensity guidance pipeline.

Figure 2 outlines Section 3 via a schematic diagram of the structural
forecasting to intensity forecasting pipeline. There are three main
subsections:

  1. (a)

Structural trajectories via ORB. First, we apply the ORB framework (McNeely et
al., 2019, 2020) to observed IR imagery to create a “structural summary”
(Figure 1) of the spatio-temporal evolution of the present and recent past TC
structure.

  2. (b)

Structural forecasting  with a deep autoregressive generative model. Next, we
propagate the observed IR structure up to 12 hours forward in time via a deep
pixel-autoregressive model, which stochastically simulates  an ensemble of
possible trajectories of IR radial profiles.

  3. (c)

Forecasting TC intensity via convolutional neural networks. Finally, we input
the observed structure, the forecasted structure, and  TC intensity up to 6
hours prior to the current time into a nowcasting model to estimate the
current intensity; we choose CNNs because they are easy to train and commonly
used for image data.  By filling in the missing $t+6$ hour and $t+12$ hour
structure, we can then extend the nowcasting model from a nowcast for time $t$
(i.e., hour 0) to a forecast at time $t+6$ hours and then to time $t+12$
hours.

Section 4 details the results of our prototype forecasting pipeline. The final
Geo IR-based TC intensity guidance provides inherent measures of uncertainty
and  insight into the potential TC structural changes that influence a given
forecast. The results in this work use proof-of-concept structural forecasting
and  a pipeline that relies solely on persistence predictors (i.e., prior
intensity  estimates) together with  observed past and simulated future radial
profiles; no environmental factors such as vertical wind shear or ocean heat
content are included at this time. We demonstrate that a purely autoregressive
prototype achieves a useful degree of forecasting accuracy.

Figure 2: TC Intensity Guidance via Structural Forecasting. Outline of the
structural forecasting to intensity guidance pipeline. (a) ORB functions are
used to quantify the evolution of spatio-temporal convective structure,
linking IR imagery to Structural Summaries. (b) We generate structural
forecasts by projecting the ORB functions into the future via a deep
autoregressive model, thereby filling in the missing +6 hour and +12 hour
structure. (c) A CNN nowcasting model then forecasts intensities at +6 to +12
hours from three sources of inputs: (i) observed structure, (ii) forecasted
structure, and (iii)  operational intensity estimates up through the current
time.

##  2 Data

Our model relies on two data sources: sequences of Geo IR imagery captured by
GOES satellites and past TC intensity. For training and verification (that is,
model selection), we use NOAA’s Hurricane Database 2 (HURDAT2; Landsea and
Franklin 2013) because that database provides the post-season best estimates
of TC intensities. For forecasting, we rely on _operational_ TC intensity
estimates, the CARQ entries from the Naval Research Laboratory’s Automated
Tropical Cyclone Forecast (ATCF) operational “A-deck” files (Sampson and
Schrader 2000) to assess model performance under real-time conditions.

GOES IR imagery is available through NOAA’s Merged IR (MERGIR) database
(Janowiak et al., 2020) at 30-minute$\times$4-km resolution over the North
Atlantic (NAL) basin from 2000–2020. For each TC, we download $\sim$2,000
km$\times$2,000 km “stamps” of IR imagery centered on the TC location at a
30-minute temporal resolution. Figure 1 (left) shows two such stamps after an
800-km radius mask is applied. For this work, we sample the 30-minute data at
2-hour resolution because of periodic corruption of the imagery in the MERGIR
database (Liu, 2021).

During training, we linearly interpolate TC location and intensity from
HURDAT2 to obtain locations and intensities for non-synoptic times; however,
model assessment is restricted to synoptic times. We include TC lifetimes
between the first synoptic time at which intensity reaches at least 35 kt and
the last synoptic time at which intensity is at least 35 kt; note that this
can result in the inclusion of TCs $<$ 35 kt if the TC decays and then re-
intensifies.

Finally, we rely on NHC’s official forecast verification to assess our model’s
performance. We also draw on the SHIPS developmental database’s 200-850-hPa
vertical wind shear values calculated within a 200-800-km annulus from the TC
center  as reference during model validation due to the  known impact of shear
on TC convective structure (DeMaria, 2018).

##  3 Methods

As outlined in Figure 2, we first construct a summary of IR structural
evolution (a; Section 33.1). We then train a stochastic autoregressive model,
which is an explicit likelihood model (of structural trajectories) that we can
use to simulate probable IR structural evolution (b; Section 33.2). Finally,
we combine observed and forecasted structure with  operational intensity
estimates up to and including the current time to provide interpretable short-
term intensity guidance, based solely on Geo IR imagery and operational
intensity estimates (c; Section 33.3).

###  3.1 Structural Trajectories via ORB

Operational forecasting of TC intensity is a human-in-the-loop process and
thus places a premium on guidance interpretability. In this spirit, the ORB
framework (Organization, Radial structure, Bulk morphology) summarizes 2-D
imagery via continuous 1-D functions to enable static visualization of spatio-
temporal patterns in TC development via Hovmöller diagrams (Hovmöller, 1949).
Our past work focused on the rich quantification of spatial information in Geo
IR imagery (McNeely et al., 2019, 2020). More recently, we demonstrated the
value of _temporal_ patterns in ORB functions (McNeely et al., 2022),
specifically the radial profile.

The radial profile of brightness temperature
$\overline{BT}(r)=\frac{1}{2\pi}\int_{0}^{2\pi}T_{b}(r,\theta)d\theta$
captures the structure of cloud-top brightness temperature ($T_{b}$) as a
function of radius $r$ from the TC center and serves as an easily
interpretable description of the depth and location of convection near the TC
core (Sanabia et al., 2014; McNeely et al., 2020). The radial profiles are
computed at 5-km resolution from 0-400 km ($d=80$) (Figure 1, center); we
denote the summary of convective structure at each time $t$ by
$\mathbf{BT}_{t}$. The _structural trajectory_ is then defined as the 24-hour
sequence of present and 12 preceding radial profiles at a 2-hour resolution:

${\color[rgb]{0,0,0}\mathbf{S}_{<t}=(\mathbf{BT}_{t-24h},\mathbf{BT}_{t-22h},\dots,\mathbf{BT}_{t})}$
(1)

We visualize such a trajectory with a Hovmöller diagram (see Figure 1, right).

McNeely et al. (2022) demonstrated a relationship between TC intensity change
and Hovmöller diagrams of radial profiles. However, the radial profile, if
averaged over all angles, will disregard asymmetry within the original 2-D
images,  which can degrade performance for cases affected by strong vertical
wind shear. In this work, we instead compute a separate radial profile for
each geographic quadrant (NE, NW, SE, SW) to capture asymmetries via the
differences between quadrants. We use geographic quadrants instead of motion-
relative or shear-relative quadrants because the directions of motion and
shear are unstable when the magnitudes of those vectors are small.

###  3.2 Structural Forecasting via Deep Autoregressive Generative Model

The crucial step in our guidance framework is the propagation of radial
profiles into the near future. The Hovmöller diagram captures the spatio-
temporal evolution of the TC over an extended period of time; that is, we can
summarize TC development by an easily-interpretable image. By treating the
structural trajectory as an image, where the y-axis corresponds to the passage
of time, forecasting radial profiles becomes equivalent to an image completion
problem. That is, we predict the missing pixels at the bottom of an image
(forecasted structure) given those at the top (observed structure). Image
completion is an active research area in machine learning; here we focus on a
state-of-the-art model in the class of pixel-autoregressive models (Van Oord
et al., 2016).

Pixel-autoregressive models impose an ordering on the pixels of an image, such
as a raster-scan ordering (left-to-right, top-to-bottom). Let each pixel in a
four-quadrant radial profile trajectory be represented by
$\mathbf{x}_{i}:=(x_{i1},x_{i2},x_{i3},x_{i4})$, where $i$ is the index in the
raster scan. The pixel-AR approach factors the joint distribution of pixel
values in the image as a product of conditionals,

$p(\mathbf{x}_{1},...,\mathbf{x}_{n})=\prod_{i=1}^{n}p(\mathbf{x}_{i}|\mathbf{x}_{i-1},...,\mathbf{x}_{1}),$
(2)

where the probability of each pixel value is conditioned on all previous
pixels in the raster scan. Then, to generate a new radial profile, one
simulates repeatedly from
$p(\mathbf{x}_{i}|\mathbf{x}_{i-1},...,\mathbf{x}_{1})$; due to the raster-
scan ordering, the distribution of a given pixel is not influenced by elements
further down the sequence, hence enforcing causality.

The challenge of how to estimate the conditional likelihoods
$p(\mathbf{x}_{i}|\mathbf{x}_{i-1},...,\mathbf{x}_{1})$ has given rise to many
flavors of pixel-autoregressive models, including PixelRNN (Van Oord et al.,
2016), PixelCNN (Van den Oord et al., 2016), PixelCNN++ (Salimans et al.,
2017), and PixelSNAIL (Chen et al., 2018). This work utilizes the last model,
PixelSNAIL. There are two main ingredients in the model: (i) causal
convolution and (ii) self-attention. Causal convolution utilizes the same
convolutional feature extraction outlined in Section 3.3 but masks each
convolution so that each element in the raster sequence only receives
information from previously generated sequences (e.g., Figure 3). Purely
convolutional models, however, are restricted to small neighborhoods of
pixels, leading to only a finite receptive field (area of the source image
involved in a given convolution), and thus struggle with long-range
dependencies in the conditional
$p(\mathbf{x}_{i}|\mathbf{x}_{i-1},...,\mathbf{x}_{1})$. PixelSNAIL, on the
other hand, features a self-attention mechanism that leads to unbounded
receptive fields with pinpointed access to information far away in the
sequence; see Chen et al. (2018) for details on the PixelSNAIL architecture.

To ensure that cloud-top temperatures remain bounded, we rescale the data to
the range ${\mathbf{X}}\in(0,1)^{4}$ and work in the logit-transformed space,
$\mathbf{Z}=\log({\mathbf{X}}/(1-{\mathbf{X}}))$; while values of $\mathbf{Z}$
are unbounded, ${\mathbf{X}}$ remains bounded in $(0,1)^{4}$ and are then
transformed back to the temperature range observed in the training data. We
model the density $p(\mathbf{z}_{i}|\mathbf{z}_{i-1},...,\mathbf{z}_{1})$ as 4
independent mixtures of logistic distributions, one for each quadrant. That
is, for pixel $i$ in quadrant $q$,

$\displaystyle p(z_{iq}|\mathbf{z}_{i-1},\dots,\mathbf{z}_{1})=$
$\displaystyle\sum_{k=1}^{K}\pi_{qk}f(z_{iq};\mu_{qk},s_{qk})$
$\displaystyle\text{where }f(z_{iq};\mu_{qk},s_{qk})=$
$\displaystyle\frac{g_{qk}(z)}{s_{qk}(1-g_{qk}(z))^{2}},$ $\displaystyle
g_{qk}(z)=$ $\displaystyle\exp{\big{(}-(z-\mu_{qk})/s_{qk}\big{)}},$
$\displaystyle\text{and }\sum_{k=1}^{K}\pi_{qk}=$ $\displaystyle 1.$

Thus, with four quadrants and $K$ mixture components, the distribution
$p(\mathbf{z}_{i}|\mathbf{z}_{i-1},...,\mathbf{z}_{1})$ has $4(3K-1)$
parameters: a mean $\mu$, a scale $s$, and a mixture coefficient $\pi$ for
each quadrant, with the constraint that the mixture coefficients in each
quadrant sum to one. With $K=3$ mixture components, this results in 32
parameters total. Each draw from the distribution is transformed back to the
bounded space via the relationship
$\mathbf{X}=\frac{1}{1+\exp{(-\mathbf{Z})}}$, then rescaled to the range of
values observed in the input radial profiles.

This autoregressive model enables stochastic simulation of structural
trajectories based on Geo IR persistence. For a given synoptic time, we can
simulate many trajectories from the observed history and then feed each
potential trajectory through the nowcasting model to obtain the associated
intensity guidance. Via multiple simulations per forecast time, an ensemble
forecast provides a measure of uncertainty in both structural trajectories and
intensities while also offering insight into cases where the model over- or
under-estimates intensity. For example, overestimates may be caused by too-low
profile temperatures or overestimated symmetry between quadrants.

Figure 3: Masking in Pixel Autoregression. Illustration of raster-scan
ordering and the causal masking. Convolutions at index $i$ only have access to
pixel values in previous rows (earlier time points, color coded by yellow),
and pixel values in the same row but to the left of pixel $x_{i}$ (same time
point, color coded by orange).

The structural forecasting model is trained on TCs from 2000-2012, with
2013-2020 withheld for testing. We train the model using input radial profiles
calculated every 2 hours but test on synoptic times. Because AR models are
likelihood-based, we can directly calculate and minimize the negative log-
likelihood (NLL), a measure of the model’s ability to generalize well on
withheld data.

###  3.3 Nowcasting TC Intensity via Convolutional Neural Networks

Traditional linear models are attractive for reasons of interpretability and
good performance in low sample size settings. However, linear models often
struggle to capture the complex, time-varying processes which drive TCs. It is
also unclear how to include the radial profile Hovmöller diagrams as inputs to
a linear model without sacrificing interpretability. In this work, we instead
consider a simple convolutional neural network to map observed IR trajectories
($\mathbf{S}_{t}$) to current intensities ($Y_{t}$). Because we treat time as
a spatial dimension in these diagrams and a structural trajectory is
represented as an image, a CNN will leverage both spatial and temporal
patterns in the data.

CNNs operate by two main elements: convolutional layers and fully-connected
layers. The convolutional layers first convolve each layer (here, each
quadrant) with a library of filters (i.e., matrices whose entries are learned
parameters); some of these filters may resemble familiar matrices, such as
gradient approximators (e.g., Sobel matrices). After each convolutional layer,
the image is pooled to reduce the image size and increase the receptive field
of the next set of convolutions. In the final step, the results of all
convolutions are passed into a fully-connected layer which approximates the
relationship between the convolutional feature map and the response.

Figure 4: Nowcasting Model Architecture. The convolutional neural network used
for nowcasting consists of three convolution-max pool layer pairs (Conv#),
fully connected layers (FC#), and a concatenation with persistence features
between regressions.

We augment the traditional CNN in two ways. First, we add two layers which
encode the radial and temporal location of pixels within the image; regular
CNNs are translation invariant, whereas patterns in TCs have different meaning
depending on their location and time of occurrence (corresponding to column
versus row index, respectively, in the Hovmöller diagram). Second, we augment
the model output with relevant TC persistence features (intensities and
intensity changes up to 6 hours prior to time $t$) before passing them to a
second fully-connected layer. The final model is then

$\displaystyle Y_{t}=f_{\text{nwcst}}$
$\displaystyle(\mathbf{S}_{<t},Y_{t-30h},$ $\displaystyle
Y_{t-24h},\ldots,Y_{t-6h},\Delta Y_{t-30h},\Delta
Y_{t-24h},\ldots,Y_{t-6h})+\epsilon_{t},$ (3)

where $\mathbf{S}_{<t}$ is the 24-h structural trajectory up to current time
(see Equation 1); the $Y$’s are intensity data at 6-h resolution; the $\Delta
Y$’s are intensity changes interpolated (ip) from 6-h data to a 2-h resolution
(e.g., $\Delta Y_{t-30h}=Y_{t-30h}-Y^{\rm ip}_{t-32h}$), and $\epsilon$ is the
prediction error.

Like the structural forecasting model, the nowcasting model is trained on TCs
from 2000-2012, here by minimizing the mean squared error. The model is
trained on data with a 2-hour resolution (rather than synoptic times alone)
with intensities linearly interpolated to those times; we do not include non-
synoptic times in the test TCs (2013-2020). The details of the CNN
architecture are given in Figure 4.

Figure 5: Case Studies: Comparison of the 6-hour (left) and 12-hour (right)
intensity forecasts (gray) from the structural forecasting model versus
observed intensities (black). Vertical wind shear also shown (gold). The solid
gray lines indicate the mean of 64 simulations at each time point. Reported
errors indicate the error of the average model prediction, not individual
simulations. The model captures the behavior of Hurricanes Jose [2017; center]
and Dorian [2019; bottom] fairly well, but struggles with both rapid weakening
events exhibited by Hurricane Nicole [2016; top] as well as low-intensity
maintenance periods in Hurricanes Nicole and Dorian.

###  3.4 From Nowcasting to Forecasting

Section 33.3 defines a nowcasting model for  estimating intensity at time $t$
(i.e., hour 0)  by training on post-season (best-track or HURDAT2) intensities
from -30 h to -6 h and imagery from -30 h to 0 h. After we have trained and
validated the nowcasting model to estimate 0-h intensities, we apply the CNN
nowcasting model to TC intensity forecasting. To  forecast intensity at time
$t+6$ h, we need the intensities at times $\leq t$ (in this work, we use
operational intensities drawn from CARQ in the A-deck files when generating TC
intensity _forecasts_) and structural trajectories at times $\leq t+6$ h
(observed at times $\leq t$ and simulated at times from $t+2$ h to $t+6$ h).
Using the structural forecasting model in Section 33.2, we simulate many
possible trajectories from times $t+2$ hr to $t+6$ hr. Each of these possible
future trajectories is then passed to the nowcasting model to obtain a
separate intensity forecast,  giving an ensemble of possible intensities.

Our proposed framework for intensity forecasts  at +6 and +12 h has two
primary benefits: (i) by providing an additional structural forecast, we
provide insight into potential TC evolution predicted by the model, such as
deepening convection or the emergence of an eye; (ii) because the structural
forecast is stochastic, we can straightforwardly assess the uncertainty in
structural evolution over time and the associated uncertainty in the intensity
forecasts.

##  4 Model Results

We first demonstrate the performance of our proposed model on specific cases
(Hurricanes Jose [2017], Nicole [2016], and Dorian [2019]) in Section 4.1,
discussing both accuracy and the insight provided by structural forecasts at
6- and 12-hour lead times. We then assess the performance of the model during
2013-2020 in the North Atlantic basin at 6- and 12-hour lead times in Section
4.2.

Figure 6: Observed and Simulated Trajectories, Hurricane Dorian [2019]: The
observed structural trajectory is shown in the top left corner. To the right,
we see 10 individual simulations of radial profiles (averaged over all
quadrants) at 12-hour lead times. All radial profiles above the black
horizontal line are observed, while profiles below the black line are
simulated. The bottom left corner shows the arithmetic mean over 64 simulated
trajectories.

###  4.1 Case Studies

We examine Hurricane Jose (2017) due to the presence of high vertical wind
shear which produces convective asymmetries not captured by the azimuthally-
averaged radial profiles (i.e., not quadrant-based) of McNeely et al. (2022).
Hurricane Nicole (2016) was selected due to undergoing two rapid
intensification and two rapid weakening events. Finally, Hurricane Dorian
(2019) was a powerful TC with many _in situ_ observations.

#### Intensities

Figure 5 shows the 6-hour forecasts based on 64 independently simulated
structural trajectories per synoptic time. Because the structural forecasts
are currently based entirely on persistence—no environmental fields, such as
200-850-hPa vertical wind shear, have been included—we expect the guidance to
be most useful in the short term (6- to 12-hour time frame). The steadier
development of Hurricanes Jose and Dorian are well-modeled, but the swift
intensity changes exhibited by Hurricane Nicole as well as the rapid
intensification period of Hurricane Dorian both prove challenging to capture.

Figure 7: Hurricane Dorian [2019] 6-h Guidance: (Left) Distribution of
forecasted intensities with observed (black) and average forecast (red)
intensities marked.  The distribution of the 64 intensity forecasts in the
ensemble is approximated by a histogram (bar plot) and by a kernel density
estimate (blue curve). (Right) Simulated profiles by quadrant with observed
profiles represented by solid black curves, and averaged simulated profiles
represented by dashed red curves. Figure 8: Hurricane Dorian [2019] 12-h
Guidance: Model bias during intensification (center row) is more exaggerated
at 12-hour lead times. (Left) Distribution of forecasted intensities with
observed (black) and average forecast (red) intensities marked.  The
distribution of the 64 intensity forecasts in the ensemble is approximated by
a histogram (bar plot) and by a kernel density estimate (blue curve). (Right)
Simulated profiles by quadrant with observed profiles represented by solid
black curves, and averaged simulated profiles represented by dashed red
curves.

Extending lead time to 12 hours increases the variation among individual
simulations, but the average simulated intensity continues to roughly track
the observed intensities. The rapid intensity change events exhibited by
Hurricane Nicole are challenging to forecast with only IR persistence.
However, the model follows Hurricane Jose’s evolution relatively well,
indicating that the model has value as-is at 12-hour lead times.

#### Diagnostics

While the end goal of intensity guidance models is ultimately prediction of TC
intensity,  our structural forecasting pipeline  adds valuable diagnostic
insight into  structural factors contributing to its predictions.

Figure 6 demonstrates the three-step (12-hour lead time) structural forecast
for Hurricane Dorian  valid for 18 UTC 27 August during a period in which it
maintained 45-kt intensity. The final 6 rows of each Hovmöller diagram are
simulated from the structural forecast model; in this figure, we average the
four quadrants for ease of visualization (see Appendix A, Figure B10, in the
supplementary materials for Hurricane Dorian structural forecasts broken down
by quadrant). Cloud-top temperature magnitude tends to be underestimated, but
the expansion of cloud coverage  during this 12-h period is captured across
most simulations.

Figure 7 demonstrates the 6-hour forecasting guidance available at individual
synoptic times. The average simulated profiles in each quadrant tend to track
observed profiles reasonably well, although they  tend to predict too flat a
curve and too symmetric an eye. Figure 8 shows the same information but for
the 12-hour lead time. Here, model biases tend to be amplified by longer lead
times. We note that the emergence of an eye is captured in trajectory (B),
even 12 hours out.

Similar figures for Hurricanes Jose and Nicole are provided in the
supplemental material. In general, the structural forecast follows the
observed profile, even at 12-hour lead times. We did not perform any data
augmentation during training (e.g,. rotation) in order to preserve dominant
geographic patterns (e.g., the prevalence of TC convection sheared eastward
and northeastward in the North Atlantic), but it is possible that augmentation
by rotating TCs would improve simulation fidelity, as it has been shown to
improve accuracy in other TC intensity forecasting applications such as
Griffin et al. (2022).

###  4.2 Model  Verification

The same models are used to produce 16 simulated trajectories with associated
intensity guidance for each synoptic time from 2013-2020 at the 6- and 12-hour
lead times. (We use 16 rather than 64 simulations when validating over the
entire 8-year period for computational reasons.) Intensity predictions
provided via averaging the 16 simulations are validated against HURDAT2 best-
track intensities, and  past TC intensity values provided as input to the
model come from operational estimates (CARQ) to emulate real-time performance.

Table 1 (left) reports  the performance of the simulated trajectories versus
forecast lead time in terms of root mean variance (RMV), mean absolute
deviation (MAD), and bias averaged over all quadrants and radii. Let
$\overline{BT}_{qi}(r)$ denote the $i^{\text{th}}$ simulated profile in
quadrant $q$ and $\tau_{q}(r)$ denote the true profile in quadrant $q$. Then,

$\displaystyle\mathrm{RMV}=$ $\displaystyle\Big{(}\frac{1}{400\mathrm{\
km}}\int_{0}^{400}\frac{1}{4n}\sum_{q=1}^{4}\sum_{i=1}^{n}(\overline{BT}_{qi}(r)-\tau_{q}(r))^{2}dr\Big{)}^{\frac{1}{2}},$
(4) $\displaystyle\mathrm{MAD}=$ $\displaystyle\frac{1}{400\mathrm{\
km}}\int_{0}^{400}\frac{1}{4n}\sum_{q=1}^{4}\sum_{i=1}^{n}\big{|}\overline{BT}_{qi}(r)-\tau_{q}(r)\big{|}dr,$
(5) $\displaystyle\mathrm{Bias}=$ $\displaystyle\frac{1}{400\mathrm{\
km}}\int_{0}^{400}\frac{1}{4n}\sum_{q=1}^{4}\sum_{i=1}^{n}\overline{BT}_{qi}(r)-\tau_{q}(r)dr.$
(6)

The above are defined for simulations at a single simulation time; to combine
over multiple simulation times, we average MAD and bias, and average RMV in
quadrature. The measures of noise (RMV and MAD) are large even for the
shortest forecast lead times; increasing the simulation size beyond 16 will
reduce the impact of this noise on the average forecast. Bias, meanwhile,
becomes steadily more negative with time  for our structural forecasts (top
left). This will lead to overestimates of intensity, as low IR temperatures
are generally associated with stronger TCs.  Persistence IR forecasts (bottom
left) offer a less biased IR forecast on average but higher overall errors in
structure at all lead times.

Table 1 (right) reports  verification statistics for intensity guidance using
the traditional definitions for root mean squared error (RMSE), mean absolute
error (MAE), and bias. As expected, the negative bias in structural forecasts
manifests as a positive bias in intensity guidance.

Trajectory  Verification
Model Lead time 2 h 4 h 6 h 8 h 10 h 12 h Structural RMV 10.5C 14.7C 17.4C
22.2C 25.1C 26.9C MAD 7.1C 10.5C 12.8C 15.6C 17.6C 19.0C Bias -0.6C -2.2C
-4.0C -6.3C -7.9C -9.3C Persistence RMV 12.0C 16.2C 19.7C 22.7C 28.2C 31.7C
MAD 8.2C 11.1C 13.6C 15.8C 18.5C 20.5C Bias -0.4C 0.6C 2.0C 3.5C 5.5C 6.8C

Intensity  Verification
Lead time 6 h 12 h RMSE 4.9kt 9.5kt MAE 3.5kt 7.1kt Bias 0.6kt 2.3kt

Table 1: Overall Model  Verification at $\mathbf{n=16}$: (Left)  Trajectory
verification of structural forecasts, compared to IR persistence forecasts
where the radial profiles are fixed at their 0 h values. Simulation noise
(root mean variance and mean absolute deviation) grows rapidly in the first 6
hours; bias increases in magnitude steadily. We note that persistence offers a
less biased IR forecast on average, but higher overall errors in structure at
all lead times. (Right) Intensity  verification vs HURDAT2 best-track
intensities from 2013-2020 at each lead time.

Shear Magnitude

| Structural | NHC OFCL |
---|---|---|---
Shear | 12-h RMSE/MAE/Bias | N
0-10kt | 9.3/7.3/-0.7 kt | 9.3/6.4/-1.1 kt | 243
10-20kt | 9.7/6.9/1.0 kt | 8.1/5.7/-0.5 kt | 509
20+kt | 8.6/6.3/2.1 kt | 7.5/5.1/-0.6 kt | 439
Total | 9.2/6.8/1.1 kt | 8.1/5.6/-0.6 kt | 1,191

Shear Direction

| Structural | NHC OFCL |
---|---|---|---
Shear Direction | 12-h RMSE/MAE/Bias | N
SW | 8.6/6.1/-0.1 kt | 7.5/5.4/-1.1 kt | 106
SE | 9.1/6.9/0.3 kt | 8.5/5.9/-0.5 kt | 440
NE | 9.3/6.6/2.2 kt | 7.9/5.4/-0.6 kt | 575
NW | 10.6/8.2/-2.0 kt | 8.9/6.1/-0.9 kt | 70
Total | 9.2/6.8/1.1 kt | 8.1/5.6/-0.6 kt | 1,191

Table 2: Intensity Guidance  Verification Relative to Shear: Model
verification binned by 200-850-hPa vertical wind shear, reported as
RMSE/MAE/Bias. (Left) The performance of the structural forecasting model does
not change meaningfully relative to wind shear magnitude, while the NHC
official forecast performs better in higher shear environments. The structural
forecast has comparable performance to the NHC official forecasts in low-shear
environments. (Right) The performance of the structural model does vary with
shear direction. Both the NHC forecasts and the structural model produce
higher errors for NW shear (6% of cases).

TC Intensity

| Structural | NHC OFCL |
---|---|---|---
Category | 12-h RMSE/MAE/Bias | N
Tropical Depression | 5.5/4.4/3.0 kt | 5.9/4.2/-3.1 kt | 112
Tropical Storm | 7.2/5.6/2.8 kt | 6.4/4.4/-0.6 kt | 567
Hurricane | 10.3/7.7/0.6 kt | 9.3/6.7/-0.6 kt | 355
Major Hurricane | 14.0/10.7/-5.5 kt | 11.8/8.4/0.9 kt | 157
Total | 9.2/6.8/1.1 kt | 8.1/5.6/-0.6 kt | 1,208

Intensity Change

| Structural | NHC OFCL |
---|---|---|---
Evolution | 12-h RMSE/MAE/Bias | N
Weakening | 12.1/8.9/7.7 kt | 7.8/5.4/2.9 kt | 263
Maintenance | 6.7/5.3/3.3 kt | 6.1/4.3/-0.0 kt | 497
Intensifying | 9.8/7.3/-5.5 kt | 10.2/7.2/-3.5 kt | 431
Total | 9.2/6.8/1.1 kt | 8.1/5.6/-0.6 kt | 1,208

Table 3: Intensity Guidance  Verification by TC Intensity: Model  verification
split out by intensity and intensity change, reported as RMSE/MAE/Bias. (Left)
Both the structural and NHC official forecasts struggle more with intense
storms, which are rarer. The structural forecast has much stronger bias, which
is expected due to the heavy influence of persistence features in the absence
of environmental predictors. (Right) Similarly, both forecasts perform best
during maintenance periods (6-hour change $\leq 5$ kt in magnitude),
overestimate during weakening, and underestimate during intensification. The
bias is more pronounced in the structural forecast due to the absence of
environmental predictors.

Tables 2 and 3 assess the performance of our intensity guidance via structural
forecasting at 12-hour lead times and compare it to the NHC’s official
forecast  verification from 2013-2019 due to availability of  verification
data at time of writing; note that this is a subset of the times reported in
Table 1, consisting of cases where both our structural forecasts and NHC
official  verification are available. Overall, the RMSE of the structural
forecast is about 1.1 kt larger than the NHC official forecast error as
computed by RMSE, and structural forecasts produce roughly twice the bias (1.1
vs -0.6 kt). The structural forecast sees unchanged MSE with increasing
200-850-hPa vertical wind shear; the bias, however, increases with increasing
wind shear (Table 2). This trend is expected, as the model does not include
wind shear as a predictor but instead relies on the positive correlation
between shear and asymmetry in IR imagery (as captured by radial profiles
computed by quadrant). The NHC official forecast error exhibits a similar, if
less pronounced, trend in bias with increasing shear. The direction of shear
seems more important, with both our model and the NHC official forecast
performing most poorly for NW shear (6% of cases) and best for SW shear (9% of
cases). The NE and SE cases dominate the overall model performance since they
comprise the remaining 85% of the data set. The disparity between different
shear magnitudes and directions could be alleviated in a model which utilizes
environmental predictors.

Table 3 demonstrates similar error trends for both official forecasts and our
structural forecasts. Errors tend to increase with TC intensity and with rate
of intensification or weakening. The structural model produces higher bias for
weaker TCs and lower bias for stronger TCs. Similarly, the structural model
tends to overestimate intensities during weakening and underestimate them
during intensification. The model errors are comparable to NHC official
forecast errors during periods of maintenance and intensification (although
bias is higher); it is periods of weakening which tend to be poorly modeled by
the structural forecast. We suspect that the inclusion of environmental
information could improve fidelity in weakening cases; see  Section 6 on
Future Work Directions for a discussion of such avenues for model improvement.

Figure 9:  SHAP Variable Importance Map for 6-Hour Intensity Forecast: (Top)
IR channels of CNN nowcasting model for Hurricane Dorian [2019] with observed
IR structure above the horizontal black lines and +6 h forecasted structure
from deep auto-regressive generative model below the horizontal lines.
(Center) Pixel-wise SHAP variable importance of IR inputs on the 6-h intensity
forecast. (Bottom Left) SHAP variable importance of VMax (linearly
interpolated operational intensity estimates) and Delta V (2-h rate of change
of operational intensity estimates) on the 6-h intensity forecast. (Bottom
Right) Aggregated SHAP values over each channel, indicating IR features
contributing to intensity forecasts to a degree comparable to persistence
features.

###  4.3  Variable Importance in Intensity Forecasts

Our model results show that structural forecasts result in 6-hour and 12-hour
intensity predictions of comparable accuracy to NHC official forecasts. For
insight into how much our model relies on IR inputs and prior intensities when
making predictions, we compute a saliency map (also known as pixel
attribution) for each input. There are varied definitions for saliency,
including occlusion-based approaches such as SHAP explainability values
(Lundberg and Lee, 2017), LIME values (Ribeiro et al., 2016), and gradient-
based approaches.

Figure 9 (center) shows a map of the SHAP importance or contribution of each
pixel of the IR observed and forecasted imagery on the 6-hour intensity
forecast for Hurricane Dorian [2019]. The bottom-left panel shows the SHAP
values for prior intensity and prior intensity change. The bottom-right panel
shows aggregated SHAP values for each input channel. From this result and a
similar analysis with SHAP variable importance maps for Hurricane Jose [2017]
and Hurricane Nicole [2016] in Appendix A and gradient-based saliency maps in
Appendix B of Supplementary Materials, we conclude that (i) IR imagery
contributes to the intensity forecasts to a degree comparable to persistence
features, (ii) forecasted infrared imagery from our deep autoregressive
generative model plays a more important role than observed past imagery in the
TC intensity forecasts, (iii) the current and past presence/absence of an eye
is generally the key feature of a storm, and (iv) the core temperatures
outside of the eye play a signifcant role for intensity forecasting.

##  5 Discussion and Conclusion

This paper demonstrates a novel  interpretable approach to short-term TC
intensity guidance  trained solely on intensity estimates up to 6 hours prior
to the current time and IR observations up to 0 h. We specifically leverage
spatial characteristics of TC convection as captured by radial  IR profiles.
By  forecasting an ensemble of +6 h and +12 h trajectories of TC IR structure
with radial profiles computed over four geographic quadrants, we obtain
reasonable estimates of future  +6 h and +12 h TC intensity while
simultaneously capturing  and enabling visualization of signals in convective
structure relevant to those future intensities. We focus on interpretable,
physically-based factors to facilitate understanding of the model’s
performance (e.g., upcoming intensification corresponds with decreasing cloud-
top temperatures in the structural forecast). The approach outlined here has
the potential for further  improvement by adopting other network architectures
for structural forecasts and by including environmental predictors provided in
real time by SHIPS guidance. Though testing on years of cases takes time, an
individual forecast for a single TC can be obtained in minutes on a single
GPU, indicating the potential for the eventual use of this model as part of
the available TC guidance suite in an operational setting.

##  6  Future Work Directions

###  6.1 Improving the Network Architecture for Structural Forecasts

The PixelSNAIL approach provides reasonable simulations of TC IR structural
evolution up to 12-hour lead times. However, there exists a wealth of
alternate deep autoregressive generative models, each of which can be designed
and trained in innumerable ways. Likewise, deep autoregressive models are not
the only generative models available. Simulation could be carried out via
vector autoregression on a low-dimensional projection of profiles (e.g.,
principal component analysis, Fourier bases, etc.), generative adversarial
networks (GANs; Creswell et al. 2018),  or transformers (e.g., temporal fusion
transformers for multihorizon forecasting (Lim et al., 2021) and
spatiotemporal transformers (Grigsby et al., 2021)). The PixelSNAIL
architecture was chosen to demonstrate the value and feasibility of structural
forecasting for intensity guidance.

###  6.2  Calibrating the Probability Distribution of Structural Forecasts

Our structural forecasts are probabilistic in nature, taking the form of
probability distributions over future structural trajectories
$\mathbf{S}_{>t}$. In the current work, we apply a standard machine learning
approach of fitting a model by minimizing a loss function (in this case the
negative log likelihood). A good probabilistic forecast, however, should be
conditionally calibrated. That is, the probability of a particular event (in
our case, specific radial profiles 6-12 hours into the future), given or
“conditional on” a particular history of evolution and other predictors,
should match the predicted probability of the same event. This is essentially
saying that draws from the forecasting model should be indistinguishable from
actual observations, if all relevant conditions are the same. Dey et al.
(2022) recently proposed a new method for adjusting or “recalibrating”
probabilistic forecasts, so that they will have his property. Indeed, one can
potentially apply their procedure sequentially to each autoregressive
component $p(Z_{iq}|\mathbf{Z}_{i-1},\dots,\mathbf{Z}_{1})$, for pixel
$i=1,\dots,n$, and quadrant $q=1,2,3,4$, so as to obtain a conditionally
calibrated density over structural trajectories $\mathbf{S}_{>t}$ given
present and past observations; see Discussion in Dey et al. (2022).

###  6.3 Inclusion of Environmental Variables

The PixelSNAIL model presented here is a purely autoregressive process; that
is, it simulates future structural features using only past IR imagery as an
input. The inclusion of environmental variables known to impact TCs such as
vertical wind shear, atmospheric moisture, or sea surface temperature may
improve the accuracy of the forward simulation of radial profiles,
particularly of structural evolution beyond 12 hours. Such factors can be
added to the PixelSNAIL architecture as additional input layers via values
provided by SHIPS which are not forecasted by the model. These inputs would
then serve as “guiderails” for simulated structural evolution with potential
to better capture the effects of such factors on profile asymmetry.  Despite
these limitations, our prototype model (which is derived solely from prior and
present TC intensity estimates and Geo IR imagery alongside forecasted TC
structure using a very simple network architecture) provides reasonable short-
term structural and intensity forecasts comparable to NHC forecasts at 6- and
12-h lead times. The inclusion of environmental variables in the nowcasting
model is likely to improve its intensity forecasts, which would then be
compared to SHIPS forecasts as well as NHC official forecasts, the latter of
which are crafted using SHIPS and other guidance.

###### Acknowledgements.

Part of this research was done as an Independent Study in Spring 2021 while
Pavel Khokhlov was a Master in Machine Learning student at Carnegie Mellon
University. We are grateful to Microsoft for providing Azure computing
resources for this work. The authors would like to thank Katerina Fragkiadaki
for a discussion on deep generative networks, and Galen Vincent for many
helpful comments on the research. This work is supported in part by NSF
DMS-2053804, NSF PHY-2020295, and the C3.ai Digital Transformation Institute.
Code to generate Geo IR radial profiles from openly available data can be
found at https://github.com/ihmcneely/ORB2sample, which draws from the MERGIR
database openly available from NASA at
https://disc.gsfc.nasa.gov/datasets/GPM_MERGIR_1/summary. The HURDAT2 best
track database is openly available from the NHC at
https://www.nhc.noaa.gov/data, while the ATCF operational best tracks (B-deck)
are openly available from the National Center for Atmospheric Research at
http://hurricanes.ral.ucar.edu/repository/. Official forecast  verification
files are openly available from the NHC at
https://www.nhc.noaa.gov/verification/. Finally, SHIPS developmental data is
openly available from the Cooperative Institute for Research in the Atmosphere
at
https://rammb.cira.colostate.edu/research/tropical_cyclones/ships/developmental_data.asp.

## References

  * Chen et al. (2018) Chen, X., N. Mishra, M. Rohaninejad, and P. Abbeel, 2018: Pixelsnail: An improved autoregressive generative model. International Conference on Machine Learning, PMLR, 864–872.
  * Combinido et al. (2018) Combinido, J. S., J. R. Mendoza, and J. Aborot, 2018: A convolutional neural network approach for estimating tropical cyclone intensity using satellite-based infrared images. 2018 24th International Conference on Pattern Recognition (ICPR), IEEE, 1474–1480.
  * Creswell et al. (2018) Creswell, A., T. White, V. Dumoulin, K. Arulkumaran, B. Sengupta, and A. A. Bharath, 2018: Generative adversarial networks: An overview. IEEE Signal Processing Magazine, 35 (1), 53–65.
  * DeMaria (2018) DeMaria, M., 2018: SHIPS developmental database file format and predictor descriptions (2018). Tech. rep., Colorado State University. URL http://rammb.cira.colostate.edu/research/tropical˙cyclones/ships/docs/ships˙predictor˙file˙2018.doc, Developmental Data.
  * DeMaria and Kaplan (1999) DeMaria, M., and J. Kaplan, 1999: An updated statistical hurricane intensity prediction scheme (SHIPS) for the Atlantic and Eastern North Pacific basins. Weather and Forecasting, 14 (3), 326–337.
  * DeMaria (2015) DeMaria, R., 2015: Automated tropical cyclone eye detection using discriminant analysis. Ph.D. thesis, Colorado State University.
  * Dey et al. (2022) Dey, B., D. Zhao, J. A. Newman, B. H. Andrews, R. Izbicki, and A. B. Lee, 2022: Calibrated predictive distributions via diagnostics for conditional coverage. arXiv preprint arXiv:2205.14568.
  * Dvorak (1975) Dvorak, V. F., 1975: Tropical cyclone intensity analysis and forecasting from satellite imagery. Monthly Weather Review, 103 (5), 420–430, 10.1175/1520-0493(1975)103¡0420:TCIAAF¿2.0.CO;2.
  * Ebert-Uphoff and Hilburn (2020) Ebert-Uphoff, I., and K. Hilburn, 2020: Evaluation, tuning, and interpretation of neural networks for working with images in meteorological applications. Bulletin of the American Meteorological Society, 101 (12), E2149–E2170.
  * Gray (1979) Gray, W. M., 1979: Hurricanes: Their formation, structure and likely role in the tropical circulation. Meteorology over the tropical oceans, 155, 218.
  * Griffin et al. (2022) Griffin, S. M., A. Wimmers, and C. S. Velden, 2022: Predicting rapid intensification in north atlantic and eastern north pacific tropical cyclones using a convolutional neural network. Weather and Forecasting.
  * Grigsby et al. (2021) Grigsby, J., Z. Wang, and Y. Qi, 2021: Long-range transformers for dynamic spatiotemporal forecasting. arXiv preprint arXiv:2109.12218.
  * Hovmöller (1949) Hovmöller, E., 1949: The trough-and-ridge diagram. Tellus, 1 (2), 62–66.
  * Hu et al. (2020) Hu, L., E. A. Ritchie, and J. S. Tyo, 2020: Short-term tropical cyclone intensity forecasting from satellite imagery based on the deviation angle variance technique. Weather and Forecasting, 35 (1), 285–298.
  * Janowiak et al. (2020) Janowiak, J., B. Joyce, and P. Xie, 2020: NCEP/CPC L3 half hourly 4km global (60S - 60N) merged IR v1. NASA Goddard Earth Sciences Data and Information Services Center, URL https://disc.gsfc.nasa.gov/datacollection/GPM˙MERGIR˙1.html, 10.5067/P4HZB9N27EKU.
  * Knaff and DeMaria (2017) Knaff, J. A., and R. T. DeMaria, 2017: Forecasting tropical cyclone eye formation and dissipation in infrared imagery. Weather and Forecasting, 32 (6), 2103–2116.
  * Knapp and Wilkins (2018) Knapp, K. R., and S. L. Wilkins, 2018: Gridded satellite (GridSat) GOES and CONUS data. Earth System Science Data, 10 (3), 1417–1425, 10.5194/essd-10-1417-2018.
  * Lee et al. (2019) Lee, J., J. Im, D.-H. Cha, H. Park, and S. Sim, 2019: Tropical cyclone intensity estimation using multi-dimensional convolutional neural networks from geostationary satellite data. Remote Sensing, 12 (1), 108\.
  * Lim et al. (2021) Lim, B., S. Ö. Arık, N. Loeff, and T. Pfister, 2021: Temporal fusion transformers for interpretable multi-horizon time series forecasting. International Journal of Forecasting, 37 (4), 1748–1764.
  * Liu (2021) Liu, Z., 2021: private communication.
  * Lundberg and Lee (2017) Lundberg, S. M., and S.-I. Lee, 2017: A unified approach to interpreting model predictions. Advances in neural information processing systems, 30.
  * McGovern et al. (2019) McGovern, A., R. Lagerquist, D. J. Gagne, G. E. Jergensen, K. L. Elmore, C. R. Homeyer, and T. Smith, 2019: Making the black box more transparent: Understanding the physical implications of machine learning. Bulletin of the American Meteorological Society, 100 (11), 2175–2199.
  * McNeely et al. (2019) McNeely, T., A. B. Lee, D. Hammerling, and K. Wood, 2019: Quantifying the spatial structure of tropical cyclone imagery.
  * McNeely et al. (2020) McNeely, T., A. B. Lee, K. M. Wood, and D. Hammerling, 2020: Unlocking GOES: A statistical framework for quantifying the evolution of convective structure in tropical cyclones. Journal of Applied Meteorology and Climatology, 59 (10), 1671–1689.
  * McNeely et al. (2022) McNeely, T., G. Vincent, A. B. Lee, R. Izbicki, and K. M. Wood, 2022: Detecting distributional differences in labeled sequence data with application to tropical cyclone satellite imagery. arXiv preprint arXiv:2202.02253.
  * Olander et al. (2021) Olander, T., A. Wimmers, C. Velden, and J. P. Kossin, 2021: Investigation of machine learning using satellite-based advanced dvorak technique analysis parameters to estimate tropical cyclone intensity. Weather and Forecasting, 36 (6), 2161–2186.
  * Olander and Velden (2007) Olander, T. L., and C. S. Velden, 2007: The advanced dvorak technique: Continued development of an objective scheme to estimate tropical cyclone intensity using geostationary infrared satellite imagery. Weather and Forecasting, 22 (2), 287–298.
  * Pradhan et al. (2017) Pradhan, R., R. S. Aygun, M. Maskey, R. Ramachandran, and D. J. Cecil, 2017: Tropical cyclone intensity estimation using a deep convolutional neural network. IEEE Transactions on Image Processing, 27 (2), 692–702.
  * Ribeiro et al. (2016) Ribeiro, M. T., S. Singh, and C. Guestrin, 2016: ” why should i trust you?” explaining the predictions of any classifier. Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining, 1135–1144.
  * Salimans et al. (2017) Salimans, T., A. Karpathy, X. Chen, and D. P. Kingma, 2017: Pixelcnn++: Improving the pixelcnn with discretized logistic mixture likelihood and other modifications. arXiv preprint arXiv:1701.05517.
  * Sanabia et al. (2014) Sanabia, E. R., B. S. Barrett, and C. M. Fine, 2014: Relationships between tropical cyclone intensity and eyewall structure as determined by radial profiles of inner-core infrared brightness temperature. Monthly Weather Review, 142 (12), 4581–4599, 10.1175/MWR-D-13-00336.1.
  * Schmit et al. (2017) Schmit, T. J., P. Griffith, M. M. Gunshor, J. M. Daniels, S. J. Goodman, and W. J. Lebair, 2017: A closer look at the ABI on the GOES-R series. Bulletin of the American Meteorological Society, 98 (4), 681–698.
  * Tian et al. (2020) Tian, W., W. Huang, L. Yi, L. Wu, and C. Wang, 2020: A cnn-based hybrid model for tropical cyclone intensity estimation in meteorological industry. IEEE Access, 8, 59 158–59 168.
  * Van den Oord et al. (2016) Van den Oord, A., N. Kalchbrenner, L. Espeholt, O. Vinyals, A. Graves, and Coauthors, 2016: Conditional image generation with pixelcnn decoders. Advances in neural information processing systems, 29.
  * Van Oord et al. (2016) Van Oord, A., N. Kalchbrenner, and K. Kavukcuoglu, 2016: Pixel recurrent neural networks. International conference on machine learning, PMLR, 1747–1756.
  * Wang et al. (2020) Wang, C., Q. Xu, X. Li, and Y. Cheng, 2020: Cnn-based tropical cyclone track forecasting from satellite infrared images. IGARSS 2020-2020 IEEE International Geoscience and Remote Sensing Symposium, IEEE, 5811–5814.
  * Zhang et al. (2021) Zhang, C.-J., X.-J. Wang, L.-M. Ma, and X.-Q. Lu, 2021: Tropical cyclone intensity classification and estimation using infrared satellite images with deep learning. IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, 14, 2070–2086.

Supplemental Material for “Structural Forecasting for Short-term Tropical
Cyclone Intensity Guidance”

[A]  Additional SHAP Variable Importance Maps for TC Intensity Forecasts

Figure 10:  SHAP Variable Importance Map for 6-Hour Intensity Forecast: (Top)
IR channels of CNN “nowcasting” model for Hurricane Jose [2017] with observed
IR structure above the horizontal black lines, and +6 hour forecasted
structure from deep auto-regressive generative model below the horizontal
lines. (Center) Pixel-wise SHAP variable importance of IR inputs on 6-hour
intensity forecast. (Bottom Left) SHAP variable importance of VMax (linearly
interpolated operational intensity estimates) and Delta V (2-h rate of change
of operational intensity estimates) on the 6-h intensity forecast. (Bottom
Right) Aggregated SHAP values over each channel showing IR features
contributing significantly to the intensity forecasts.

Figure 11:  IR Channels and SHAP Variable Importance Map for 6-Hour Intensity
Forecast: As figure 10, but for Hurricane Nicole [2016].

[B]

Input Saliency for Forecasting Model Our model results in Section 4 showed
that structural forecasts result  in 12-hour intensity predictions of
comparable accuracy to NHC  official forecasts.  Here we apply a simple
gradient-based approach to provide some insight as to how much the model
relies on different IR inputs when making predictions. The gradient describes
how much a feature contributes to the model response $Y$. More specifically,
we define the saliency $S_{i}(\mathbf{x})$ of the $i^{th}$ pixel or feature by

$S_{i}(\mathbf{x})=\Bigg{\lvert}\ \frac{\partial Y}{\partial
x_{i}}\Big{|}_{\mathbf{x}}\ \Bigg{\rvert},$ (7)

where $\mathbf{x}$ denotes the total input.

In order to visualize the overall impact of each input channel (four IR
quadrants, the radius channel, the time channel, the observed  prior
intensity, and the observed  prior intensity change) on the  forecasted future
intensity, we  “aggregate” the saliency, summing over all pixels in each
channel. Figure 12 shows the saliency aggregated by channel over time for each
of our three example TCs. Note that because  prior intensity/intensity change
are not included in the convolutional layers, they are linear, and thus have a
fixed saliency; because the model is nonlinear in the other channels, the
saliency varies over time with the IR inputs. Of particular note is that the
aggregated saliency of the IR input channels is comparable to the persistence
features, indicating that the model does not simply rely on persistence to
make its predictions but instead makes use of the structural forecasts. Figure
13 shows the same values for IR channels with the mean and trend removed,
demonstrating that the model tends to rely more heavily on convective
structure in the southern quadrants, and particularly the southeast quadrant.

Figure 12: Saliency per input layer in the model. Saliency values over time
indicate that the model for TC intensity forecasts utilizes image inputs to a
degree comparable to  prior intensity values. Figure 13: Trend- and mean-
removed saliency for image inputs. Comparing the saliency values over time
indicates 1) the southern quadrants are more heavily utilized by the
forecasting model, and 2) the degree to which the model relies on one quadrant
over another is not constant (e.g. Hurricane Jose [2017] around Sept. 14,
2017).

[C] Additional Forecast Materials for Case Studies

Figure 14: Hurricane Jose [2017] 6-hr Guidance: Hurricane Jose was subjected
to vertical wind shear out of the west/northwest due to Hurricane Irma; the
structural forecasts tend to underestimate temperatures in the NW quadrant and
thus overestimate TC intensity. (Left) Distribution of forecasted intensities
with observed (black) and average forecast (red) intensities marked. (Right)
Simulated profiles by quadrant with observed profiles represented by solid
black curves, and averaged simulated profiles represented by dashed red
curves. Figure 15: Hurricane Jose [2017] 12-hr Guidance: At 12-hour lead
times, this TC’s evolution is still well modeled save for a handful of
individual overestimates. (Left) Distribution of forecasted intensities with
observed (black) and average forecast (red) intensities marked. (Right)
Simulated profiles by quadrant with observed profiles represented by solid
black curves, and averaged simulated profiles represented by dashed red
curves. Figure 16: Hurricane Nicole [2016] 6-hr Guidance: Hurricane Nicole
underwent several rapid intenity change events. At 6 UTC 7 October, the
structural forecast models the cloud top temperatures well, but this is
insufficient to predict the extreme change from intensification to weakening
even at 6-hour lead times. (Left) Distribution of forecasted intensities with
observed (black) and average forecast (red) intensities marked. (Right)
Simulated profiles by quadrant with observed profiles represented by solid
black curves, and averaged simulated profiles represented by dashed red
curves. Figure 17: Hurricane Nicole [2016] 12-hr Guidance: Model biases during
the quick shifts from intensification to weakening and during stead
intensification are exacerbated at 12-hour lead times. (Left) Distribution of
forecasted intensities with observed (black) and average forecast (red)
intensities marked. (Right) Simulated profiles by quadrant with observed
profiles represented by solid black curves, and averaged simulated profiles
represented by dashed red curves. Figure 18: Hurricane Nicole [2016] 12-hr
Structural Forecasts: The observed structural trajectory is shown in the top
left corner of each row. To the right, we see 10 individual simulations of
radial profiles (averaged over all quadrants) at 12-hour lead times. All
radial profiles above the black horizontal line are observed, while profiles
below the black line are simulated. The bottom left corner shows the average
simulation over 64 simulated trajectories. Figure 19: Hurricane Nicole [2016]
Forecasts By Quadrant (B): As Figure 18, but broken down by quadrant for
example (B) only. The observed structural trajectory is shown in the top left
corner. To the right, we see four individual simulations of radial profiles by
quadrant at 12-hour lead times. All radial profiles above the black horizontal
line are observed, while profiles below the black line are simulated. The
bottom left corner shows the average simulation over 64 simulated
trajectories. Figure 20: Hurricane Jose [2017] 12-hr Structural Forecasts: The
observed structural trajectory is shown in the top left corner of each row. To
the right, we see 10 individual simulations of radial profiles (averaged over
all quadrants) at 12-hour lead times. All radial profiles above the black
horizontal line are observed, while profiles below the black line are
simulated. The bottom left corner, shows the average simulation over 64
simulated trajectories. Figure 21: Hurricane Jose [2017] Forecasts By Quadrant
(A): As Figure 20, but broken down by quadrant for example (A) only. The
observed structural trajectory is shown in the top left corner. To the right,
we see four individual simulations of radial profiles by quadrant at 12-hour
lead times. All radial profiles above the black horizontal line are observed,
while profiles below the black line are simulated. The bottom left corner
shows the average simulation over 64 simulated trajectories. Figure 22:
Hurricane Dorian [2019] 12-hr Structural Forecasts: The observed structural
trajectory is shown in the top left corner of each row. To the right, we see
10 individual simulations of radial profiles (averaged over all quadrants) at
12-hour lead times. All radial profiles above the black horizontal line are
observed, while profiles below the black line are simulated. The bottom left
corner, shows the average simulation over 64 simulated trajectories. Figure
23: Hurricane Dorian [2019] Forecasts By Quadrant (A): As Figure 22, but
broken down by quadrant for example (A) only. The observed structural
trajectory is shown in the top left corner. To the right, we see four
individual simulations of radial profiles by quadrant at 12-hour lead times.
All radial profiles above the black horizontal line are observed, while
profiles below the black line are simulated. The bottom left corner shows the
average simulation over 64 simulated trajectories.