Thermospheric Mass Density Modelling during Geomagnetic Quiet and Weakly Disturbed Time

Abstract

Atmospheric drag stands out as the predominant non-gravitational force acting on satellites in Low Earth Orbit (LEO), with altitudes below 2000 km. This drag exhibits a strong dependence on the thermospheric mass density (TMD), a parameter of vital significance in the realms of orbit determination, prediction, collision avoidance, and re-entry forecasting. A multitude of empirical TMD models have been developed, incorporating contemporary data sources, including TMD measurements obtained through onboard accelerometers on LEO satellites. This paper delves into three different TMD modelling techniques, specifically, Fourier series, spherical harmonics, and artificial neural networks (ANNs), during periods of geomagnetic quiescence. The TMD data utilised for modelling and evaluation are derived from three distinct LEO satellites: GOCE (at an altitude of approximately 250 km), CHAMP (around 400 km), and GRACE (around 500 km), spanning the years 2002 to 2013. The consistent utilisation of these TMD data sets allows for a clear performance assessment of the different modelling approaches. Subsequent research will shift its focus to TMD modelling during geomagnetic disturbances, while the present work can serve as a foundation for disentangling TMD variations stemming from geomagnetic activity. Furthermore, this study undertakes precise TMD modelling during geomagnetic quiescence using data obtained from the GRACE (at an altitude of approximately 500 km), CHAMP (around 400 km), and GOCE (roughly 250 km) satellites, covering the period from 2002 to 2013. It employs three distinct methods, namely Fourier analysis, spherical harmonics (SH) analysis, and the artificial neural network (ANN) technique, which are subsequently compared to identify the most suitable methodology for TMD modelling. Additionally, various combinations of time and coordinate representations are scrutinised within the context of TMD modelling. Our results show that the precision of low-order Fourier-based models can be enhanced by up to 10 % through the utilisation of geocentric solar magnetic coordinates. Both the Fourier- and SH-based models exhibit limitations in approximating the vertical gradient of TMD. Conversely, the ANN-based model possesses the capacity to capture vertical TMD variability without manifesting sensitivity to variations in time and coordinate inputs.

1. Introduction

Low Earth Orbit (LEO) is an important altitude region for Earth observation, collision avoidance, and re-entry prediction. However, the increasing number of space objects has posed a tremendous challenge to collision avoidance and re-entry prediction in the LEO environment, especially below the altitude of 600 km. The aerodynamics force is the largest non-gravitational force experienced by LEO spacecraft [1]. Meanwhile, this force is one of the largest error sources for precise orbit determination and orbit prediction, partly due to the temporal and spatial variability of thermospheric mass density (TMD) driven by various factors, including solar, magnetic, and lower atmospheric forcing [2]. It is also called the atmospheric drag since it dominantly acts in the opposite direction of the object’s relative motion with respect to the atmosphere.

The upper thermosphere in the region of 200–600 km is a complicated non-linear system coupled with the ionosphere, troposphere, and magnetosphere. The thermospheric models can be generally categorised into physical models and empirical models. The physical models solve the fluid equation by numerically discretising over the region of interest, e.g., the Thermosphere Ionosphere Electrodynamics General Circulation Model (TIE-GCM) [3,4]. The past two decades have witnessed a resurgence of many new empirical TMD models, e.g., DTM-2013 [5] and JB2008 [6]. These empirical models capture the statistically average dynamics of the thermosphere in a parameterised formulation. Extensive reviews of physical and empirical models can be found in Qian and Solomon [7], Emmert [8], and He et al. [9].

The empirical models have been developed using the accelerometer-derived TMD from the Earth observation satellites in LEO [10,11,12], such as the GRACE (Gravity Recovery and Climate Experiment), CHAMP (Challenging Minisatellite Payload), and GOCE (Gravity field and steady-state Ocean Circulation Explorer) satellites. For instance, Liu et al. [13] applied a linear TMD profile in their model for the altitude range of 350–420 km. This linear assumption cannot be satisfied across the whole thermosphere. Assuming a non-linear TMD profile, an updated version incorporating the solar wind merging electric field was developed by Xiong et al. [14]. Meanwhile, many advanced techniques have been attempted in TMD modelling, such as Fourier analysis [13], spherical harmonics (SH) analysis [15], principal component analysis [16], and artificial neural network (ANN) analysis [17,18].

Although many thermospheric models have been developed, further effort is still needed for the following reasons: (1) It is difficult to consistently evaluate the performance of empirical TMD models because of the hidden biases in empirical models [19], as well as the discrepancies in the reference TMD data sets used to construct the models [9]. Usually, the reference TMD data sets used for modelling are different from those used in the model evaluations. For instance, the high-latitude atmospheric neutral density model developed by Yamazaki et al. [15] using GRACE and CHAMP TMD data sets [11] may not be reliably compared with the JB2008 model, which was developed from a different reference TMD data set provided by Bowman et al. [6]. (2) Due to the previous issue, few studies have been performed to determine the pros and cons of the methodologies used in TMD modelling. (3) No one has investigated the impact of different time and coordinate representations on TMD modelling. Many solar and magnetic coordinates have been proposed to better interpret the measurements of magnetometers and to investigate the thermosphere–ionosphere system [20]. The variability of TMD shows a strong dependence on solar and magnetic activity.

TMD modelling during geomagnetic quiet times is fundamental to TMD prediction. On the one hand, the geomagnetic quiet and weakly disturbed periods covers most of historical time. On the other hand, it is the first step in separating the TMD response to severe space events by removing TMD variations under quiet time. Therefore, the main objective of this study is to develop and comprehensively evaluate empirical TMD models based on Fourier, SH, and ANN analysis. This preliminary study only focuses on TMD modelling during geomagnetic quiet and weakly disturbed time. The structure of the paper is organised as follows. The TMD data sets and preprocessing procedures are first outlined in Section 2. The modelling methodology is documented in Section 3. The analysis and discussion of the results are presented in Section 4, followed by the conclusions and future work in Section 5.

2. TMD Data Sets

The data sets used in the present study covered an altitude of 200–550 km. The accelerometer-derived TMD from the GRACE (2002–2010), CHAMP (2002–2010), and GOCE (2009–2013) satellites were used to develop empirical TMD models in this study. Managed by the German GeoForschungsZentrum (GFZ) Potsdam, the CHAMP satellite was operated at the altitude of ∼450 km. The primary aim of the CHAMP satellite was to investigate the Earth’s gravity and geomagnetic field recovery. As a joint mission between the National Aeronautics and Space Administration (NASA) and the German Aerospace Center (DLR), the GRACE twin-satellite mission was launched into a near-polar and circular orbit with a nominal altitude of 500 km and separation of 220 km. GOCE was launched by the European Space Agency (ESA) into a very low sun-synchronous orbit at ∼280 km. The TMD data sets derived from the CHAMP and GRACE satellites have been made freely available by Mehta et al. [12]. Note that the TMD data of Mehta et al. [12] were re-scaled from those of Sutton [10] using a different gas–surface interaction model. The reader is directed to these references for more details regarding the computation and data processing methods used.

These TMD data sets were first calibrated to account for the hidden biases of 10–20%, as documented by Mehta et al. [12] and He et al. [9]. To determine the most proper indices for further modelling, the space weather indices used in the modelling were also analysed based on their Pearson correlation coefficients with the TMD.

2.1. Data Reprocessing

The discrepancies between different TMD data sets derived from multiple satellites have been noted in many previous studies, e.g., [15,21]. These discrepancies originate from the different handling of neutral wind and the use of different physical drag coefficient models. An inter-calibration procedure of TMD data sets was performed by comparing with the NRLMSISE-00 model to minimise the discrepancies of TMD data sets from different satellites. The NRLMSISE-00 model was used in the calibration due to its capability in physically capturing the annual and semiannual oscillations compared to the post hoc TMD modulation used in other empirical models [22], e.g., JB2008. Consequently, the ACC-TMD $\rho \left(h\right)$ at the altitude of h was scaled by

$$\tilde{\rho }\left(h\right)=〈\frac{{\rho }_m\left(h\right)}{\rho \left(h\right)}〉·\rho \left(h\right),$$

(1)

where ${\rho }_m$ is the TMD output from NRLMSISE-00; ${\rho }_m/\rho$ is the model-to-observation TMD ratio; and $\langle ·\rangle$ denotes the average operator over the years of 2002–2013.

Note that the calibration procedure undertaken here is different from the vertical TMD normalisation in previous studies, e.g., [15,23,24]. Moreover, this procedure may not change the final results since the TMD data during space weather events were identified and removed, as shown in the next section. Also, TMD data were flagged as outliers if the TMD ratio was beyond the range of 3·IQR [9]. As a result, up to 0.2% of the TMD data sets during high solar activity were excluded. The exclusion rate was closer to zero during other times for all satellites. Only TMD data sets that passed this quality control are used hereafter.

The yearly TMD ratios in 2002–2013 for the four satellites are statistically illustrated in Figure 1. Consistent with the results given by He et al. [9], the yearly mean TMD ratios marked by green crosses show a solar-dependent trend, which most likely resulted from the estimation of the drag coefficient. The annual mean values of the TMD ratios for these four satellites (green plus markers) show a strong dependence on the solar activity from 2002 to 2010. The STDs of the TMD ratios in Figure 1b,d were less affected by the solar conditions, which suggests that the STD may be a more appropriate metric of precision in the TMD comparisons.

2.2. Response of TMD to Space Weather Indices

Many different space weather indices have been proposed and used in thermosphere modelling. This study evaluated the correlation between TMD and ∼40 space weather indices. These indices include: (a) hourly solar wind parameters (e.g., inter-planetary magnetic field, solar wind velocity, and solar wind pressure); (b) 3h geomagnetic indices (e.g., $K_p$, $a_p$, $Dst$, and $a_m$); and (c) daily solar indices (e.g., solar flux at the wavelengths of 30, 15, 10.7, 8, and 3.2 cm) [9].

The six solar indices with the highest Pearson correlation coefficient are illustrated in Figure 2. The time lag is the length of days before or after the current epoch of TMD data. The TMD data sets from GRACE-A/B, CHAMP, and GOCE were daily averaged and normalised to the altitude of $400$ km using NRLMSISE-00. The analysis presented here may not be affected by such TMD normalisation during the geomagnetic quiet and weakly disturbed time. Note that all other TMD data sets were scaled to that of GRACE-A in order to remove the hidden discrepancies, as discussed in He et al. [9]. Figure 2 shows that $P_{10.7}=(F_{10.7}+{\overline{F}}_{10.7})/2$ has the best correlation with daily mean TMD.

The $K_p$ index from the NASA (National Aeronautics and Space Administration) OMNI2 (Operating Missions as Nodes on the Internet) data set also shows the highest correlation among geomagnetic indices (Figure 3). The Pearson correlation coefficients for geomagnetic indices are much lower than those of the solar indices for two main reasons. First, geomagnetic activity has a short-term (e.g., hourly) influence on TMD and hence does not correlate well with solar flux activity in the long term. The other reason is the fact that the TMD modelling in this study was performed during geomagnetic quiet and weakly disturbed time without geomagnetic storms and solar flares. The geomagnetic quiet time is hereafter identified by $Dst\ge -50 \mathrm{n}\mathrm{T}$ and $a_p\le 32 \mathrm{n}\mathrm{T}$ (corresponding to $K_p<4+$), with an X-ray flux no larger than 1.0 × 10⁻⁵ W m⁻² (corresponding to X-class and M-class solar flares).

There is difficulty in accurately predicting the aerodynamic force results from the unpredictability and irregularity of thermospheric mass density (TMD), which is excited by many factors, e.g., the upward-propagating solar tides and solar radiation absorption [25]. These responses can be observed in many variables, such as neutral temperature, atmospheric composition, and the strength of the solar and magnetic field.

Current atmospheric models cannot represent the TMD distribution and variability as accurately as required, even though TMD models have adopted one single index or a combination of different indices to represent the effect of solar and geomagnetic activities. At the moment, the accuracy of the empirical models is restricted to around 15–30% of the absolute density value [1,26], and the RMS error can reach up to 100% [27] during extreme space weather conditions, e.g., geomagnetic storms and solar flares. Improving the knowledge of TMD not only helps in understanding thermospheric dynamics, but also benefits the OD and OP required in re-entry prediction and space situational awareness.

3. Modelling Methods

This study developed three different empirical TMD models. Only the $P_{10.7}$ and $K_p$ indices were used from the correlations discussed in Section 2.2. Furthermore, the evaluation of these new empirical models was based on the natural logarithm of TMD, which is more appropriate in the representation of TMD [15]. Note that the model coefficients are represented by $a_i$ for clarity, but their values are different in each model. An equivalent error weight was assumed for the TMD data sets of the four satellites.

In addition to the commonly used geographic coordinates (geodetic latitude and longitude), solar and magnetic coordinates were also used in this study, such as the centred dipole (CD), eccentric dipole (ED), geocentric solar magnetic (GSM), solar magnetic (SM), quasi-dipole (QD), and modified apex (MA) coordinates, to determine the optimal time and coordinate representations in TMD modelling. These are termed as ‘general’ latitudes and longitudes hereafter. Likewise, the hour of day ($HOD$) measured by universal time (UT), local time (LT), local solar time (LST), or magnetic local time (MLT) was investigated in this study. We refer the readers to Laundal and Richmond [28] and He [29] for the detailed definitions of these time and coordinate representations.

3.1. Fourier Analysis

As the TMD derived from the four satellites covering the altitudes of 200–550 km was used in this study, a Fourier-based model $\rho (P_{10.7},K_p,\phi ,\lambda ,h,DOY,HOD)$ was developed to consider different reference heights:

$$ln\rho \left(h_0\right)=ln{\rho }_0+f_1\left(P_{10.7}\right)+f_2\left(K_p\right)+f_3\left(\phi \right)+f_4\left(\lambda \right)+f_5\left(HOD\right)+f_6\left(DOY\right),$$

(2)

where $\phi$ and $\lambda$ are the geographic/solar/geomagnetic latitude and longitude, respectively (see [28]), which are termed as ‘general’ latitudes and longitudes hereafter. $HOD$ is the hour of the day, and $DOY$ is the day of the year, which can be measured by universal time (UT), local time (LT), local solar time (LST), or magnetic local time (MLT), as elaborated in Laundal and Richmond [28]. For instance, the $\phi$, $\lambda$, and $HOD$ used by Xiong et al. [14] were the geographic coordinates and MLT, respectively. h is the altitude, and ${\rho }_0$ is the TMD at the reference altitude of $h_0$. The terms $f_i(i=1-7)$ in the form of trigonometric functions (see Appendix B) represent the vertical, solar activity, geomagnetic, latitudinal, longitudinal, intra-diurnal, and intra-annual variations in TMD, respectively. $h_0$ is the reference heights of 250, 300, 400, 500, and 550 km. The log-TMD at the altitude of h was linearly interpolated from the nearest two reference heights.

This study was performed during geomagnetic quiet and weakly disturbed time, but the $K_p$ index was still used in the model generation to improve the representation of the models. As clarified by Yamazaki et al. [15], $a_p=0$ ($K_p=0$ in this study) was applied in the model evaluation, representing the geomagnetic quiet time. Due to the potential rank deficiency in the design matrix $\mathit{B}$ and the large size of the utilised data records (1.163 × 10⁷), obtaining an accurate and convergent solution heavily relied on the selection of a suitable initial value for the coefficients. Initially, the Levenberg–Marquardt non-linear algorithm [30] was employed to calculate the initial value, expressed as follows:

$$\mathit{a}={\left({\mathit{B}}^{\top }\mathit{B}+\beta {\mathit{I}}_{n_a}\right)}^{-1}{\mathit{B}}^{\top }\mathit{V},$$

(3)

where $\beta$ is a non-negative scalar factor, called the damping factor; ${\mathit{I}}_{n_a}$ is an $n_a\times n_a$ identity matrix; and $n_a$ is the number of model coefficients. As the value of $\beta$ in the aforementioned equation approaches zero, it leads the Levenberg–Marquardt estimation to converge towards the regular least-squares solution. Conversely, a larger value of $\beta$ causes it to approximate the gradient-descent method.

3.2. Spherical Harmonics Analysis

Similar to the Horizontal Wind Model (HWM) [31,32] and other empirical models based on SH analysis, e.g., [33], the HANDY model [15] was developed using SH expansion and split into two parts—a geomagnetic quiet model and a disturbed model. This quiet model was calibrated from NRLMSISE-00 only to an altitude of $400$ km. The TMD model based on SH analysis in this study is slightly different from both the NRLMSISE-00 and HANDY models, as follows:

$$ln\rho =\sum _{i=1}^{n_1}{a}_0\left(i\right)h^i+G\left(P_{10.7},{\overline{P}}_{10.7},K_p,\phi ,\lambda ,DOY,UT,HOD\right),$$

(4)

where $UT$ is UTC hours, and $a_0$ is the constant coefficients of the altitude-dependent variation. The function of $G(P_{10.7},{\overline{P}}_{10.7},K_p,\phi ,\lambda ,DOY,UT,HOD)$ is given in Appendix B.

An altitude-dependent version of the SH-based model was proposed in this study as follows:

$$ln\rho \left(h_0\right)=ln{\rho }_0+G\left(P_{10.7},{\overline{P}}_{10.7},K_p,\phi ,\lambda ,DOY,UT,HOD\right),$$

(5)

where $h_0$ is the reference heights of 250, 300, 400, 500, and 550 km. The log-TMD at the altitude of h was linearly interpolated from the two nearest reference heights. The function of G based on spherical harmonics can be found in Appendix B, and the model coefficients were estimated using the Levenberg–Marquardt algorithm, which was also used in the development of NRLMSISE-00 [34]. The same algorithm was used to consistently estimate the unknown parameters in all models.

The SH-based models in this study were derived from NRLMSISE-00, although some differences exist. Firstly, the SH analysis was directly applied to the number density of each atmospheric composition in NRLMSISE-00. Secondly, NRLMSISE-00 was based on a theoretical profile of atmospheric temperature, which was not taken into account in this study.

3.3. Artificial Neural Networks

The ANN technique has been widely applied to deal with classification, regression, and pattern recognition tasks in Earth science, such as by Chen et al. [35]. A classic ANN consists of three components: an input layer, hidden layers, and an output layer. An ANN model with a large number of hidden layers (depth) may have a better fitting performance and is also known as a deep neural network. This study aimed to compare the classic ANN techniques with the other fitting methods mentioned above; state-of-the-art deep learning models such as the residual network [36] were not considered.

In this study, the training, testing, and validation processes of the deep learning model were crucial for achieving accurate and reliable predictions. The Levenberg–Marquardt algorithm was employed to iteratively estimate the coefficients of the ANN models [37]. The TMD data sets were randomly partitioned into three segments: training (70%), validation (15%), and test (15%) in order to prevent over-training (over-fitting) and under-training (under-fitting). Over-training occurs when the ANN model becomes excessively specialised to the training data and loses its ability to generalise to unseen data. On the other hand, under-training happens when the ANN model fails to capture the underlying relationships due to insufficient network capacity [17]. During the training process, ANN models are fine-tuned using the training data set, which comprises pairs of input vectors and corresponding output vectors. This allows the model to learn the patterns and relationships in the data. Meanwhile, a validation data set is used to optimise the hyper-parameters of the deep learning model. Techniques such as early stopping, based on the performance on the validation data set, can be employed to prevent over-fitting and enhance generalisation. The test data set, which is separate from the training and validation sets, is used to evaluate the final performance of the trained deep learning model. By assessing the model’s accuracy and predictive capability on unseen data, it provides an unbiased assessment of the model’s performance in real-world scenarios. The proper partitioning of the data into training, validation, and test sets, along with the effective use of regularisation techniques, ensures that the deep learning model achieves reliable and robust predictions. The ANN algorithm was implemented in the built-in MATLAB Deep Learning Toolbox using the default threshold and settings.

In the present research, a single-hidden-layer model, ANN-1, with different numbers of neurons ranging from 5 to 100 was first built and examined in order to find the optimal structure of the ANN, as listed in

Table 1. Details of the Fourier-, SH-, and ANN-based models. ANN-1 and ANN-2 are single-hidden-layer ANN models, and ANN-3 has three hidden layers. The number of neurons in the hidden layer(s) is given in the fifth column.

Model	Inputs	No. Inputs	No. Coefficients	No. Neurons
Fourier	$P_{10.7}$, $K_p$, h, $\phi$, $\lambda$, $HOD$, $DOY$	7	170
SH-1	$P_{10.7}$, $K_p$, h, $\phi$, $\lambda$, $HOD$, $DOY$	7	102
SH-2	$P_{10.7}$, $K_p$, h, $\phi$, $\lambda$, $HOD$, $DOY$	7	300
ANN-1	$P_{10.7}$,$K_p$, h, $sin2\phi$, $cos2\phi$, $sin\lambda$, $cos\lambda$, $sin\left(\frac{2\pi }{24}HOD\right)$, $cos\left(\frac{2\pi }{24}HOD\right)$, $sin\left(\frac{2\pi }{365}DOY\right)$, $cos\left(\frac{2\pi }{365}DOY\right)$	11	66	5
			131	10
			261	20
			651	50
			1301	100
ANN-2	$P_{10.7}$, $K_p$, h, $sin2m\phi$, $cos2m\phi$ ($m=1-6$), $sinn\lambda$, $cosn\lambda$ ($n=1-4$), $sin\left(\frac{2\pi k}{24}HOD\right)$, $cos\left(\frac{2\pi k}{24}HOD\right)$ ($k=1-3$), $sin\left(\frac{2\pi d}{365}DOY\right)$, $cos\left(\frac{2\pi d}{365}DOY\right)$ ($d=1-2$)	33	176	5
			351	10
			701	20
			1751	50
			3501	100
ANN-3	$P_{10.7}$, $K_p$, h, $sin2\phi$, $cos2\phi$, $sin\lambda$, $cos\lambda$, $sin\left(\frac{2\pi }{24}HOD\right)$, $cos\left(\frac{2\pi }{24}HOD\right)$, $sin\left(\frac{2\pi }{365}DOY\right)$, $cos\left(\frac{2\pi }{365}DOY\right)$	11	291	10-10-5
			506	15-15-5
			591	15-15-10
			881	20-20-10
			991	20-20-15

. Moreover, the harmonics of latitude and longitude with different orders, as used in the Fourier-based model, were also assessed in the modelling of the ANN-2 model. For the purpose of simplifying the ANN structure, a deep ANN model with the classic three hidden layers was adopted and labelled as ANN-3. The Pearson correlation coefficient and principal component analysis were used in the preprocessing of the ANN models to examine the statistical inter-dependence of the inputs listed in Table 1.

Due to the considerable computational burden of ANN modelling, only the combinations of GEO-LST, GSM-LT, and QD-LST were examined. This selection was determined by considering the statistical results of the two previous types of models. Only $P_{10.7}$ and $K_p$ were used as the space weather indices in these three ANN-based models. Again, $K_p=0$ was evaluated in the model comparison, representing the quiet geomagnetic condition.

The inputs of the ANN-based TMD model for storm time developed by Chen et al. [18] include the altitude h for the vertical variation of TMD, geographic latitude $\phi$ and longitude $\lambda$ for the spatial variations, $cos\phi$ for the latitudinal symmetry in the global distribution of TMD, $cos\left(\frac{2\pi }{24}HOD\right)$ for the diurnal TMD variation, and $cos\left[\frac{2\pi }{365}(DOY-81)\right]$ for the seasonal TMD variation. However, discontinuity in the longitudinal TMD distribution can be found around the longitudes of $\pm 180{}^{\circ }$ due to the range of longitudes adopted in this study ($-180{}^{\circ }\le \lambda \le +180{}^{\circ }$) and the periodic nature of longitude. A feasible solution is extending the longitude range to, e.g., [$-240{}^{\circ }$,$+240{}^{\circ }$], but this cannot perfectly solve the discontinuity problem. Therefore, sine and cosine functions of general latitudes and longitudes were input into the ANN-based models. Table 1 gives the detailed inputs, outputs, and structural of the three ANN-based models investigated in this study.

4. Results and Discussion

In this study, the model performances discussed previously were evaluated using three metrics. First, the a posteriori STD of unit weight (${\widehat{\sigma }}_0$) was considered, which can be calculated from

$${\widehat{\sigma }}_0=\sqrt{\frac{{\mathit{V}}^{\top }\mathit{V}}{n-n_a}}.$$

(6)

where $\mathit{V}$ is the observed-minus-predicted log TMD residuals, n is the number of observations, and $n_a$ is the number of model coefficients.

TMD models with different orders have different numbers of model coefficients. Generally, the models with more coefficients are expected to have better performance; hence, it is not wise to evaluate models with different coefficients just by the traditional metrics such as the mean, STD, and root-mean-square (RMS) error. The ${\widehat{\sigma }}_0$ used in this study approaches the RMS as $n\gg n_a$ and includes both systematic biases and random errors. From the equation of ${\widehat{\sigma }}_0$, the model degrees of freedom (the number of model coefficients $n-a$) were considered.

The other two selected metrics used in the model evaluation include the mean TMD ratio, represented as

$$\overline{r}=〈\frac{\rho }{\tilde{\rho }}〉,$$

(7)

and the Pearson correlation coefficient, represented as

$$R=\frac{{\sigma }_{\rho ,\tilde{\rho }}^2}{{\sigma }_{\rho }·{\sigma }_{\tilde{\rho }}},$$

(8)

where $\rho$ and $\tilde{\rho }$ are the prediction and observation of TMD, respectively. The mean TMD ratio is a classic model bias metric.

4.1. Fourier-Based Model

The Fourier model was designed to better capture the vertical variability of TMD by applying altitude-dependent modelling at 250, 300, 400, 500, and $550$ km. The fitting results ${\widehat{\sigma }}_0$ for selected coordinate–time frames are presented in

Table 2. The a posteriori STD of unit weight (${\widehat{\sigma }}_0$ of the Fourier model for the four satellites in the unit of $ln(\mathrm{k}\mathrm{g} {\mathrm{m}}^{-3})$. The model was built at five reference heights of 250, 300, 400, 500, and $550$ km. For simplicity, only the results of five coordinate and time frames are presented.

Coordinate–Time	GRACE-A	GRACE-B	CHAMP	GOCE	Overall
GEO-LST	0.3338	0.3712	0.2890	0.1354	0.3099
GEO-MLT	0.3410	0.3765	0.2842	0.1465	0.3133
GSM-LT	0.3030	0.3390	0.2353	0.1140	0.2742
GSM-LST	0.3011	0.3353	0.2260	0.1193	0.2704
QD-LST	0.3331	0.3705	0.2891	0.1363	0.3095

. As the magnetic coordinate frame (QD, MA, etc.) showed almost the same performance, only the result of QD is presented in this table. The GSM-LT coordinate combination provided the best fitting result with time frames of LT or LST. A 10% improvement in ${\widehat{\sigma }}_0$ in comparison with other combinations of time and coordinate frames can be observed.

Figure 4 presents a scatter plot between the Fourier model predictions and the TMD observations for the GRACE, CHAMP, and GOCE satellites. The lines in red depict the linear regression result. The figure illustrates a consistent slope of regression lines for the four satellites compared to an altitude-dependent trend observed from the Fourier model based on a linear or quadratic polynomial of altitude (not shown here). This result may indicate that the vertical gradient of TMD across more than one scale height range (>100 km) cannot be well approximated by the polynomials.

The usage of the GSM coordinates significantly changed the longitudinal and HOD variations because the x-axis in the GSM coordinate system points to the Sun. The global maximum of the TMD will therefore appear around 0° of the GSM longitude, indicating that the GSM longitude is correlated with HOD (e.g., LT used here). Hence, the selection of the orders was considered reasonable for the Fourier-1 model. Lower orders $(1,1,1,4,6,2,2)$ were considered in the Fourier model to avoid multiplying the number of model coefficients and over-fitting the results.

4.2. SH-Based Models

The two schemes of the SH-based model are summarised in Table 1. The input UT used in Equations (4) and (5) was implemented by different time systems, such as LT and LST. The fitting results of the SH model for all satellites are given in

Table 3. The a posteriori STD of unit weight (${\widehat{\sigma }}_0$, unit: $ln(\mathrm{k}\mathrm{g} {\mathrm{m}}^{-3})$) of the SH-1 model. Values in bold show the best fitting result. MLT for GEO, GSM, SM, and DIP frames was derived from the QD longitude.

Coordinates	Time
	LT	LST	MLT
GEO	0.3147	0.3146	0.3173
GSM	0.3102	0.3103	0.3110
SM	0.3185	0.3184	0.3173
DIP	0.3242	0.3241	0.3274
CD	0.3162	0.3162	0.3162
ED	0.3169	0.3168	0.3165
QD	0.3174	0.3173	0.3165
MA	0.3174	0.3173	0.3165

and

Table 4. The a posteriori STD of unit weight (${\widehat{\sigma }}_0$, unit: $ln(\mathrm{k}\mathrm{g} {\mathrm{m}}^{-3})$) of the SH-2 model. The model was built at five reference heights of 200, 300, and $500$ km.

Coordinate–Time	GRACE-A	GRACE-B	CHAMP	GOCE	Overall
GEO-LT	0.2716	0.3085	0.2261	0.1592	0.2557
GEO-LST	0.2711	0.3079	0.2248	0.1630	0.2554
GEO-MLT	0.2745	0.3114	0.2264	0.1639	0.2580
GSM-LT	0.2677	0.3032	0.2211	0.1580	0.2514
GSM-LST	0.2678	0.3033	0.2211	0.1583	0.2515
QD-LST	0.2762	0.3130	0.2287	0.1614	0.2594

. The GSM coordinates showed the best fitting result again, and QD and MA had almost the same performance. In comparison with the improvement of nearly 10% for Fourier-1 (39 coefficients) when using the GSM coordinates, the SH-1 model (102 coefficients) showed a statistical improvement of only 1%. This implies that the consideration of GSM in TMD modelling becomes less important if a sufficiently complex model (based on the number of model coefficients) is used.

Note that it is not wise to directly compare the NRLMSISE-00 with newly developed empirical TMD models without re-estimating the model scales. In particular, the same or calibrated reference TMD data must be simultaneously used in the model generation and evaluation. Otherwise, the statistical result may be affected by the biases of the empirical models. Additionally, the SH-based model generated in this study can only represent the performance of the SH methodology rather than that of NRLMSISE-00 itself. This is because many features of NRLMSISE-00 were not adopted in this study, e.g., the vertical variation in TMD was based on the profile of atmospheric temperature in NRLMSISE-00.

4.3. ANN-Based Models

For the ANN-1 model with a lower number of neurons, GSM-LT still showed an improvement of nearly 8%. However, ANN-1 models with different time and coordinate combinations reached a similar precision as the neuron number approached 100. This result shows that the ANN-based model with sufficient numbers of neurons can adapt to the different mappings associated with GEO and GSM/QD coordinates and the time transformation. In other words, an ANN model with sufficient neurons in the hidden layer may be less sensitive to the inputs of the time and coordinates. In addition, the residuals of ANN-1 follow an approximated Gaussian distribution. An example for QD-LST is illustrated in Figure 5, and an obvious improvement can be observed, particularly for the GOCE satellite at the lowest altitude in the bottom panels.

Table 5. The a posteriori STD of unit weight (${\widehat{\sigma }}_0$, unit: $ln(\mathrm{k}\mathrm{g} {\mathrm{m}}^{-3})$); mean TMD ratio ($\overline{\rho }$); and Pearson correlation coefficient (R) of ANN-1 model.

Coordinate–Time	Neurons in Hidden Layer	${\widehat{\mathit{\sigma }}}_0$	$\overline{\mathit{r}}$	R
GEO-LST	5	0.2718	1.0392	0.9881
	10	0.2419	1.0307	0.9882
	20	0.2255	1.0267	0.9927
	50	0.2085	1.0226	0.9948
	100	0.2019	1.0211	0.9954
GSM-LT	5	0.2501	1.0332	0.9922
	10	0.2442	1.0315	0.9878
	20	0.2234	1.0283	0.9943
	50	0.2117	1.0233	0.9949
	100	0.2053	1.0218	0.9952
QD-LST	5	0.2656	1.0374	0.9882
	10	0.2393	1.0302	0.9902
	20	0.2252	1.0264	0.9936
	50	0.2111	1.0256	0.9946
	100	0.2028	1.0226	0.9952

and

Table 6. The a posteriori STD of unit weight (${\widehat{\sigma }}_0$, unit: $ln(\mathrm{k}\mathrm{g} {\mathrm{m}}^{-3})$); mean TMD ratio ($\overline{\rho }$); and Pearson correlation coefficient (R) of ANN-2 model.

Coordinate–Time	Neurons in Hidden Layers	${\widehat{\mathit{\sigma }}}_0$	$\overline{\mathit{r}}$	R
GEO-LST	5	0.2426	1.0309	0.9918
	10	0.2202	1.0251	0.9931
	20	0.2117	1.0229	0.9948
	50	0.2027	1.0213	0.9952
	100	0.1920	1.0199	0.9955
GSM-LT	5	0.2391	1.0302	0.9906
	10	0.2216	1.0259	0.9931
	20	0.2125	1.0234	0.9946
	50	0.2013	1.0208	0.9951
	100	0.1904	1.0189	0.9950
QD-LST	5	0.2447	1.0316	0.9911
	10	0.2392	1.0307	0.9898
	20	0.2143	1.0241	0.9948
	50	0.2006	1.0207	0.9951
	100	0.1959	1.0199	0.9950

give the fitting results of the ANN-1 and ANN-2 models, respectively, with different inputs of time and coordinate frames. An improvement of 3–10% can be found for ANN-2. Similar results for the three-hidden-layer ANN-3 model are presented in

Table 7. The a posteriori STD of unit weight (${\widehat{\sigma }}_0$, unit: $ln(\mathrm{k}\mathrm{g} {\mathrm{m}}^{-3})$); mean TMD ratio ($\overline{\rho }$); and Pearson correlation coefficient (R) of ANN-3 model.

Coordinate–Time	Neurons in Hidden Layers	${\widehat{\mathit{\sigma }}}_0$	$\overline{\mathit{r}}$	R
GEO-LST	10-10-5	0.2159	1.0242	0.9949
	15-15-5	0.2077	1.0224	0.9951
	15-15-10	0.2047	1.0218	0.9953
	20-20-10	0.2032	1.0217	0.9956
	20-20-15	0.2032	1.0208	0.9954
GEO-MLT	10-10-5	0.2216	1.0255	0.9950
	15-15-5	0.2114	1.0238	0.9953
	15-15-10	0.2106	1.0230	0.9953
	20-20-10	0.2049	1.0239	0.9955
	20-20-15	0.2032	1.0212	0.9957
GSM-LST	10-10-5	0.2199	1.0253	0.9942
	15-15-5	0.2100	1.0229	0.9952
	15-15-10	0.2087	1.0227	0.9951
	20-20-10	0.2038	1.0216	0.9953
	20-20-15	0.2032	1.0202	0.9954
QD-LST	10-10-5	0.2166	1.0235	0.9943
	15-15-5	0.2084	1.0221	0.9951
	15-15-10	0.2076	1.0240	0.9952
	20-20-10	0.2041	1.0221	0.9954
	20-20-15	0.2032	1.0219	0.9953

. The ANN-3 model performed the best among the three types of ANN-based model given the equivalent number of model coefficients, e.g., ANN-1(50), ANN-2(20), and ANN-3(20-20-10)—the numbers inside the parentheses indicate the number of neurons in the hidden layer(s). The QD latitude versus LST distribution of the three ANN-based models is given in Figure 6. A two-cell equatorial mass density anomaly (EMA) structure, i.e., TMD peaks at mid-latitudes and a trough near the equator, was observed by the CHAMP mission [13].

The crests in equatorial mass density anomaly (EMA), a two-cell structure [13], were reproduced at the QD latitude of 30° during solar maximum, as expected in Figure 6a,c,e. Similarly, an equatorial maximum in TMD near midnight can be observed during solar minimum in panels (b), (d), and (f). This indicates that the ANN-1, ANN-2, and ANN-3 models can effectively capture the EMA and MDM. However, it should be noted that the peaks in TMD around noon during the solar minimum period (right column) display latitudinally asymmetric distributions for the ANN-1 and ANN-2 models. Furthermore, there is a noticeable difference exceeding 15% among the three models during solar minimum in the right column of Figure 6. This discrepancy can be attributed to the limited learning capacity of the shallow neural networks employed by the ANN-1 and ANN-2 models. Therefore, only the ANN-3 model was considered in the following evaluation due to its better performance, robustness, and stability in the process of training.

Figure 7 shows comparisons of the Fourier-3 (GSM-LT), SH-1 (QD-LST), and ANN-3 (QD-LST) models with the reference TMD data sets over 2002–2013. The ANN-based model had the highest precision and lowest bias among the three example empirical models. Since dominant variations in TMD are attributed to solar activity [15,38], the performance of the models showed a strong dependency on the solar activity. In this research, the ${\widehat{\sigma }}_0$ for all models was larger during low solar activity (around 2007) than during high solar activity. The spikes in ${\widehat{\sigma }}_0$ at the beginning of 2002 and 2012 were caused by the data gap during these years. Interestingly, the ANN-based model did not appear to be affected by this and still showed good consistency.

5. Conclusions

This study delved into the modelling of thermospheric mass density (TMD) during geomagnetic quiet and weakly disturbed time, employing accelerometer-derived TMD observations from four Low Earth Orbit (LEO) satellites: GRACE (∼500 km), CHAMP (∼400 km), and GOCE (∼250 km), spanning the years 2002 to 2013. Various temporal and coordinate representations were scrutinised within the context of empirical TMD models. Furthermore, we explored three modelling methods, namely Fourier analysis, spherical harmonics (SH) analysis, and ANN techniques, to assess their suitability for global TMD modelling. A comprehensive evaluation and validation of these models was conducted to ascertain the optimal method for TMD modelling. The result showed the superiority of the Geocentric Solar Magnetic (GSM) coordinate system in the empirical TMD models. Both Fourier- and SH-based models exhibited limitations in approximating the vertical gradient of TMD. In contrast, the ANN-based model demonstrated a capacity to capture vertical TMD variability without exhibiting sensitivity to the input parameters of time and coordinates.

A significant challenge in TMD modelling lies in accurately approximating the vertical TMD gradient. Two potential solutions emerge: altitude-dependent modelling, which entails constructing multiple models at distinct reference heights, and ANN modelling. The results of this research indicate that the ANN-based models outperformed the Fourier- and SH-based models on three fronts: (1) the ANN-based models adeptly captured vertical TMD variability without necessitating explicit altitude dependency; (2) a sufficiently complex ANN structure, marked by an adequate number of hidden layers and neurons, proved insensitive to variations in time and coordinate frames; (3) the ANN-based models attained higher accuracy when compared to the other two model types.

Furthermore, given that TMD measurements from LEO satellites are obtainable solely along their trajectories, certain spatial features of TMD, such as equatorial mass anomalies (EMAs), are challenging to capture using satellite drag-derived TMD exclusively. In future research, the pursuit of improved metrics may be necessary to address the limitations of existing TMD modelling techniques, particularly when dealing with LEO-satellite-derived measurements.