# Global 4D Ionospheric STEC Prediction based on DeepONet for GNSS Rays

Dijia Cai, Zenghui Shi, Haiyang Fu, Huan Liu, Hongyi Qian, Yun Sui, Feng Xu,
and Ya-Qiu Jin,
Manuscript received: Corresponding authors: Haiyang Fu (e-mail:
haiyang_fu@fudan.edu.cn)Dijia Cai, Zenghui Shi, Haiyang Fu, Hongyi Qian, Yun
Sui, Yaqiu Jin are with the Key Laboratory for Information Science of
Electromagnetic Waves (MoE), school of information science and engineering,
Fudan University, Shanghai 200433, China. Huan Liu is in the school of
mathematics, Shanghai University of Finance and Economics, Shanghai 200433,
China. This work is supported by National Key Research and Development Program
of China (2021YFA0717300). This work is also supported by the National Science
Foundation (42074189; 62231010).

###### Abstract

The ionosphere is a vitally dynamic charged particle region in the Earth’s
upper atmosphere, playing a crucial role in applications such as radio
communication and satellite navigation. The Slant Total Electron Contents
(STEC) is an important parameter for characterizing wave propagation,
representing the integrated electron density along the ray of radio signals
passing through the ionosphere. The accurate prediction of STEC is essential
for mitigating the ionospheric impact particularly on Global Navigation
Satellite Systems (GNSS). In this work, we propose a high-precision STEC
prediction model named DeepONet-STEC, which learns nonlinear operators to
predict the 4D temporal-spatial integrated parameter for specified ground
station - satellite ray path globally. As a demonstration, we validate the
performance of the model based on GNSS observation data for global and US-CORS
regimes under ionospheric quiet and storm conditions. The DeepONet-STEC model
results show that the three-day 72 hour prediction in quiet periods could
achieve high accuracy using observation data by the Precise Point Positioning
(PPP) with temporal resolution $\rm{30\,s}$. Under active solar magnetic storm
periods, the DeepONet-STEC also demonstrated its robustness and superiority
than traditional deep learning methods. This work presents a neural operator
regression architecture for predicting the 4D temporal-spatial ionospheric
parameter for satellite navigation system performance, which may be further
extended for various space applications and beyond.

###### Index Terms:

Space Ionospheric Parameter Prediction, High 4D Temporal-Spatial Resolution,
Deep Operator Network, Global Navigation Satellite System Data

##  I Introduction

The ionosphere, spanning from 60 km to 1000 km above the Earth, constitutes
the ionized particles of the upper atmosphere[1]. Comprised of plasma
electrons, ions and neutral components, the ionosphere exhibits complex and
dynamic nature influenced by solar radiation, the Earth’s magnetic field, and
the atmospheric activity, which holds great significance particularly in the
realms of radio communications, satellite navigation and L-band synthetic
aperture radar (SAR)[2, 3]. When electromagnetic (EM) signals such as Global
Navigation Satellite Systems (GNSS) propagate in the ionosphere[4], the EM
wave encounters diverse effects such as dispersion, refraction, and group
delay, producing a certain degree of interference which leads to amplified
positioning errors[2, 5, 6] from several centimeters to hundred meters in
severe ionospheric conditions [7]. As global space ionospheric weather and
GNSS service rapidly growing, the development of highly precise and low-
latency global ionospheric sensing become imperative in order to obtain high
temporal-spatial (4D) resolution accurate estimation and fast prediction of
ionospheric properties for various applications.

Ionospheric sensing based on GNSS data have been developed as one of the
dominant source due to global constellation, massive stations and high
accuracy positioning, navigation and timing (PNT) service. The global
ionospheric sensing based on GNSS data has been evolving from traditional two
dimensional (2D) modeling of the Vertical Total Electron Contents(VTEC) using
spherical Harmonics function fitting[8, 9], and Neural Network (NN)
interpolation[10] to three dimensional (3D) ionospheric tomography [11] . The
altitude and temporal resolution of the ionospheric tomography cannot be
neglected in order to capture high temporal-spatial resolution of ionospheric
parameter dynamics. To satisfy high GNSS application, the Slant Total Electron
Contents (STEC) estimation has been widely used in the GNSS society such as in
Wide Area Augmentation System (WAAS) using the Kriging model [12] and multiple
kernal model [13] to express the spatial ionospheric correlation under
ionospheric scintillation. However, the STEC interpolation [13] actually could
not predict the future temporal state. The present ionospheric parameter
estimation is actually lacking of achieving estimation and prediction with
high temporal-spatial resolution.

Indeed, it has been pursued to predict the temporally varying ionosphere from
short several hours to long term as another topic. The predicting method
includes both traditional statistical method and machine learning methods. In
recently years, due to its applicability of machine learning to nonlinearity,
several works have been done for ionospheric VTEC prediction, including
recurrent neural network (RNN) [14], long short-term memory (LSTM) neural
networks[15, 16], LSTM with convolutional neural networks[17], and encoder-
decoder ConvLSTMcite [18] ande etc. There are few methods to predict STEC
directly. Recently Zhu et al.[19] proposed an XGBoost-based method to predict
the single-station STEC for high accuracy Precise Point Positionin(PPP)
service. Although these methods can predict VTEC/STEC, the spatial-temporal
prediction cannot be simultaneously achieved for any satellite-station ray
path with high accuracy and resolution.

The existing ionospheric estimation and prediction can not satisfy the high 4D
temporal-spatial resolution of global ionospheric parameter, which is possible
to improve. On one hand, existing VTEC map is actually a low accurate method.
The mapping function is usually used for converting STEC to VTEC. But the
mapping function cannot characterize the STEC variation in different azimuths
between the user and satellites. The Ionospheric VTEC models are established
for all GNSS satellites as a whole and can only provide an averaged solution.
But this is not the optimal solution for each individual satellite[20]. The
conversion process from VTEC to STEC will also introduce errors by the mapping
function. The tomographic modeling from electron density to calculate STEC
suffers from meshing errors and expensive computational cost. These factors
hinder significant improvements in the accuracy of estimating STEC. On the
other hand, prevailing STEC estimation approaches primarily focus on spatial
interpolation techniques, with limited emphasis on temporal prediction. The
drawback of current exisiting methodologies lacks a unified framework for
achieving 4D temporal-spatial STEC estimation, which encompasses constructing
models based on past STEC observations to estimate STEC for the present and
future, and along any ray between satellites and stations within the domain.

In recent years, with the development of deep learning, neural operators have
gradually played an important for predicting the complex dynamics. The
operator regression methods approximate mappings between infinite function
spaces and considered as discretization invariant. Some of the operator
regression includes the Fourier neural operator (FNO)[21], the graph kernal
network (GKN) [22] and the deep neural operator(DeepONet) [23]. The neural
operator learns representations between functions and work on any
discretization of inputs to a meshless resolution [24], while the performance
of standard neural network such as CNN can degrade when data resolution during
deployment changes from model training[25]. Recently, the FNO operator based
prediction model such FourCastNet [26] has achieved substantial progress for
accurate medium-range global weather forecasting, which triggers several
atmospheric prediction model based on machine learning [27] with capability
superior than traditional methods. In addition, prediction methods for VTEC
map using deep learning methods have also been attempted [28, 29], including
detecting pre-earthquake anomalies[30]. This motivates our attempt to further
achieve the 4D ionospheric temporally-spatial prediction based on advanced
machine learning methods.

In this paper, we developed a 4D temporal-spatial prediction framework
DeepONet-STEC for predicting ionospheric parameter for station-satellite ray
path based on deep operator regression. The advantage of this model is capable
of predicting the future 10 days STEC for any specified ground station-
satellite rays by capturing representations between functions from data. The
main contributions of this work can be summarized as follows: 1) We proposed a
novel 4D ionospheric parameter prediction framework based on neural operator
regression. 2) The proposed framework provides simultaneously high accuracy
spatial estimation and temporal prediction of the STEC values for global GNSS
data. 3) The innovative Kernel method is adopted to construct the input
function from the real observation data for the DeepONet-STEC model. The
structure of this paper is organized as following. After introduction in
Section I, the description of the proposed 4D DeepONet-STEC model will be
explained in detail in Section II. In Section III, we will present the process
of dataset generation, model training, prediction results based on simulated
and observation data. Section IV will draw conclusions with discussion.

##  II Prediction Model

In this section, we present the modeling approach for the STEC 4D prediction
problem mentioned above, emphasizing two key methods used in the modeling
process: the Kernel Methods (KM) and the principles of DeepONet. The
construction process of the 4D DeepONet-STEC model is then detailed, along
with the techniques employed for data processing.

###  II-A Problem statement

Figure 1: Illustration of the 4D DeepONet-STEC architecture. Using
$\\{(\tilde{\textbf{x}}_{i},\tilde{t}_{i},\tilde{y}_{i})\\}_{i=1}^{M}$ to
train, the model can predict given ray $(\textbf{x},t)\in\mathcal{D}$ to get
the output STEC value $y$. The 4D DeepONet-STEC comprises two main components:
the Branch network and the Trunk network. Both neural networks adopt a fully
connected structure. The Branch network takes the output value of the $u$
function at fixed point location, representing the mapped STEC value, as its
input. The input for the Trunk network includes the coordinates of the
corresponding rays in the observed data, as well as sine and cosine encoded
time. An activation function ReLu is applied to the output of each hidden
layer. The branch network has the input layer with 300 neurons, 30 hidden
layer with 64 neurons, and the output layer with 300 neurons.

In ionospheric applications, the observations are often limited due to the
limitation of observation such as GNSS. Typically, only partial observable
STEC value $y$ along the $i$-th ray can be measured from a restricted number
of satellite $s$ and receiver $r$ reference pairs x at a specific time $t$,
where x represents the coordinates of the satellite and receiver.
Consequently, there arises a need to predict the STEC value
$y\in\mathbb{R}^{*}$ at any temporal-spatial point
$(\textbf{x},t)\in\mathcal{D}$, therefore we develop the 4D DeepONet-STEC
model based on the operator neural network.

Let all observed rays
$\\{(\tilde{\textbf{x}}_{i},\tilde{t}_{i},\tilde{y}_{i})\\}_{i=1}^{M}$, be the
data set, with each observed sample
$(\tilde{\textbf{x}}_{i},\tilde{t}_{i},\tilde{y}_{i})\in\tilde{\mathcal{D}}\times\tilde{\mathbb{R}}^{*}$
and the subscript $i$ denotes the $i$-th ray. To evaluate the generalization
performance of the 4D DeepONet-STEC model, the training set and test set are
created by collecting partial data from the data set, where the training set
is denoted by
$\mathcal{T}:=\\{(\tilde{\textbf{x}}_{i}^{(0)},\tilde{t}_{i}^{(0)},\tilde{y}_{i}^{(0)})\\}_{i=1}^{N}$.

In contrast with conventional neural networks to approximate functions,
DeepONet approximates linear and nonlinear operators. The model comprises two
deep neural networks: one network that encodes the discrete input function
space (i.e., branch net) and one that encodes the output function space (i.e.,
trunk net).

Furthermore, we create two subsets from the training set $\mathcal{T}$, which
are denoted by $\mathcal{U}$ and $\mathcal{V}$, respectively. The first subset
$\mathcal{U}:=\\{(\tilde{\textbf{x}}_{i}^{(1)},\tilde{t}_{i}^{(1)},\tilde{y}_{i}^{(1)})\\}_{i=1}^{m}$
consisting of $m$ observed samples is used to construct the input function $u$
of the 4D DeepONet-STEC model, and the second subset
$\mathcal{V}:=\\{(\tilde{\textbf{x}}_{i}^{(2)},\tilde{t}_{i}^{(2)},\tilde{y}_{i}^{(2)})\\}_{i=1}^{h}$
consisting of $h$ observed samples will be employed to train the model.
Obviously,
$(\tilde{\textbf{x}}_{i}^{(j)},\tilde{t}_{i}^{(j)})\in\tilde{\mathcal{D}}$ for
$j=0,1,2$ with $\tilde{\mathcal{D}}$ being a subset of $\mathcal{D}$.

Therefore, the problem can be reformulated as follows: (I) Constructing the
input function $u:\tilde{\mathcal{D}}\rightarrow\tilde{\mathbb{R}}^{*}$
satisfying
$u(\tilde{\textbf{x}}_{i}^{(1)},\tilde{t}_{i}^{(1)})=\tilde{y}_{i}^{(1)}$ for
$i=1,\ldots,m$, i.e., the STEC value at a given time on a given path can be
obtained through the function $u$; (II) Define the mapping operator
$G:u\rightarrow v$ using DeepONet, and learn the optimal hyper-parameters of
the DeepONet by minimizing the loss
$\sum_{i=1}^{h}|G(u)(\tilde{\textbf{x}}_{i}^{(2)},\tilde{t}_{i}^{(2)})-\tilde{y}_{i}^{(2)}|^{2}$;
(III) Once the optimal hyper-parameters of the DeepONet are obtained, then the
STEC value $y$ at any path and any time $(\textbf{x},t)\in\mathcal{D}$ can be
predicted through the network output $G(u)(\textbf{x},t)$.

###  II-B Constructing mapping operator $G$ using DeepONet

We use DeepONet to construct the operator $G$, which is a deep neural network-
based model that can learn both linear and nonlinear operators. The DeepONet
is proposed based on the universal approximation theorem of operators, which
states that a neural network with a single hidden layer can accurately
approximate any nonlinear continuous operator. More specifically, given two
functions $u$ and $v$ defined on different spaces as follows

$u:\tilde{\mathcal{D}}\ni(\tilde{\textbf{x}},\tilde{t})\rightarrow
u(\tilde{\textbf{x}},\tilde{t})\in\tilde{\mathbb{R}}^{*}$ (1)

and

$v:\mathcal{D}\ni(\textbf{x},t)\rightarrow v(\textbf{x},t)\in\mathbb{R}^{*}$
(2)

DeepONet can train the mapping operator $G$:

$G:u\rightarrow v$ (3)

In terms of concrete implementation, DeepONet consists of two sub-networks:
one for encoding the input function at a fixed number of sensors (branch
network), and another for encoding the locations for the output functions
(trunk network)[23].

The branch network takes the function evaluations
$[u(\tilde{\textbf{x}}_{1}^{(1)},\tilde{t}_{1}^{(1)}),u(\tilde{\textbf{x}}_{2}^{(1)},\tilde{t}_{2}^{(1)}),\cdots,u(\tilde{\textbf{x}}_{m}^{(1)},\tilde{t}_{m}^{(1)})]$
as the input, and the output of the branch network is
$[\beta_{1}(u),\beta_{2}(u),\cdots,\beta_{p}(u)]$, where $p$ is the number of
neurons. The trunk network takes all the rays from
$\\{(\tilde{\textbf{x}}_{1}^{(2)},\tilde{t}_{1}^{(2)}),(\tilde{\textbf{x}}_{2}^{(2)},\tilde{t}_{2}^{(2)}),\cdots,(\tilde{\textbf{x}}_{h}^{(2)},\tilde{t}_{h}^{(2)})\\}$
as the input and the outputs are
$\\{[\tau_{1}(\tilde{\textbf{x}}_{i}^{(2)},\tilde{t}_{i}^{(2)}),\tau_{2}(\tilde{\textbf{x}}_{i}^{(2)},\tilde{t}_{i}^{(2)}),\cdots,\tau_{p}(\tilde{\textbf{x}}_{i}^{(2)},\tilde{t}_{i}^{(2)})]\\}^{h}_{i=1}$.
Then, by taking the inner product of trunk and branch outputs, the output of
DeepONet is:

$G(u)(\tilde{\textbf{x}}_{i}^{(2)},\tilde{t}_{i}^{(2)})=\sum_{k=1}^{p}\beta_{k}(u)\tau_{k}(\tilde{\textbf{x}}_{i}^{(2)},\tilde{t}_{i}^{(2)}),~{}~{}i=1,\ldots,h$
(4)

We use the following loss function to train the DeepONet, which can be
minimized by many well-developed gradient-based optimization methods:

$L=\sum_{i=1}^{{h}}|{G(u)(\tilde{\textbf{x}}_{i}^{(2)},\tilde{t}_{i}^{(2)})-\tilde{y}_{i}^{(2)}}|^{2}$
(5)

For any pair $(\textbf{x},t)\in\mathcal{D}$ as the input, the outputs of the
trunk network is
$[\tau_{1}(\textbf{x},t),\tau_{2}(\textbf{x},t),\cdots,\tau_{p}(\textbf{x},t)]$.
Therefore, we can obtain the predicted STEC value
$y=G(u)(\textbf{x},t)=\sum_{k=1}^{p}\beta_{k}(u)\tau_{k}(\textbf{x},t)$ for
any temporal-spatial pair $(\textbf{x},t)$ provided that the optimal hyper-
parameters of the DeepONet are obtained after training process.

###  II-C Constructing input function $u$ using Kernel Method

Kernel methods are mathematical functions that operate on vectors in the
original space and yield the inner product of vectors in the feature space
[31, 32]. In this paper, we use the kernel method to construct the input
function $u$. Distinct from Gaussian Random Field (GRF) and Chebyshev
polynomials in DeepONet[23] due to lacking the observation data, our work uses
the Representer Theorem[33] to construct the input function $u$, which can
achieve preliminary fitting of the observation data.

Based on the kernel method, there exists a feature map
$\Phi(\cdot):\mathcal{Z}\rightarrow\mathcal{Z}^{\prime}$, where $\mathcal{Z}$
is a nonempty set that the input vectors are taken from. The feature map
transforms the vector in original space $\mathcal{Z}$ to the feature space
$\mathcal{Z}^{\prime}$. Suppose $\boldsymbol{x}_{i}$ and $\boldsymbol{x}_{j}$
are two random vectors in $\mathcal{Z}$, then the kernel function
corresponding to the feature map $\Phi$ is defined as a function that maps
$\mathcal{Z}\times\mathcal{Z}\rightarrow\mathbb{R}$ satisfying [31]:

$\kappa(\boldsymbol{x}_{i},\boldsymbol{x}_{j})=\langle\Phi(\boldsymbol{x}_{i}),\Phi(\boldsymbol{x}_{j})\rangle$
(6)

There are several widely used kernel functions, such as the Gaussian kernel
function (Radial Basis Function, RBF) [34]

$\kappa_{\text{RBF}}({\boldsymbol{x}}_{i},{\boldsymbol{x}}_{j})=\exp(-\frac{\lVert{\boldsymbol{x}}_{i}-{\boldsymbol{x}}_{j}\rVert_{2}^{2}}{2\sigma^{2}})$
(7)

and the Mat$\acute{\text{e}}$rn kernel function[8]

$\kappa_{\text{Mat$\acute{\text{e}}$rn}}({\boldsymbol{x}}_{i},{\boldsymbol{x}}_{j})=(1+\frac{\sqrt{3}\lVert{\boldsymbol{x}}_{i}-{\boldsymbol{x}}_{j}\rVert_{1}}{\sigma})\cdot\exp(-\frac{\sqrt{3}\lVert{\boldsymbol{x}}_{i}-{\boldsymbol{x}}_{j}\rVert_{1}}{\sigma})$
(8)

where $\|\cdot\|_{2}$ and $\|\cdot\|_{1}$ denote the $L_{2}$ norm and $L_{1}$
norm, respectively, $\sigma$ is the width of the kernel function as the hyper-
parameter describing the correlation between any two sampled ray pairs
${\boldsymbol{x}}_{i}=(\tilde{\textbf{x}}_{i},\tilde{t}_{i})$ and
${\boldsymbol{x}}_{j}=(\tilde{\textbf{x}}_{j},\tilde{t}_{j})$ in our model.

Based on Representer Theorem, given a kernel function $\kappa$ defined on
Reproducing Kernel Hilbert Space (RKHS) $\mathcal{H}$, the solutions of a
large class of optimization problems can be expressed as kernel expansions
over the sample points. In 4D STEC prediction problem, the input function $u$
can be constructed that satisfies

$u^{\ast}=\operatorname*{arg\,min}_{u\in\mathcal{H}}\lVert
u(\textbf{X},\textbf{T})-\textbf{Y}\rVert_{2}^{2}$ (9)

and for any $(\tilde{\textbf{x}},\tilde{t})\in\tilde{\mathcal{D}}$, $u^{\ast}$
can be expressed as

$\displaystyle
u^{\ast}(\tilde{\textbf{x}},\tilde{t})=\kappa\big{(}[\tilde{\textbf{x}},\tilde{t}],[\textbf{X},\textbf{T}]\big{)}\kappa\big{(}[\textbf{X},\textbf{T}],[\textbf{X},\textbf{T}]\big{)}^{-1}\textbf{Y}$
(10)

where $\textbf{X}=[\textbf{x}_{1}^{(1)},\cdots,\textbf{x}_{m}^{(1)}]^{\top}$,
$\textbf{T}=[t_{1}^{(1)},\cdots,t_{m}^{(1)}]^{\top}$,
$\textbf{Y}=[y_{1}^{(1)},\cdots,y_{m}^{(1)}]^{\top}$. Therefore, when we
specify the form of the kernel function $\kappa$ and the data characteristics,
then the input function $u$ can be obtained.

###  II-D Preprocessing

####  II-D1 Temporal Encoding

In the STEC 4D estimation problem, without a suitable encoding, the variance
between adjacent time $t$ is not obvious and is overwhelmed by large values.
In the input domain, we specify time $t$ into year (Year, YY), day of year
(Day Of Year, DOY), and second of day (Second Of Day, SOD), namely as
$t=(\text{YY, DOY, SOD})$. DOY refers to the sequential count of days within a
calendar year from the beginning of the year. It is commonly sine-cosine-
encoded[35]

$\displaystyle\text{SOD}_{s}=\sin(\text{SOD}\cdot 2\pi/P)$ (11)
$\displaystyle\text{SOD}_{c}=\cos(\text{SOD}\cdot 2\pi/P)$

where $\text{SOD}_{s}$ and $\text{SOD}_{c}$ represent the sine and cosine
encoding of SOD, respectively, and $P$ represents the period, which is taken
as $3600\times 24$ seconds for one day here. The large and non-periodic values
of time data is avoided by encoding time $t$ into two orthogonal periodic
components.

####  II-D2 High Dimensional Sobol Sampling

When constructing the branch network input, it is necessary to obtain the
input function
$\\{u(\tilde{\textbf{x}}_{i}^{(1)},\tilde{t}_{i}^{(1)})\\}^{m}_{i=1}$ at $m$
scattered temporal-spatial sample points from the observation domain
$\tilde{\mathcal{D}}$. However, due to the fact that the domain
$\tilde{\mathcal{D}}$ in the 4D STEC estimation problem is high-dimensional,
the sampling is required to ensure the distribution of these $m$ sampling
points as uniform as possible.

To achieve high-dimensional sampling, an efficient uniform method-the Sobol
sampling [36] is adopted to deal with this difficulty. The key idea behind the
Sobol sequence is to use a set of primitive polynomials that exhibit desirable
properties when generating each component of the points. These polynomials are
carefully chosen to have a balance between low-discrepancy and efficient
computation.

(a) US station before sampling

(b) US station after sampling

Figure 2: US-CORS station distribution for the training (red), validation
(blue) and test (green) data for (a) all available sites; (b) the down-
sampling sites.

####  II-D3 Adaptive Inverse Frequency Weighting Optimization

Since the distribution of the train data is imbalanced, the Adaptive Inverse
Frequency Weighting (AIFW) method is adopted to optimize [37]. The AIFW method
handles the imbalanced training data and assigns weights to each data point
based on the frequency of its target value, which enables models to adjust
weights for sparse large value data.

Algorithm 1 4D DeepONet-STEC

0: a training set $\mathcal{T}$ consisting of $N$ observation data and a
random given ray $(\textbf{x},t)\in\mathcal{D}$ in real space.

0: the estimated STEC data $G(u)(\textbf{x},t)$ at $(\textbf{x},t)$.

1: Pre-sample observation data and construct the input function $u$, i.e., in
Eq. (10) [Process C]

2: Fix $m$ sample points $(\tilde{\textbf{x}}_{m}^{(1)},\tilde{t}_{m}^{(1)})$
in the current space [Process D2] and encode $\tilde{t}_{m}^{(1)}$ for the
sample points $(\tilde{\textbf{x}}_{m}^{(1)},\tilde{t}_{m}^{(1)})$ [Process
D1]

3: Feed preliminary estimated STEC values gained into the Branch network and
select input $(\textbf{x},t)$ into the Trunk network after temporal encoding

4: Use the Branch network output $\beta_{k}(u)$ and Trunk network output
$\tau_{k}(\textbf{x},t)$ to obtain the network output in Eq. (4). [Process B]

5: Train the model using weighted loss in Eq. (13) to obtain the operator
$G(u)(\cdot,\cdot)$. [Process D3]

6: return $G(u)(\textbf{x},t)$

For the training set $\mathcal{T}$, we introduce the label space
$\mathcal{Y}$, where we divide $\mathcal{Y}$ into $B$ groups (bins) with equal
intervals, i.e., $[y_{0},y_{1}),[y_{1},y_{2}),\cdots,[y_{B-1},y_{B})$.
$\mathcal{Y}=\\{\textbf{y}_{i}\\}_{i=1}^{B}$ with
$\textbf{y}_{i}=[y_{i-1},y_{i})$. We denote $b\in\\{0,\cdots,B-1\\}$ as the
group index of the target value. The defined bins reflect a minimum resolution
we care for grouping data in a regression task. We define $\delta y\triangleq
y_{b+1}-y_{b}=1$, showing a minimum difference of 1 TECU is of interest for
our problem. Finally, we could define each interval $\textbf{y}_{i}$ as the
target value $y_{i}$ in each bin, $\mathcal{P}(\textbf{y}_{i})$ is as the
number of the target value within this bin, then the weight $\omega_{i}$ for
${y_{i}}$ can be expressed as

$\omega_{i}=\big{(}\frac{y_{i}}{\mathcal{P}(\textbf{y}_{i})}\big{)}^{\lambda}$
(12)

where $\lambda$ represents the control factor of the weighting decay
amplitude. In this paper, $\lambda$ was chosen as 0.05 by trial and error. By
adding weights into Eq. (5) of DeepONet, the modified loss function becomes

$L=\sum_{i=1}^{{h}}|G(u)(\tilde{\textbf{x}}_{i}^{(2)},\tilde{t}_{i}^{(2)})-y_{i}^{(2)}|^{2}\cdot\frac{w_{i}}{\sum\limits_{i=1}^{h}w_{i}}$
(13)

Figure 3: Global station distribution for the training, validation and test
data after down-sampling. Tree test sites (green triangle) and four validation
sites (blue triangle) are selected globally, respectively, while the red
square refers to the training sites.

The 4D DeepONet-STEC prediction model is illustrated in Figure 1. Using
$\\{(\tilde{\textbf{x}}_{i},\tilde{t}_{i},\tilde{y}_{i})\\}_{i=1}^{M}$ to
train, the model can input random giving ray $(\textbf{x},t)\in\mathcal{D}$ to
get the output STEC value $y$. Specifically, assuming the time period to
predict is $\textbf{T}_{0}$ and $\textbf{T}_{0}\in[0,\textbf{T}]$, the input
space during training is defined as
$\tilde{\mathcal{D}}_{input}=\tilde{\mathcal{X}}\times[0,\textbf{T}-\textbf{T}_{0}]$,
while the output space is defined as
$\tilde{\mathcal{D}}_{output}=\tilde{\mathcal{X}}\times[\textbf{T}-\textbf{T}_{0},\textbf{T}]$.
The architecture consists of two key parts. The first part involves
constructing the input function $u$, with the input space defined as
$\tilde{\mathcal{D}}$ and the output space as $\mathcal{D}$. During the
network training, the available data is limited to observations, necessitating
the partitioning of $\tilde{\mathcal{D}}$. The input function $u$ is
constructed using the kernel function labeled Process C, utilizing observation
data to map a given ray in the observation space to STEC. After Process D1 and
Process D2, this serves as the input for the second part, which trains the
operator $G$ to extend the range of applicability and accuracy of the input
function $u$ as Process B and Process D3. As the final generated function
$G(u)(\cdot,\cdot)$ is the dot product of the outputs from the Branch and
Trunk networks, both networks require an equal number of neurons in their
output layers. Using the optimized function $G(u)(\cdot,\cdot)$ obtained from
the training, the STEC estimate $G(u)(\textbf{x},t)$ can be obtained. Thus,
the constructed 4D DeepONet-STEC model enables 4D STEC estimation. The 4D
DeepONet-STEC algorithmic flow is depicted in Algorithm 1.

(a) Kp&Dst index (Quiet Period)

(b) All rays at one GODE station (Quiet Period)

(c) Kp&Dst index (Storm Period)

(d) All rays at one GODE station (Storm Period)

Figure 4: Temporal dataset of the geomagnetic index Kp&Dst and retrieved STEC
value at a single station (GODE) during solar ionospheric quiet (a-b) and
storm (c-d) time periods.

##  III Validation Results and Analysis

To verify and validate the accuracy and robustness of the 4D DeepONet-STEC
model, we perform a series of tests using both the simulation datasets and
real observation datasets under global and regional GNSS station geometry
scenarios.

###  III-A Validation Setup

The validation sites were selected based on US-CORS (Continuously Operating
Reference Stations) GNSS stations in Fig. 2 and global IGS (International GNSS
service) as illustrated in Fig. 3 to ensure representative coverage. To assess
the performance of the proposed 4D DeepONet-STEC model, the whole number of
stations is divided into the training sites (red square) , the validation
sites (blue triangle) and the test sites (green triangle) in Fig. 2(b) and
Fig. 3. By employing this selection of sites and training data, this model
aims to simulate real-world scenarios for users at arbitrary locations within
the region of interest.

In order to verify the performance of the model at different solar activity
for diverse ionospheric conditions, we choose two global dataset to test the
model: the quiet period in Fig. 4(a) and the storm period to test the model in
Fig. 4(c). The quiet time period spans from June 1, 2020, to June 30, 2020 and
the storm time period spans from February 1, 2023 to March 31, 2023 based on
the magnetic storm $\rm{Kp}$ and $\rm{Dst}$ index.

In quiet periods, the training dataset for the train stations were utilized
during June 1 to June 27, 2020. The validation stations without training are
used to optimize the train model performance by making prediction from June 25
to June 27, 2020. The test sites were selected to predict from June 28 to June
30, 2020. Similarly, in storm periods from February 1, 2023 to March 31, 2023,
the test days is enhanced to 10 days from March 22 to April 1, 2023. The
training and validation dataset is from February 1 to March 21, 2023. The
ratio of the training and test days for quiet periods is $27:3$ and the ratio
of the training and test days for magnetic storm periods is $49:10$,
respectively. respectively.

For the simulation data generation, we use the NeQuick2 model [38]. The
NeQuick model is a three-dimensional and time dependent ionospheric electron
density, which has been adapted for Galileo satellite navigation single-
frequency ionospheric corrections. The NeQuick2 model can provide the electron
density along any ground-to-satellite straight line ray-path and the
corresponding slant Total Electron Content (STEC) by numerical integration.
The expression for the Epstein layer function of the Nequick2 model is shown
as

$N_{Epstein}=\frac{4N_{max}}{\big{(}1+\exp(\frac{h-h_{max}}{B})\big{)}^{2}}\exp(\frac{h-h_{max}}{B})$
(14)

where $h$ is the height, $h_{max}$ is the layer peak height, $B$ is the
thickness parameter, and $N_{max}$ is the layer peak electron density.

The NeQuick-STEC simulator is used to generate rays by extracting satellite
coordinates for specified time periods using precision ephemeris files. The
NeQuick2 model was then applied to these rays, for selected time as inputs, to
generate simulated STEC values for each ray, forming the synthetic STEC
dataset [13], which can provide efficient test for evaluating the developed
model.

For the real observation dataset generation, the undifferenced and uncombined
precise point positioning (UC-PPP) with known fixed station coordinates is
used to extract the STEC values from GNSS observables [39, 40]. For
electromagnetic wave frequency $f_{i}$, the GNSS observation equations can be
written as [39]:

$\displaystyle
P^{s}_{r,f_{i}}=\rho^{s}_{r}+(dt_{r}-dt^{s})+T_{r}^{s}+I_{r,f_{i}}^{s}$ (15)
$\displaystyle+(d_{r,f_{i}}-d^{s}_{f_{i}})+\epsilon^{s}_{r}$

$\displaystyle
L^{s}_{r,f_{i}}=\rho^{s}_{r}+(dt_{r}-dt^{s})+T_{r}^{s}-{I_{r,f_{i}}^{s}}$ (16)
$\displaystyle+(b_{r,f_{i}}-b^{s}_{f_{i}})+\lambda
N^{s}_{r,f_{i}}+\eta^{s}_{r}$

where the superscript $s$ and the subscript $r$ denotes a satellite and a
receiver, respectively. $P^{s}_{r}$ and $L^{s}_{r}$ denotes the pseudo-range
and carrier phase observable, respectively. $\rho^{s}_{r}$ is the geometric
distance between the receiver $r$ to the satellite $s$. Here, $dt_{r}$ and
$dt^{s}$ are the receiver and satellite clock offsets, respectively.
$d_{r,f_{i}}$ and $b_{r,f_{i}}$ are the receiver code and phase biases, and
$d^{s}_{f_{i}}$ and $b^{s}_{f_{i}}$ are the satellite code and phase biases
(in meters). Here, $I_{r}^{s}$ is the line of sight (LOS) slant ionospheric
delay and $T^{s}_{r}$ is troposphere delay, $N^{s}_{r}$ is the phase
ambiguity, $\lambda$ is the wavelength and $\epsilon_{p}$ is the unmodelled
errors for pseudo-range and carrier phase observable. The ionospheric STEC is
estimated by the dual-frequency undifferenced and uncombined UCPPP method
using both pseudo-range and carrier phase observables[41]. The DCBs of the
receivers and satellites are separated by ionospheric single layer modeling
(SLM), yielding the final unbiased STECs. The epoch temporal resolution is
$30\,\rm{s}$.

TABLE I: List of Selected IGS Station for Test and Validation Station | Region | Name | Latitude[∘] | Longitude[∘]
---|---|---|---|---
Test | USA | CLRE | 42.72 | -82.59
KYTI | 38.42 | -83.74
| ALAS | 34.47 | -87.86
| Global | TIXG | 71.63 | 128.87
| GODE | 39.02 | -76.83
| MAYG | -12.78 | 45.26
Validation | USA | OHCD | 41.54 | -81.63
GAST | 35.31 | -81.18
| INLP | 41.58 | -86.69
| TN30 | 35.55 | -87.57
| Global | FALK | -51.69 | -57.87
| AUCK | -36.6 | 174.83
| SCH2 | 54.83 | -66.83
| YSSK | 47.03 | 142.72

####  III-A1 US region data generation

The latitude and longitude range for the North American region was set as
[33∘N, 43∘N] and [80∘W, 90∘W], consisting of 300 GNSS stations, respectively.
To address the large number of stations for high computational cost, a sparse
downsampling process was employed. Specifically, the entire region was divided
into a 0.77${}^{\circ}\times$0.77∘ grid patch unit with a number of
13$\times$13 patches. Only one station is selected from each grid patch. This
downsampling process reduces the dataset size while maintaining spatial
diversity. Within the total number of stations 137 after downsampling, three
stations were chosen for testing, while four stations were designated for
validation. The distribution of the test, validation, and training stations in
the North American region is illustrated in Fig. 2(b).

(a) US CORS observation STEC dataset of single-station, single-satellite
between June 1-27 (Training) and June 28-30 (Test)

(b) US CORS predicted (red) and observed (blue) STEC between June 28-30 (only
test comparison)

Figure 5: US CORS observation dataset predicted STEC of single-station,
single-satellite (G08) for three test stations for (a) one-month data and (b)
only the three-day prediction period. The blue dots represents the
observation, while the red dots represents the predicted STEC values in quiet
periods.

(a) CLRE

(b) KYTI

(c) ALAS

Figure 6: Up: US CORS observation data prediction result by the DeepONet-STEC
model for three test station-satellite rays, where the color counts the number
of rays denoting the correlation between the observation and predicted value
within the prediction period 3 days. Down: the DeepONet-STEC model prediction
result comparison with observation within 3 days in quiet periods.

####  III-A2 Global data generation

To generate the global dataset, the entire Earth was considered with latitudes
from -90∘ to 90∘ and longitudes from -180∘ to 180∘. In order to handle the
large number of available stations worldwide, similar downsampling process was
applied to divide the global stations into a 9${}^{\circ}\times$9∘ grid patch
unit with a total number of 800 patches. Then, we averagely select 114 sites
from original 534 sites in total. The resulting distribution of selected
stations across the global region forms the basis of the global dataset used
in this study as shown in Fig. 3. The maximum STEC value is 74 TECU.

Fig. 4(b) shows the retrieved STEC measurement data by UCPPP for the global
dataset during the quiet period for June $1-30$, 2020. Fig. 4(d) shows the
retrieved STEC measurement data by UCPPP during the storm period from Feb 1 to
Mar 31, 2023 for different GPS satellites at the cutoff elevation angel
$15^{\rm o}$. The time resolution for each ray data is $60s$. Based on high
temporal resolution historical data, the proposed DeepONet-STEC model can
predict STEC for the next 72 hours for instance with each minute interval. The
maximum STEC value in quiet periods is 42 TECU in Fig. 4(b) and the maximum
STEC value in storm periods is 243 TECU in Fig. 4d.

To evaluate the prediction performance of the 4D DeepONet-STEC model, we
define the Root Mean Square Error (RMSE) and $R^{2}$ metrics to quantify the
accuracy and goodness-of-fit of the model’s predictions. Furthermore, to
assess the model’s suitability for navigation applications, the Quality
Assessment (QA) metrics are introduced [42], which provides additional metric
for the model’s capability to achieve high-precision and reliable navigation
systems.

$RMSE=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(y_{i}-\hat{y_{i}})^{2}}$ (17)

$\it{R^{2}}=1-\frac{\sum_{i=1}^{N}\left(y_{i}-\hat{y_{i}}\right)^{2}}{\sum_{i=1}^{N}\left(y_{i}-\bar{y}\right)^{2}}$
(18)

$\it{QA}=\frac{num(\left|y_{i}-\hat{y_{i}}\right|<\theta)}{N}\times 100\%$
(19)

$\it{MAPE}=\frac{1}{N}\sum_{i=1}^{N}\left|\frac{y_{i}-\hat{y_{i}}}{\hat{y_{i}}}\right|$
(20)

where $y_{i}$ represents the $i$-th prediction value, $\hat{y}_{i}$ represents
the $i$-th true value, $N$ represents the total number of data for test
stations. In QA metrics, $\theta$ represents the scale of quality assessment
and $num(\mathcal{A})$ represents the number of individuals satisfying
condition $\mathcal{A}$. We denoted QA value as QA03 for $\theta=0.3$ TECU and
QA10 for $\theta=1.0$ TECU. For mean absolute percentage error (MAPE), we
calculated the MAPE value during the predicting period as shown in Table II,
III and IV, respectively.

###  III-B Simulated Data Results in Quiet Periods

(a) Global simulated STEC dataset of single-station, single-satellite between
June 1-27 (Training) and June 28-30 (Test)

(b) Global simulated dataset predicted (red) and observed (blue) STEC between
June 28-30 (only test comparison)

Figure 7: Global simulated dataset predicted STEC of single-station, single-
satellite (G08) for three test stations. The blue dots represents the
observation, while the red dots represents the predicted STEC values in quiet
periods.

US CORS observation STEC dateset of single-station, single-satellite between
June 1-30; US CORS predicted (red) and observed (blue) STEC between June
28-30.

(a) TIXG

(b) GODE

(c) MAYG

Figure 8: Global simulated dataset correlation between the predicted value and the model truth based on the global simulated 27 days dataset over all satellite-station rays for the future three days in quiet periods. TABLE II: Statistical prediction results of the simulated data based on the test stations in quiet period Region | Station | RMSE [TECU] | $R^{2}$ | MAPE [%] | QA03 [%] | QA10 [%]
---|---|---|---|---|---|---
USA | CLRE | 0.1347 | 0.9992 | 1.36 | 95.95 | 100.00
KYTI | 0.1366 | 0.9993 | 1.39 | 95.53 | 100.00
ALAS | 0.1559 | 0.9994 | 1.44 | 94.93 | 99.83
Global | TIXG | 0.4989 | 0.9695 | 3.54 | 46.49 | 94.85
GODE | 0.2596 | 0.9981 | 3.45 | 79.01 | 99.62
MAYG | 0.6014 | 0.9960 | 5.78 | 45.78 | 90.91

The result shows that the model’s predicted STEC RMSE error for the US region
is at $\sim 0.15$ based on simulation data as shown in Table II, the disparity
between the predicted STEC and true STEC is very trivial, and the differences
in latitudes and longitudes of the stations in the US region are negligible
that can be regarded as a quasi-baseline. Therefore, only the simulated STEC
results of global stations are presented below.

Fig. 7 represents the predicted STEC of single-station, single-satellite (G08)
within three days for the cutoff elevation angel $15^{o}$ based on global
30-day simulated dataset for three test stations (TIXG, GODE, MAYG). Based on
the simulation STEC data, the overall trend of single station single satellite
within 30 days of single satellite is relatively smooth and periodic. The
simulated results show that the predicted results within June 28-30 in the red
curve in good agreement with the simulation true STEC values in blue, which
demonstrates that the model learns the variability of the STEC data with time
at different regimes, indicating good generalisation based on the Nequick2
simulated data.

The simulation data prediction results for the USA and global region data are
summarized in Table II. In the North American region, the CLRE station and
KYTI station achieved results with RMSE less than 0.14 TECU. For the two
stations, the QA03 value is higher than 95% and QA10 of 100%. On the other
hand, the ALAS station exhibited relatively lower accuracy with an RMSE of
0.15 TECU, $R^{2}$ value of 0.9994, QA03 of 94.93%, and QA10 of 99.83%. The
reason for the degraded performance at ALAS can be attributed to its location
in the southern region of the North American region with fewer stations
available. The global MAPE $3.5\%\sim 5.7\%$ is higher than that in the USA
simulated dataset $1.3\%\sim 1.4\%$.

(a) Global observation dataset predicted STEC of single-station between June
1-30

(b) Global observation dataset predicted STEC of single-station between June
28-30 (only test comparison)

Figure 9: Global observation dataset predicted STEC of single-station, single-
satellite (G08) for three test stations. The blue dots represents the
observation, while the red dots represents the predicted STEC values in quiet
periods.

(a) TIXG

(b) GODE

(c) MAYG

Figure 10: Up: Global observation data prediction result by the DeepONet-STEC
model, where the color counts the number of rays denoting the correlation
between the observation and predicted value for all rays within the prediction
period 3 days. Down: the model DeepONet-STEC prediction result comparison with
observation within 3 days in periods.

Fig. 8. presents a correlation between the model predicted STEC and simulated
true STEC value within three days. In the Global region, the GODE station
exhibited high accuracy with an RMSE of 0.2596 TECU, $R^{2}$ value of 0.9981,
QA03 of 79.01%, and QA10 of 99.62%. The performance can be attributed to its
location, which falls within the densely populated region of stations. The
availability of nearby stations contributes to improved data coverage and more
precise predictions. In contrast, the TIXG station and MAYG station displayed
relatively higher RMSE values of 0.4989 TECU and 0.6014 TECU. The limited data
coverage near the equator and at high latitudes leads to low accuracy and high
uncertainty. For the global simulation data, with a wider area and sparser
station distribution, the model still performs well, which demonstrates that
the model has good generalization.

The prediction result of the 4D DeepONet-STEC model based on the simulation
data clearly demonstrates its superiority in accurately predicting the STEC
variations. The selected stations within the North American and global regions
have consistently exhibited high accuracy, with RMSE values within 0.7 TECU
and $R^{2}$ indices exceeding 0.96.

###  III-C Observation Data Results in Quiet Periods

TABLE III: Statistical prediction results of the observation data based on the test stations in quiet periods Region | station | RMSE [TECU] | $R^{2}$ | MAPE [%] | QA03 [%] | QA10 [%]
---|---|---|---|---|---|---
USA | CLRE | 1.3322 | 0.8894 | 14.23 | 23.50 | 63.93
KYTI | 1.4220 | 0.8890 | 11.13 | 24.36 | 62.53
ALAS | 1.3964 | 0.9073 | 9.33 | 23.21 | 61.80
Global | TIXG | 1.4728 | 0.7584 | 16.20 | 18.65 | 55.76
GODE | 1.2460 | 0.9097 | 9.79 | 19.56 | 58.94
MAYG | 1.7371 | 0.8461 | 26.64 | 14.48 | 44.28

First of all, we present the DeepONet-STEC prediction results for the US CORS
observation data. Fig. 5. represents the 30-day observation STEC value versus
the 28-30-day model-predicted STEC value for each station in the single-
satellite (G08) case based on the US CORS observation data. In Fig. 6, the
overall change of the observation STEC data at single station for all
satellite rays within 30 days of single satellite varies periodically each
day. The unsmooth and fluctuating characteristics of the observation STEC data
compared to the Nequick2 simulation STEC data are related to the quality of
the station observation data, as well as the complex real-time conditions of
the ionosphere. For different stations, the model predictions fluctuate
differently, while for multiple days of data from the same station, there is a
pattern of variation in the model predictions, but the model performs slightly
worse on individual data with large values. The results in the USA region
suggest that the model can learn the disparity between stations and the
patterns of the change in the STEC data over time.

Secondly, we present the DeepONet-STEC prediction results for the Global
observation data. Fig. 9. represents the 30-day observation STEC value versus
the 28-30-day model-predicted STEC value for each station in the single-
satellite (G08) case in the global region. From this scatterplot in Fig. 10 ,
the observation STEC data, the overall change is generally periodic with large
fluctuations, while the STEC value varies at different stations at the same
day. The large latitude and longitude span and the uneven distribution of the
global stations lead to a more complex dataset, which increases the difficulty
of the model learning. The results show that in many test cases, especially
for small values, the red dots overlap with the blue dots, indicating that the
model-predicted STEC values do not differ much from the observed STEC values.
The model still performs slightly worse for individual data with large values.
For the three test stations at sparse distributed locations, the model
predictions are still within acceptable limits.

The observation data prediction results for the two regions are summarized in
Table III. Fig. 6. present a comparison between the prediction value obtained
from the model and the observation data in the USA region. In the USA region,
the performance of the three stations is basically similar, with RMSE values
around 1.3-1.4 TECU, $R^{2}$ around 0.89, QA03 around 23%, and QA10 around
62%. From the single-station scatter plots, it can be noted that the
observation data are not as smooth as the simulation data, and there is a
certain degree of fluctuation in STEC value from day to day. The global
observation dataset MAPE up to $\sim 26.6\%$ is higher than that in the USA
observation dataset $9.3\%\sim 14.2\%$.

The observation date results demonstrate that the model’s performance in the
USA regional areas is stable.

Figure 11: Global prediction results during the validation time span from
March 12 to March 21, 2023 using the magnetic storm dataset at two validation
stations. Up: SCH2 (Northern hemisphere); Down: FALK (Southern hemisphere).

Figure 12: Global prediction results from February 1 to March 21, 2023 at the
validation station SCH2 and FALK using the magnetic storm dateset. The blue
curve is the observation and the red curve is the predicted STEC value between
March 12 to March 21, 2023.

Figure 13: Global storm observation data prediction results by three methods
during the time span March 12 to 21, 2023 at the test GODE station. Up: 4D
DeepONet-STEC model, Middle: ANN model, Down: RF model.

Figure 14: Predicted STEC for single satellite at the test GODE station using
the magnetic storm dataset. Up: 4D DeepONet-STEC model. Middle: ANN model.
Down: RF model. The blue curve is the observation and the red curve is the
predicted STEC between March 22 and March 31, 2023.

Fig. 10. present a comparison between the predicted results obtained from the
DeepONnet-STEC model and observation values of observation data in the global
region. In the Global region, the GODE station in the Global region also
exhibited high accuracy with an RMSE of 1.2460 TECU, $R^{2}$ value of 0.9097,
even better than small region result. The TIXG station shows good result as
small region result, where RMSE value is 1.4728 TECU and $R^{2}$ value is
0.7584.The MAYG station is slightly underperforming with RMSE values RMSE
value of 1.7371 TECU and $R^{2}$ value of 0.8461. The single-site, single-
satellite results can be noticed to show that there are different trends in
STEC value changes at different location stations around the global area, and
the data are all subject to fluctuations. This indicates that the predictive
ability of the model is still credible for the observation data. For the MAYG
station, it is noted that the simulated and observed data are different, which
can indicate that there exist disparity between the model and observed data,
which further validates the robustness of the model.

The prediction result of observation data of the 4D DeepONet-STEC model
performance also shows high accuracy in STEC prediction problem. The selected
stations within the North American and Global regions have consistently
exhibited high accuracy, with RMSE values within 1.8 TECU and $R^{2}$ indices
exceeding 0.75. Overall, due to the complexity of the observation data, the
performance of the model on observation dataset is not as good as that on
simulation dataset, but the value error of the model prediction results is
generally small, which proves that the model can learn the whole range of STEC
temporal and spatial patterns of change in a small observation dataset.

Figure 15: Global storm observation data prediction result by three methods
during the time span March 12 to 21, 2023 at the test TIXG station. Up: 4D
DeepONet-STEC model. Middle: ANN model. Down: RF model.

Figure 16: Predicted STEC for single satellite at the test TIXG station using
the magnetic storm dataset. Up: 4D DeepONet-STEC model. Middle: ANN model.
Down: RF model. The blue curve is the observation and the red curve is the
predicted STEC between March 22 and March 31, 2023.

Figure 17: Global storm observation data prediction result at the test MAYG
station by three methods during the time span March 12 to 21, 2023 using the
magnetic storm dataset . Up: 4D DeepONet-STEC model. Middle: ANN model. Down:
RF model.

Figure 18: Predicted STEC for single satellite at the test MAYG station using
the magnetic storm dataset. Up: 4D DeepONet-STEC model. Middle: ANN model.
Down: RF model. The blue curve is the observation and the red curve is the
predicted STEC between March 22 and March 31, 2023.

(a) DeepONet-STEC result between March 22-31 in storm dataset

(b) ANN result between March 22-31 in storm dataset

(c) Random Forest result between March 22-31 in storm dataset

Figure 19: Global storm observation dataset predicted STEC of single-station,
single-satellite (G08) for three test stations between March 22-31. The blue
dots represents the observation, while the red dots represents the predicted
STEC values by three models in storm periods.

###  III-D Global Observation Data Results in Storm Periods

In order to evaluate the performance of the DeepONet-STEC model, we compare
the DeepONet-STEC with the other deep learning methods of random forest (RF)
and ANN. Both methods will directly learn the mapping function from the input
value to the output value. Since the 4D DeepONet-STEC uses two networks with
16 and 30 layers respectively to learn, in order to compare with 4D DeepONet-
STEC under relatively equal conditions, the network architecture of ANN is set
to 46 layers, while the maximum depth of the tree of RF is limited to 46
layers.

Here, we choose the two-month magnetic storm period data from February 1, 2023
to April 1, 2023 for validation as shown in Fig. 4(c)&(d). Based on the the
geomagnetic index Kp and Dst, the two storms occur from February 25 to March
14 with the minimum $\rm{Dst}=-138$, 2023 and March 22 to 31, 2023 with the
minimum $\rm{Dst}=-184$. The training and validation data is from Feb 1 to
March 21,2023 and the test data is from March 22 to April 1, 2023.

In Fig. 11 and Fig. 12, the prediction results at the validation stations are
analyzed for the magnetic storm period data here. The training data does not
contain validation stations, and the model prediction from validation stations
from March 12 to March 21 will be recorded. The model results at two
representative validation stations SCH2 in the northern hemisphere and FALK in
the southern hemisphere, are presented in Fig. 11. The results at two
validation stations show that the $R^{2}$ coefficient is greater than 80% ,
that captures the STEC variance well. For validation station SCH2, there are
missing data for some dates because STEC data were not extracted for every
station on every day, but still our model has good results. The results at the
validation stations show that our model has exhibited good performance,
therefore we further analyze the model performance at the test stations.

Figs. 13-18 represent the two-month observation STEC value versus the model-
predicted STEC value of the single-satellite (G08) at three test stations
(TIXG, MAYG, GODE) based on the global observation storm data using DeepONet-
STEC, ANN, and Random Forest. The DeepONet-STEC captures the trends much
better than the RF and ANN. Table IV compared the results of three method at
TIXG, GODE and MAYG, respectively. The high latitude station (TIXG) in Fig. 15
and Fig. 16 performs the worst among the three sites. The $R^{2}$ coefficient
at the GODE in Fig. 13 and Fig. 14 and MAYG station in Fig. 17 and Fig. 18
reaches to a good value in Table IV for DeepONet-STEC even during the storm
period and degraded quickly at the high latitude TIXG site. The value of RMSE
depends on the data retrieval accuracy for the PPP algorithm during the
magnetic storm, which will be another subject to further investigate in the
future.

TABLE IV: Three method prediction results comparison of the observation data based on the test stations in storm periods Station | Model | RMSE[TECU] | $R^{2}$ | MAPE[%] | QA03[%] | QA10[%]
---|---|---|---|---|---|---
TIXG | DeepONet-STEC | 11.0743 | 0.5674 | 46.72 | 3.28 | 11.45
ANN | 16.0717 | 0.0889 | 98.36 | 0.97 | 3.30
RF | 15.2634 | 0.1783 | 88.17 | 1.42 | 4.60
GODE | DeepONet-STEC | 9.4456 | 0.7771 | 29.93 | 2.01 | 6.83
ANN | 18.9082 | 0.1069 | 88.76 | 0.98 | 3.29
RF | 19.7382 | 0.0268 | 91.98 | 1.21 | 4.07
MAYG | DeepONet-STEC | 10.8742 | 0.8991 | 22.78 | 2.93 | 10.11
ANN | 40.1250 | -0.3742 | 148.38 | 0.60 | 2.10
RF | 40.4694 | -0.3979 | 152.76 | 0.40 | 1.34

It it noted that for active solar activities during magnetic storms, the STEC
values are much larger and fluctuate more widely than that during quiet
periods, reaching levels greater than 100 TECU. Because the magnetic storms in
the predicted time period are stronger than that in the time period of the
training set, there appear some extreme data at individual stations such as
MAYG. The prediction results by DeepONet for large value data shows relative
smooth trend compared to the observation data.

The magnified results of the DeepONet-STEC during the prediction $10$ days
period between March 22-31 are compared in Fig. 19 for three test stations
using the global observation storm dataset. The results show that due to the
complexity of the training data during the magnetic storms, the prediction
results of the Random Forest model fluctuate a lot, and the ANN model gives
unstable prediction results. The DeepONet-STEC model maintains good stable
predictions, which proves the robustness of our DeepONet-STEC model for global
magnetic storm periods.

Overall, the 4D DeepONet-STEC model can relatively well learn the different
STEC data characteristics during calm periods versus magnetic storms, whereas
ANN and Random Forest methods cannot effectively capture the complex changing
characteristics of STEC at different stations over long time periods.

##  IV Conclusion and Discussion

In this paper, a deep learning-based framework, the 4D DeepONet-STEC model,
has been proposed for predicting ionospheric STEC based on the deep operator
network using global GNSS data. The 4D DeepONet-STEC model first adopts the
deep operator network techniques to achieve high 4D temporal-spatial
resolution of the STEC values at any given satellite-station ray to enhance
the accuracy and performance for providing accurate ionospheric corrections.
The innovative Kernel method is adopted to construct the input function from
the real observation data for the DeepONet-STEC model.

The experimental results obtained from Nequick2 simulation data and
observation data demonstrated the effectiveness and robustness of the 4D
DeepONet-STEC model. The observation data has been extracted by the UCPPP with
time resolution $30\,\rm{s}$. The model achieved a relatively low accuracy
RMSE $\sim 1.5$ TECU and high $R^{2}\sim 0.85$ values for global observation
data in quiet period data of one month, which indicates accurate prediction
and good correlation between predicted and true values. The QA metrics,
specifically the QA03 and QA10 metrics, revealed that the absolute errors
decreases within acceptable percentages, which are specified to apply for
providing reliable ionospheric corrections for positioning applications. In
addition, the DeepONet-STEC prediction tested on data from two-month magnetic
storm period data performs better for high solar activity scenarios in
comparison with ANN and random forest model.

In conclusion, the 4D DeepONet-STEC model demonstrates strong potential for
accurately predicting ionospheric TEC, enabling improved positioning accuracy
in navigation applications. The novelty of this work presents a framework for
4D estimation of ionospheric STEC based on the deep operator network. The
model’s performance evaluation, supported by comprehensive analysis and
visualization, validates its effectiveness and reliability. Future work can
focus on further refining and optimizing the 4D DeepONet-STEC model to enhance
its accuracy and applicability across different geographical regions and
varying ionospheric conditions. Additionally, incorporating real-time
observation data into the model can improve its adaptability and
responsiveness to dynamic ionospheric changes. This work based on machine
learning pave the way for further advancements in ionospheric remote sensing
and space system applications by exploring advanced artificial intelligence.

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|  Dijia Cai received the Bachelor’s degree in National University of Defense
Technology in 2022. She is currently studying the Master’s degree in
electromagnetic field and microwave technology at Fudan University. Her
research interests include ionospheric modeling and machine learning.
---|---
|  Zenghui Shi received his B.S. degree in Electronic Information Engineering
from Xiamen University in 2020. He received his M.S. degree in electromagnetic
field and microwave technology at Fudan University and Shanghai Innovation
Center for BeiDou Intelligent Application in 2023. His research interests
include ionospheric modeling and spatial and temporal prediction of
ionospheric correction quantities.
---|---
|  Haiyang Fu received the B.S. with Honor and M.S. degrees in power mechanics
and engineering from Harbin Institute of Technology, Harbin, China, in 2006
and 2008, respectively and received the Ph.D. degree in electrical engineering
from Virginia Polytechnic Institute and State University, Blacksburg, USA, in
2012. In 2013, she joined in school of information science and engineering at
Fudan University as an assistant professor. Since 2023, she has been a
professor in the Key Laboratory of Information Science of Electromagnetic
Waves, Fudan University, Shanghai, China. She has been awarded as the URSI
(International Union of Radio Science) Young Scientist Award in 2017 and
currently served as the Chair of Woman in Radio Science Chapter of URSI-China
(2024 $\sim$ ). She has served as the deputy session chair of Commission
C4.1/D5.1 on Space Research COSPAR (2018-2022). Her current research focuses
on ionospheric physics and remote sensing with artificial intelligence.
---|---
|  Huan Liu is currently a research associate in the School of Mathematics,
Shanghai University of Finance and Economics, Shanghai, China. He received the
B.S. degree from the Central South University, Changsha, China in 2014, and
Ph.D. degree in computational mathematics from Shandong University, Jinan,
China in 2019. From 2020 to 2022, he was a postdoc research fellow in the
School of Mathematical Science, Fudan University, Shanghai, China. His current
research interests include computational methods for partial differential
equations, scientific and engineering computing and inverse problems.
---|---
|  Hongyi Qian received the B.Sc. from the School of Communication and
Information Engineering, Shanghai University in 2014 and received the MS. from
the School of information science and technology, Fudan University, Shanghai,
in 2023. He is currently an Algorithm Engineer at China Mobile Research
Institute, Shanghai, China, in the area of global navigation satellite system
(GNSS) algorithms. Nowadays, he focuses on the ionospheric modelling in PPP-
RTK algorithm.
---|---
|  Yun Sui received the Bachelor’s degree in information Engineering from
Nanjing University of Aeronautics and Astronautics, Jiangsu In 2013, and her
Master’s degree in Electromagnetic Field and Microwave Technology from Nanjing
University of Aeronautics and Astronautics, Jiangsu in 2016. She is currently
studying for her doctor’s degree in electromagnetic field and microwave
technology at Fudan University. From 2016 to 2018, she worked as an assistant
engineer in Shanghai Key Laboratory of Electromagnetism. Her research
interests include ionospheric modeling, GNSS positioning, and electromagnetic
big data.
---|---
|  Feng Xu (Senior Member, IEEE) received the B.E. degree (Hons.) in
information engineering from Southeast University, Nanjing, China, in 2003,
and the Ph.D. degree (Hons.) in electronic engineering from Fudan University,
Shanghai, China, in 2008. He is currently a Professor and the Vice Dean of the
School of Information Science and Technology and the Vice Director of the
electromagnetic waves (MoE) Key Laboratory for Information Science of
Electromagnetic Waves. His research interests include electromagnetic
scattering modeling, SAR information retrieval, and radar system development.
Dr. Xu was a recipient of the second-class National Nature Science Award of
China in 2011, the 2014 Early Career Award of the IEEE Geoscience and Remote
Sensing Society, and the 2007 SUMMA Graduate Fellowship in the advanced
electromagnetics area. He is a Topic Associate Editor of the IEEE TRANSACTIONS
OF GEOSCIENCE AND REMOTE SENSING and was an Associate Editor of the IEEE
GEOSCIENCE AND REMOTE SENSING LETTERS (2014–2021). He is the Founding Chair of
the IEEE GRSS Shanghai Chapter and an IEEE GRSS AdCom Member.
---|---
|  Ya-Qiu Jin received the bachelor’s degree in electrical engineering and
computer science from Peking University, Beijing, China, in 1970, and the
M.S., E.E., and Ph.D. degrees in electrical engineering and computer science
from the Massachusetts Institute of Technology, Cambridge, MA, USA, in 1982,
1983, and 1985, respectively. In 1985, he joined Atmospheric Environmental
Research, Inc., Cambridge, as a Research Scientist. From 1986 to 1987, he was
a Research Associate Fellow at the City University of New York, New York, NY,
USA. In 1993, he joined the University of York, York, U.K., as a Visiting
Professor, sponsored by the U.K. Royal Society. He is currently the Te-Pin
Professor and the Director of the Key Laboratory for Information Science of
Electromagnetic Waves (MoE), Fudan University, Shanghai, China. He has
authored more than 720 papers in refereed journals and conference proceedings
and 14 books, including Polarimetric Scattering and SAR Information Retrieval
(Wiley and IEEE, 2013), Theory and Approach of Information Retrievals From
Electromagnetic Scattering and Remote Sensing (Springer, 2005), and
Electromagnetic Scattering Modelling for Quantitative Remote Sensing (World
Scientfic, 1994). His research interests include electromagnetic scattering
and radiative transfer in complex natural media, microwave satellite-borne
remote sensing, as well as theoretical modeling, information retrieval and
applications in earth terrain and planetary surfaces, and computational
electromagnetics. Dr. Jin was awarded by the Senior Research Associateship in
NOAA/NESDIS by the USA National Research Council, in 1996. He was a recipient
of the IEEE GRSS Distinguished Achievement Award (2015), the IEEE GRSS
Education Award (2010), the China National Science Prize (1993 and 2011), the
Shanghai Sci/Tech Gong-Cheng Award (2015), and the first-grade MoE Science
Prizes (1992, 1996, and 2009) among many other prizes. He is the Academician
of the Chinese Academy of Sciences, and a fellow of the The World Academy of
Sciences, and the International Academy of Astronautics. He was a Co-Chair of
TPC for IGARSS2011 in Vancouver, BC, Canada, and the Co-General Chair of
IGARSS2016 in Beijing, China. He was the Associate Editor of the IEEE
TRANSACTIONSON GEOSCIENCE AND REMOTE SENSING from 2005 to 2012, the member of
the IEEE GRSS AdCom, and the Chair of the IEEE Fellow Evaluation of GRSS from
2009 to 2011. He is an IEEE GRSS Distinguished Speaker and an Associate Editor
of the IEEE ACCESS.
---|---