Introduction

The ionosphere plays a significant role in affecting the propagation of radio signals by causing refraction and delay, thereby limiting the precision of Global Navigation Satellite Systems (GNSS) positioning (Jakowski et al. 2008; Warnant et al. 2009; Grewal et al. 2020). To enable high precision GNSS, it is vital to correct for the effect of the ionosphere. One widely adopted method is the utilization of dual-frequency GNSS measurements. By simultaneously receiving GNSS signals from two different frequencies, it becomes possible to estimate and compensate for the ionospheric delay, leading to improved precision in GNSS positioning and navigation applications. However, for single frequency users, the ionospheric correction is typically derived using ionospheric models and algorithms. These models incorporate data from multiple GNSS stations, either on a global or regional scale, to estimate and predict the ionospheric delay at specific locations and times. Although not as accurate as dual-frequency measurements, these corrections still provide valuable information for mitigating ionospheric effects and enhancing positioning accuracy for single frequency GNSS users. Without ionospheric correction, the ionospheric delay can cause positioning errors of about five meters or more (Subirana et al. 2013).

Global ionospheric models (GIMs) are maps that provide estimates of the ionospheric delays for any given location on the Earth's surface. GIMs are useful for precise positioning applications that require ionospheric correction over a wide area. Generating a GIM involves collecting GNSS data from a global network of stations, which is then used to estimate the total electron content (TEC) of the ionosphere. Subsequently, the TEC values are interpolated to create an electron density map (Yang et al. 2021) which provides estimates of the ionospheric delay (or corrections) for any given location on the Earth's surface.

Regional or local ionospheric models can also be generated to support GNSS precise positioning applications that require ionospheric corrections over a smaller area. These models provide a higher level of spatial resolution compared to the GIMs, thereby improving the accuracy of ionospheric correction for local applications. Local ionospheric models enable fast and accurate precise point positioning (PPP) (Li et al. 2011, 2022; Banville et al. 2014; Dao et al. 2022). The accuracy of ionospheric delay estimates obtained from local models relies on the number and spatial density of GNSS receivers in the network. Regional or local modelling can be achieved through various techniques, including the function fitting methods such as polynomial functions or grid-based approaches (Gao and Liu 2002; Tao and Jan 2016; Banville et al. 2022).

The accuracy of the ionospheric corrections in the low latitude region presents significant variability, primarily due to the highly dynamic nature of the ionosphere. The rapid changes in electron density in a short distance may cause large differences in ionospheric delays through the network (Datta-Barua et al. 2010; Rungraengwajiake et al. 2015; Budtho et al. 2018). This could bring potential errors in the ionospheric correction model (Parkinson and Spilker 1996; Teunissen et al. 2013; Rovira-Garcia et al. 2020). A robust Kalman filter can be applied to detect, identify, and adapt measured errors at individual stations (Zhang et al. 2015; Lotfy et al. 2022). Our study primarily focuses on the detection of outliers within a bilinear model through all stations in the local network.

Various techniques are available for outlier detection. The most common methods involve utilizing statistical parameters such as standard deviation (SD), median absolute deviation (MAD), and interquartile range (IQR) (Yang et al. 2019). The SD threshold technique is well-suited for data that follow a normal distribution, while MAD is more applicable for heavy-tailed distributions such as the Laplace distribution (Lu and Chang 2022). While the empirical rule in the normal distribution is commonly used in practical statistics, the corresponding rule for the Laplace distribution, which is more representative of the ionospheric errors (Marini‐Pereira et al. 2020) and Ma et al. (2022), has not been previously discussed. We here present a new method for outlier detection of ionospheric correction in local models utilizing an empirical calculation to establish thresholds for residuals. Our goal is to enhance the accuracy of ionospheric models by detecting and removing outliers that may affect their reliability and robustness.

Methodology

Observational data

To assess the accuracy of the ionospheric models in low-latitude regions in Australia, we investigated GNSS data of Northern Territory (NT) network. In addition, for comparative analysis, we examined a similar configuration in mid-latitude regions located in Western Australia (WA). Figure 1 shows a map of the GNSS stations installed across Australia, with the two areas investigated in this study defined by the red boxes. The ionospheric delays were computed from GPS data only using the Ginan software.

Fig. 1
figure 1

Map of GNSS receivers (blue stars) over Australia, the red rectangles are locations of two local network investigated in this study

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Ginan software

Ginan is a GNSS analysis centre software developed by Geoscience Australia aimed at providing a range of real-time corrections to enhance GNSS positioning accuracy. It is a critical component of the Australian Government’s National Positioning Infrastructure Capability (NPIC). The Ginan software toolkit and service offer a range of correction products that enable PPP for users and industries across Australia. Ginan is regularly updated with new features and capabilities. This is an open-source software which is accessible through the Geoscience Australia’s GitHub page (Ginan 2023).

We computed the ionospheric delay for each satellite and receiver path using the PPP solution in the Ginan processing software which has been covered in the paper Dao et al. (2022). The stochastic model used to calculate the slant TEC is briefly presented in Table 1. The ionospheric module computes the ionospheric delay (as the slant total electron content—STEC) for each receiver-satellite path for the chosen input stations. Epoch intervals were set at 30 s, and the thin shell ionosphere height was set at 350 km above the Earth's surface. The output ionospheric delay (slant TEC or STEC) was then used for mapping the regional ionospheric model. The unit of the ionospheric delay is TECU, where one TECU approximately corresponds to a delay of 16 cm on the GPS L1 signal frequency.

Table 1 The stochastic model used to calculate the slant TEC in Ginan
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Ionospheric delay outputs from Ginan

For further analysis, we selected the STEC values with a variance below 0.3 TECU2, corresponding to an error of 0.5 TECU. Figure 2 (left) shows the STEC and its variance obtained from all stations within the NT network on July 01, 2021. To remove the receiver hardware bias in each measurement, the between-satellite single difference ionospheric delay (SD STEC) was computed for each epoch. This involves subtracting the measurement from each visible satellite with respect to a chosen reference satellite. The reference satellite is the most commonly visible satellite within the local region, if more than one satellite occurs with the highest common, we select the 1st one in order. Similarly, the single difference of STEC variance was obtained by aggregating the variance from each visible satellite with respect to the reference satellite. The SD STEC corresponds with each satellite-receiver path, and contains the different satellite biases of that satellite and the reference satellite.

Fig. 2
figure 2

Ionospheric delays (STEC) and its variance output from the Ginan software (left panels), single difference STEC (SD STEC) and its variance calculated for each satellite at each epoch (right panels) on July 01, 2021

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Figure 2 (right) displays the SD STEC and its corresponding variance, which aligns with the data shown in the left figure. They are used for creating the map across the local network to facilitate further analysis. It is important to note that the calculation of SD STEC relies on the choice of the reference satellite. Accordingly, as the reference satellite changes, the values of the SD STEC also change, resulting in discontinuities within the data (top left).

The processing steps involved in local ionospheric corrections mapping are depicted in the flowchart in Fig. 3. First, the STEC output for all stations was computed by the Ginan software. Subsequently, the SD STEC of each measurement in the network was calculated to eliminate the receiver hardware bias. The SD STEC was then used to construct the local ionospheric corrections map and compute residuals for outlier detection and identification. In this stage, we applied statistical thresholds to identify and remove outliers from the data, resulting in a refined dataset that was used to generate the local ionospheric map, providing estimated corrections for users. The final step involved evaluating the accuracy of the local mapping approach.

Fig. 3
figure 3

Flow chart for the data processing and outlier detection in local mapping

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Ionospheric correction model using the bilinear regression function.

We retrieve the ionospheric bilinear regression model for a local network as follow:

The distance matrix (HSD) is built from n station coordinates in the network that link to the satellite, with (lat, lon) is the latitude and longitude of each station,

$$H_{SD} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 1 & {lat_{sta1} } & {lon_{sta1} } \\ 1 & {lat_{sta2} } & {lon_{sta2} } \\ \end{array} } \\ {\begin{array}{*{20}c} { \ldots } & \ldots & \ldots \\ { \ldots } & \ldots & \ldots \\ { 1} & {lat_{sta\_n} } & {lon_{sta\_n} } \\ \end{array} } \\ \end{array} } \right]$$
(1)

Using least-squares estimation, the linear interpolation coefficients (ionospheric delay, latitude, longitude) for the bilinear interpolation at each epoch are then computed as:

$$\left[ {\begin{array}{*{20}c} {I_{0} } \\ {I_{lat} } \\ {I_{lon} } \\ \end{array} } \right]{ } = \left( {H_{SD}^{T} Q_{SD} H_{SD} } \right)^{ - 1} H_{SD}^{T} Q_{SD} I_{SD}$$
(2)

where the weight matrix (QSD) is calculated from SD STEC variance:

$$Q_{SD} = diag(var_{SD} )^{ - 1}$$
(3)

The ionospheric delay (ISD) from the network is used to build the model:

$$I_{SD} = \left[ {\begin{array}{*{20}c} {I_{SD,sta1} } \\ {I_{SD,sta2} } \\ {\begin{array}{*{20}c} \ldots \\ \ldots \\ {I_{SD,sta\_n} } \\ \end{array} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 1 & {lat_{sta1} } & {lon_{sta1} } \\ 1 & {lat_{sta2} } & {lon_{sta2} } \\ \end{array} } \\ {\begin{array}{*{20}c} { \ldots } & \ldots & \ldots \\ { \ldots } & \ldots & \ldots \\ { 1} & {lat_{sta\_n} } & {lon_{sta\_n} } \\ \end{array} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {I_{0} } \\ {I_{lat} } \\ {I_{lon} } \\ \end{array} } \right]$$
(4)

Using the bilinear regression function mapping, the SD STEC at any location can be predicted as:

$$I_{{pre_{SD} }} = \left[ {\begin{array}{*{20}c} 1 & {lat_{pre} } & {lon_{pre} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {I_{0} } \\ {I_{lat} } \\ {I_{lon} } \\ \end{array} } \right]$$
(6)

The model residual is computed by the difference of SD STEC from model and observed from stations in the network used to build the model:

$$res = I_{{pre_{SD} }} - I_{{obs_{SD} }}$$
(5)

The model accuracy is computed by the difference of SD STEC from model and observed from the testing stations that not used to build the model:

$$acc = |I_{{pre_{SD} }} - I_{{test_{SD} }} |$$
(7)

Statistical study of modelling residual for outlier detection

Modelling residuals

The SD STEC obtained from the local NT network is utilised. Figure 4 (left) shows the distribution of the GNSS station network in the NT region, where the red stars represent the stations utilised to construct the ionospheric map. The Darwin (DARW) station, indicated by a blue star, is not included in generation of the map. However, it can be utilized later to evaluate the map’s performance. On the other hand, Fig. 4 (right) provides an illustration for a local map of the ith satellite over the NT region. For each satellite, a local map is modelled at every epoch using a single layer ionospheric model mapped at a height of 350 km above the Earth's surface.

Fig. 4
figure 4

Location of the testing station Darwin (DARW) and the remaining stations (red stars) in the local map (left), the illustration of ionospheric mapping for a satellite Si (right)

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In this examination, we constructed a local ionospheric map of a satellite using data from a minimum of five stations automatically. Therefore, in situations where the number of stations receiving data from a satellite falls below five, no ionospheric map is available for that satellite. This results in an “unavailable satellite” or “unavailable local map”. If there are no local ionospheric correction maps available during a specific time, there will be no corrections provided to users.

The local model runs over the area for a 5-min interval, enabling faster processing while maintaining appropriate adaptation to the changes in the ionosphere. To detect outliers, we first create the local map from the observational ionospheric delays from the network stations. Then, we calculate the difference between the map and the observed data at all stations within the network. This is the residuals of the model. By analysing the distribution of these residuals, an appropriate threshold for identifying outliers can be determined.

In this study, we collected the residuals data statistically in four months: January, April, July, and October in 2021. These months were characterized by quiet conditions, free from major geomagnetic storms or disturbances. Figure 5 illustrates the distribution of the residuals observed in the local ionospheric model. Notably, the residuals exhibit a Laplace distribution which is represented by the probability density function (red lines). The Laplace distribution is a double exponential distribution, centred in the middle and has fatter tails compared to the normal distribution.

Fig. 5
figure 5

Distributions of the ionospheric residuals in logarithm scale processed in four months in 2021. The red lines are the probability density function of Laplace distribution within -5 to 5 TECU of residuals

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Empirical rule for Laplace distribution.

The empirical rule, also known as the “68–95-99.7” rule or the “three-sigma” rule, is a statistical guideline for a normal distribution. This rule was originally introduced and discovered by Abraham de Moivre in 1733 (Glen 2023). It is predicted that 68% of observations will fall within the first standard deviation (µ ± σ); 95% within the first two standard deviations (µ ± 2σ); and 99.7% within the first three standard deviations (µ ± 3σ).

The Laplace distribution has heavier tails than the normal distribution. Therefore, the empirical rule does not directly apply to the Laplace distribution. We calculate the percentile function of Laplace distribution as follows.

The function of the Laplace distribution is:

$$f\left( {x{|}\mu ,b} \right) = \frac{1}{2b}{\text{exp}}\left( { - \frac{{\left| {x - \mu } \right|}}{b}} \right)$$
(8)

where μ is the location parameter which is the median of data; b is scale parameter, sometimes referred to as the "diversity”. In this distribution, b is the medium absolute deviation (MAD), which is calculated as:

$$b = MAD = median\left( {\left| {x_{i} - median\left( x \right)} \right|} \right)$$
(9)

The percentile function of Laplace distribution is: \(\mu + bln\left( {2F} \right)\) for \({\text{F}} \le \frac{1}{2}\).

  • \({\text{Percentile function 2F }} = { 5}\% :{\text{ the quantile is}}\;bln\left( {0.05} \right) = \mu + 3b\)

  • \({\text{Percentile function 2F }} = { 3}\% :{\text{ the quantile is}}\;bln\left( {0.03} \right) = \mu + 3.5b\)

  • \({\text{Percentile function 2F }} = \, 0.{3}\% :{\text{ the quantile is}}\;bln\left( {0.003} \right) = \mu + 5.8b\)

Based on this calculation, the thresholds of μ ± 3b, μ ± 3.5b, and μ ± 5.8b are used to remove outliers and ensure that at least 95%, 97%, or 99.7% of the data falls within the thresholds, respectively. For brevity, we will refer to these thresholds as 3b, 3.5b, and 5.8b.

In practical terms, the ionosphere exhibits daily variations, with electron density levels typically higher during the day and lower at night. To account for these variations, we collected data for each hour throughout the day to ensure comprehensive coverage. Figure 6 shows the distribution of residuals for each hour over 30 days in January 2021. This process was repeated for the remaining three months in 2021, representing the four seasons, in the region. The scale parameter, represented as b, of residuals varies over time. As shown in Fig. 6, the shapes of the distributions change throughout the day, with varying b values observed in each subfigure. The vertical dash lines in red, cyan, and black represent the corresponding thresholds of 3b, 3.5b and 5.8b, respectively. For further analysis, we selected the 3.5b threshold, for the retention of approximately 97% of the data, to construct the ionospheric models. By selecting this threshold, we attempted to strike a balance between retaining a substantial amount of data for the ionospheric model yet mitigating the impact of outliers on the model accuracy.

Fig. 6
figure 6

The distribution of residuals for each hour in January 2021 which shows the changing of distribution sharp over time. The red lines are the probability density function of Laplace distribution for each hour in GPST (GPST: GPS time is based on atomic clocks and has no correction with respect to proper time variations of the Earth's rotation. In 2021, UTs = GPST + 18 s). The time leap between GPST and UTs in 2021 is 18 s. The Local Time in this area is LTs = UTs + 9:30

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We computed the hourly variation of the scale b and median μ for the different seasons (January, April, July, and November) in 2021, shown in Fig. 7. Outlier detection was performed based on the changes of μ and b. Moreover, the statistical results in this method can be used to set up a relative threshold for detecting outliners in NT network during similar low solar activity periods.

Fig. 7
figure 7

The 1-h variations of median μ and scale b taken for 30 days of four months (Jan, Apr, Jul, and Nov) in 2021 mapping in NT network

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Local model assessment

Accuracy and availability of regional model

We applied the statistical data of μ and b for each month to detect outliers. In Fig. 4, the DARW station represented by a blue star was intentionally excluded in the mapping process. This station was reserved for evaluating the accuracy of the local model both before and after removal of outliers. Figure 8 shows the accuracy of the local model on July 01, 2021, comparing the original data with the data obtained after applying the designated thresholds. Prior to outlier removal, the root mean square error (RMSE) of the original data was 1.19 TECU. However, after applying 5.8b, 3.5b and 3b thresholds to remove outliers, the model error improved to 0.95, 0.69, and 0.62 TECU, respectively. Therefore, selecting the right percentile threshold is based on the network configuration and desired accuracy that we aim to achieve.

Fig. 8
figure 8

Accuracy of ionospheric delays before (first panel) and after applying the thresholds of 5.8b, 3.5b and 3b (three last panels). The accuracy of the testing stations DAWN was measured and showed as rmse in each case

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Removing outliers using the thresholds of 3b and 3.5b brings an improved accuracy to the model. However, this may also decrease the number of available satellites or available local ionospheric correction maps at each epoch. It is noted that each station in the NT GNSS stations network can receive signals up to 13 GPS satellites at the same time, and one satellite can be chosen as a reference satellite to compute the SD STEC between satellites. Therefore, at any given time, the NT network can create up to 12 local maps for SD STEC using GPS data.

To calculate the number of available satellites at a given epoch, we need to count the number of satellites for which at least five stations in the NT network received the signals. The process is as follow:

  • For each satellite, count the number of stations that have received its signal at the given epoch.

  • If the count is at least 5, consider the satellite as an available satellite.

  • Repeat the above steps for all satellites in the network.

  • Count the total number of available satellites at a given epoch.

Figure 9 illustrates the number of satellites that are available for mapping at each epoch during the day for the original data and after applying the 5.8b, 3.5b, and 3b thresholds. The y-axis represents the number of available satellites, ranging from 0 to 12.

Fig. 9
figure 9

The number of maps availability at all epochs during the entire day of July 01, 2021. The blue, green, and red colors present the number of available satellites from the network, after applied 5.8b, 3.5b, and 3b thresholds, respectively

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As presented, utilizing the percentile function to eliminate outliers can enhance the accuracy of the model. However, this approach may lead to a reduction in available satellites or local ionospheric correction maps, especially in networks with few receivers and in active ionospheric regions like the NT network. Our data indicates that filtering out unreliable or erroneous satellite data can improve positioning accuracy for GNSS measurements. For example, in the NT network on July 1, 2021, applying a 3.5b threshold resulted in a 0.2% reduction in observations, but increased ionospheric correction accuracy by 42–0.69% TECU. Applying a 3b threshold led to a 0.4% reduction, but increased accuracy by 52% to about 0.62 TECU. Therefore, removing a small number of satellite data to enhance ionospheric accuracy has a positive impact on precise positioning.

We calculated the percentage of available ionospheric maps for each epoch of four months and displayed in Fig. 10. The x-axis represents the number of available maps per epoch, and the y-axis displays the corresponding percentage. In the NT network, the ionospheric delays from 1 to 13 satellites per epoch are observed, with the most common value centered around 9 (indicated by blue markers). Initially, the percentage of epochs with only 1–3 available satellites is nearly 0%. However, when a threshold of 3.5b is applied, the number of available satellites decreases. The green marker line illustrates that, after applying the threshold, the maximum number of available satellites is 6–7, with a percentage exceeding 20%. The percentages of epochs with 0–3 available satellites are all below 5%, but higher than the original values. Notably, around 2.5% of epochs exhibit no available satellites, indicating the percentage of cases where no corrections are available using this local network.

Fig. 10
figure 10

The percentage of available maps at each epoch before and after applying the 3.5b threshold for 4 months in 2021. Applying an outlier detection threshold (green points) decreases the number of available satellites compared with the original data (blue points)

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Accuracy of the local model in mid and low latitudes

We evaluate accuracy of the local maps by using testing stations within the network, i.e., by removing them from the network prior to performing the mapping, the same as with the Darwin station in the case above. The accuracy is calculated by finding the difference of SD STEC between the prediction from the map and the observation of the testing station. In this study, we used five testing stations located in different areas of the network. We also compared the results over NT with another network in Western Australia (WA). The WA local map has a similar network configuration to NT, but it is in the middle latitudes (30–35° S). Figure 11 displays the maps of both NT and WA networks, with the red stars indicating the receiver locations and color circles indicating the testing stations that were alternatively selected for evaluation.

Fig. 11
figure 11

Maps of testing stations in NT and WA networks. The red stars are receiver coordinates with their names next by. The name within the circle is the testing station used to evaluate local model

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Figure 12 shows the accuracy of the local model for five testing stations in the NT network after adapting with outlier detection. The results present different accuracy based on the location of the test station. For instance, when the testing station is KUNU, the local map is then created using the remaining stations. Since KUNU has few neighbouring stations, the model's accuracy for this test station is low. On the other hand, GROT is located outside the local map, but two nearby stations contribute to a higher accuracy model compared to KUNU. For stations within the network, the accuracy is better, ranging from 0.5 to 1 TECU in the NT region.

Fig. 12
figure 12

Mean accuracy of five testing stations over NT in different seasons (Jan, Apr, Jul, and Oct 2021). The mean accuracy of the testing station depends on its location which is located inside (DARW, KAT1, and LARR) or outside (KUNU and GROT) the network. Accuracy in Jan 2021 is worse than in other months because of the lower number of GNSS stations in the network

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In contrast to the NT region, the local map performs more effectively in the mid-latitude region (WA), as shown by the results in Fig. 13. The accuracy of the model for stations located within and outside of the network shows insignificant differences. In the mid-latitude region, the model shows an accuracy of approximately 0.5 TECU for all test stations.

Fig. 13
figure 13

Mean accuracy of five testing stations over WA (KELN, NCLF, WAGN, KALG, and RAVN) in different seasons (Jan, Apr, Jul, and Oct 2021). The mean accuracy of the model for those stations in this network is within 0.2 to 0.6 TECU

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To examine model accuracy before and after applying the 3.5b threshold for outlier detection, the mean accuracy of two testing stations DARW (for NT network) and RAVN (for WA network) have been selected for all data processing periods. The accuracy improvement is calculated by:

$$\% improvement = 100\% \frac{{\left| {Accuracy _{after} - Accuracy _{before} } \right|}}{{Accuracy _{before} }}$$
(10)

Figure 14 presents the number of days corresponding to the percentage that improved mean accuracy. As shown in Fig. 14 (left), the accuracy of the NT local model improves by up to 60%. For most days, the improvement is between 20 to 55%. On the other hand, the WA local network shows up to 50% improvement.

Fig. 14
figure 14

The distribution of accuracy improving by using 3.5b threshold for DARW station (NT network) and RAVN station (WA network)

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The results of this study demonstrate that the presented outlier detection methods can be applied to other latitudes. However, it should be noted that the ionosphere in mid-latitudes tends to be more stable compared to low-latitudes, which makes it easier to construct accurate local maps even with a sparse GNSS stations network. In low latitudes, the establishment of a well-distributed GNSS network with an adequate number of receivers becomes crucial for achieving higher accuracy for local ionospheric mapping. For example, in the NT network, the limited number of stations (approximately 9–10) within the region and the absence of additional stations around KUNU contribute to reduced accuracy in that specific location. The low-latitude region is known for its highly variable ionosphere and spatial gradients resulting from the equatorial anomaly, which can significantly impact mapping errors. The findings from the KUNU station, evaluated using the local map over the NT network, demonstrated the importance of having enough data and an appropriate network configuration of stations for achieving accurate ionospheric correction mapping in low-latitude regions.

Our data analysis using bilinear ionospheric models revealed that the residuals formed by Laplace distribution. Regional mapping of Marini‐Pereira et al. (2020) which used groupings of measurements in time and space, triangulation, linear interpolation, and smoothing also demonstrated that the normal distribution does not accurately represent the model errors. Ma et al. (2022) investigated the errors of both the slant TEC and the derived TEC and found that they follow Laplace distribution rather than Gaussian distribution. Therefore, it is worth to consider data distribution before applying threshold to remove an appropriate percentile of outliers for that distribution.

Conclusion

Detecting outliers in ionospheric local mapping is essential for achieving improved accuracy, especially in low-latitude regions. This study presents a method for detecting outliers in ionospheric corrections, levering statistical residuals derived from the ionospheric local model. New thresholds for outlier detection have been proposed by assigning weight numbers to account for the extended tail of the Laplace distribution. By applying statistical thresholds of μ + 5.8b, μ + 3.5b, and μ + 3b, it becomes possible to detect and eliminate outliers in the model. These thresholds are designed to preserve 99.7%, 97%, or 95% of the values within the respective thresholds. Consequently, modelling outliers can be effectively identified and removed, leading to enhanced accuracy in the ionospheric corrections. Applying modelling thresholds to remove outliers can cause data loss in some circumstances. However, it ultimately leads to a more accurate model. In low-latitude regions, approximately 2.5% of epochs exhibit no available satellites i.e., no correction data can be made using this local network, when the 3.5b threshold is applied, but model accuracy during the testing period justifies this sacrifice. Moreover, the threshold for outlier detection in the ionospheric model is based on the statistical data of b. Therefore, increasing the amount of data can potentially lead an enhanced accuracy of the model. Future research should focus on exploring different scenarios, such as during periods of geomagnetic disturbance and high solar activity, to evaluate the performance of the outlier detection method in these specific conditions.

In conditions of low solar activity and quiet conditions, of the application of a 3.5b threshold for detecting and removing outliers can improve the accuracy of the model. This improvement is consistent in both low-latitude and mid-latitude regions, with the accuracy improvement of up to 50%. Based on data observation at each epoch using a 3.5b threshold, the mean accuracy of the regional model is approximately 0.5 TECU for mid-latitude regions (WA). In low-latitude regions, such as the NT, the mean accuracy ranges from 0.5 to 2.5 TECU. These results highlight the effectiveness of the 3.5b threshold in reducing modelling outliers and improving the overall accuracy of the local ionospheric model. To detect outliers for future data, regarding the increasing solar activities, updating the statistical location and scale parameters could bring better accuracy in local models.