The dayside region is characterized by x>0 (θ<90°). Because the CL–MPC dataset contains nightside crossings, the amount of usable MPCs reduces to 137 295 after they are filtered out. Since all mentioned magnetopause models rely on the IMF Bz component, MPCs with missing IMF data cannot be used for model comparison. This leaves 133 045 usable MPCs. For each crossing, the magnetopause distance rmod is calculated for the four models with their respective formula. The observed magnetopause distance by the spacecraft is the magnitude of the positional vector of the magnetopause crossing and denoted as robs. To determine the accuracy of each model, the prediction error 5Δr=robs-rmod is calculated for each crossing. Histograms of the resulting Δr distributions are shown in Fig. . Notably, the four models perform rather equally with a common root-mean-square error (RMSE) of slightly above 1RE. The mean values of the Sh98 and Lin10 model are close to zero, indicating good magnetopause prediction capabilities on average. For Liu15, the positive mean error implies a small underestimation of the model, meaning that the magnetopause is more expanded than the model assumes for given input parameters. Similarly, the Ng22 model underpredicts the real magnetopause extend by almost 0.5RE. The standard deviation of of the prediction error is about 1RE for all models, mostly contributing to the RMSE. Figure  shows Δr over the zenith angle in a heatmap for the four models. The red dots with error bars indicate the mean and standard deviation of the prediction error for a 4.5° wide bin. The heatmap reveals larger deviations from zero and an increased error for larger zenith angles for every model, especially for θ>60°. These findings imply that the models exhibit reduced predictive accuracy in regions approaching the terminator and possibly on the nightside, consistently estimating the magnetopause to be at larger distances to Earth than it actually is, on average. As discussed for Fig. , the models deviate the most in the nightside low latitude regime, suggesting that the shape of the nightside magnetopause is not fully understood (partly due to little available data) or fluctuates largely despite similar conditions. Regardless of the differences of the models in that region, none of the models is able to make accurate predictions. For lower zenith angles (θ<60°), the models on average slightly underpredict the magnetopause distance, especially the Ng22 model, resulting in the high mean Δr value. The Lin10 model seems to be the most accurate for smaller zenith angles.

In this study, magnetopause crossings with θ<30° are considered subsolar. A zenith angle smaller than 30° is also chosen by, e.g., to characterize the subsolar region, since then the magnetopause distance rMP is very close to the subsolar standoff distance r0 due to the approximately spherical shape of the magnetopause for low zenith angles. The number of usable MPCs is reduced the same way as for the full dayside magnetopause, resulting in 33 845 usable MPCs. Again, Δr is calculated for every crossing and every model. Figure  depicts the resulting distributions. The RMSE decreases for all models compared to the full dayside. Lin10 undergoes the largest improvement, while the Ng22 performance does not change much. Looking in the region of θ<30° in Fig.  further explains the positive mean value for every model (underestimation of the magnetopause distance). For the subsolar region, some models clearly perform better than others. While the standard deviation is similar for all models, the mean deviation drives the difference in RMSE. Interestingly, the standard deviation remains rather high at roughly 0.8RE with an improvement of approx. 20 % compared to the full dayside. The findings suggest, that albeit the different models make similar predictions regarding the subsolar standoff distance, they still are not able to quantify it accurately.

computes the equivalent subsolar standoff distance for every MPC and calculates Δr with respect to the Sh98 model. Since in the subsolar region, r0≈rMP, the results are comparable specifically in that region. gets a mean deviation of 0.36RE and a standard deviation of 0.76RE, which are nearly the same as in our investigation, although the analysis methods differ slightly. The performance comparison shows that in spite of more than two decades of advancement in magnetopause modeling, there remains little measurable improvement in the models' ability to accurately predict the magnetopause location.

used a machine learning approach to find out which parameters are the most important for magnetopause modeling. However, the authors used MPCs from the whole dayside, a small dataset and a limited selection of parameters. Therefore, the same concept is applied again in this work on an extensive dataset.

ML algorithms are often treated as black boxes, where the internal decision-making process remains opaque. While the outputs can be insightful and empirically validated, the underlying reasons for these results are frequently unclear. This lack of interpretability makes it difficult to establish meaningful connections between input and output parameters, and thereby to extract insights into, for example, underlying physical relationships. To address the challenge of interpretability in machine learning models, particularly those treated as black boxes, various explainable AI (XAI) techniques have been developed. One of the most widely used approaches is based on Shapley values, a concept originally derived from cooperative game theory. Introduced by , Shapley values provide a principled method to fairly distribute the ”payout” among players based on their individual contributions to the total outcome. In the context of machine learning, this framework has been adapted to quantify the contribution of each input feature to a model's prediction. By considering all possible combinations of features, Shapley values offer a theoretically grounded and model-agnostic approach to explain how and to what extent each input feature influences the output, thereby enabling a deeper understanding of the learned relationships and supporting scientific interpretation of complex models. However, the exact computation of Shapley values is computationally expensive, as it requires evaluating all possible subsets of input features - an operation with factorial complexity relative to the number of features. This becomes impractical for models with even a moderate number of inputs. To address this challenge, SHAP (SHapley Additive exPlanations), a Python package introduced by , provides efficient algorithms and approximations for estimating Shapley values in a tractable manner. The obtained value is called the SHAP value and is used in the place of the Shapley value in the following. SHAP performs well with tree based random forest ML algorithms . Therefore, the Python library XGBoost is used to train a regressor model. The training data consists of 80 % of all MPCs, with the target value being the observed magnetopause distance robs. All but three hyperparameters of the XGB model are kept at their default values. The adjusted hyperparameters are the number of trees in the random forest (n_estimators=1000), the depth, i.e. number of decisions, of each tree (max_depth=6), and the contribution of each tree, which also influences how conservative the model is and how it generalizes (learning_rate=0.05). The SHAP value is then evaluated for every parameter per available MPC.

To get an overview of which parameters might be important for future magnetopause models, a regressor with 18 parameters is trained. The parameters are the dynamic and magnetic pressure, the AE and SYMH indices, the clock and cone angle, the components and the absolute value of the solar wind velocity and the IMF, the ion temperature, the Na/Np ratio, the dipole tilt angle and the F10.7 solar flux. The zenith angle is limited to 30° to focus on the subsolar region. Because some parameters are missing for certain MPCs, the total number of usable crossings reduces to 18 787. Especially the Na/Np ratio decreases the amount of usable MPCs, because it is only available for less than 50 % of all detected magnetopause crossings. Figure  shows a beeswarm plot of the obtained SHAP values for every parameter. The parameters are ordered in decreasing importance from top to bottom, where importance is quantified by the mean absolute value of all SHAP values per parameter and denoted as S(i) where i is the respective parameter. The color assigned to each data point reflects the value of its underlying parameter. As one can expect, the dynamic pressure is ranked as the most important parameter with S(pdyn)=0.53RE. On its own, it accounts for a difference in r0 exceeding 5RE, with higher values leading to magnetopause compression (indicated by low SHAP values), and lower values associated with expansion (high SHAP values). This is in good agreement with the literature and empirical models.

The following two parameters are the AE and SYMH indices (S(AE)=0.14RE, S(SYMH)=0.1RE), both capable of chaning the predicted value of r0 up to about 2RE. As a general trend, the magnetopause moves inward for high parameter values. While the Dst and SYMH index have been used to model Earth's magnetosphere and references therein, they did not get much attention for magnetopause modeling. investigate the influence of Dst* on the magnetopause standoff distance and find that the standoff distance decreases for increasing Dst*. A recent neural network model by uses Dst* as an input and exhibits the same dependency. Both findings agree with the SHAP values of the SYMH index in this investigation. To the best of our knowledge, the AE index has not been previously used in modeling the magnetopause location. However, our analysis clearly demonstrates that it may be a significant parameter that can potentially improve the model results.

The next important parameter is the F10.7 solar flux (S(F10.7)=0.09RE), which is a direct proxy of the solar activity. Long-term variations of the magnetopause due to the solar cycle are investigated by . The authors draw the conclusion that the F10.7 index does correlate with the magnetopause location. The derived SHAP values result in a similar conclusion.

Interestingly, the IMF magnitude B seems to be more important (S(B)=0.08RE) than the magnetic pressure, which is derived from it (S(pmag)=0.05RE). Larger IMF strengths and therefore higher magnetic pressure shift the pressure balance between the sum of dynamic and magnetic pressure in the magnetosheath and the magnetic pressure from Earth's magnetic field closer to Earth, hence the negative SHAP values for high parameter values. This is consistent with magnetopause models such as Lin10 and Ng22, which use the sum of the dynamic and magnetic pressure. Although the magnetic pressure is significantly smaller than the dynamic pressure and is considered negligible by , the SHAP values from our analysis suggest that it exerts a non-negligible influence – particularly when the IMF magnitude is used directly instead of the magnetic pressure.

Earth's dipole tilt angle γ is also ranked as an important parameter with an importance of S(γ)=0.07RE, justifying its use in many magnetopause models. Based on the SHAP values, our analysis underscores the importance of including the dipole tilt angle in magnetopause models. The effect of the tilt angle is larger in the high latitude and cusp regions, further away from the subsolar point. Still, SHAP values show that the tilt angle is also a crucial parameter for the low-latitude regime. examines the influence of different IMF parameters on the shape and location of the magnetopause. One conclusion is that the IMF cone angle does not impact the subsolar standoff distance. However, the SHAP values suggest the opposite, with the parameter ranking fairly high, scoring an importance value of S(θcone)=0.06RE. Also, the feature value is systematic across most of the SHAP values. We therefore conclude that the cone angle has the potential to further enhance the quality of future magnetopause models. finds that the cone angle is able to explain extreme deviations from theoretical to observed r0, further underlining the importance of the parameter. In combination with the IMF magnitude, the two parameters cover the influence of the IMF in greater extend than the Sh98 and Lin10 models, which only use the Bz component.

The Na/Np ratio is the only parameter covering the alpha particles in the solar wind. Previously discussed plasma parameters refer to protons only. Despite their lower abundance in the solar wind (∼5%), alpha particles are nearly four times heavier than protons and thus can exert a non-negligible influence on the overall dynamic pressure of the solar wind, which is usually neglected. High values for Na/Np have the potential to decrease the subsolar standoff distance by more than 0.5RE as indicated by the SHAP values. claim that the observed dependence of the magnetopause location on solar wind speed cannot be attributed to the contribution of alpha particles to the upstream pressure. Although subtle, the SHAP analysis ranks this parameter among the top contributors, assigning it an importance of S(Na/Np)=0.06RE. A crucial restriction is that including the Na/Np ratio roughly halves the amount of usable MPCs, since the parameter is available for only about 40 % of all crossings from the OMNI 1 min high resolution database. Using a lower time resolution (five minute or one hour averages) ups the percentage of usable MPCs, but decreases the significance of the parameter. It is therefore decided to keep the one minute time resolution to better reflect the impact of the parameter on the magnetopause position.

All remaining parameters have an importance score of less than 0.05RE and are considered less important for the magnetopause standoff distance. Furthermore, the ordering of features with low SHAP importance can fluctuate based on the underlying XGBoost model, or rather the composition of the training data, which are randomly selected 80 % of all MPCs. Still, the IMF Bz component and the clock angle rank low across different XGBoost models. Both parameters are used in previous magnetopause models or standoff distance parameterizations; especially the Bz component, which is generally believed to be an important parameter by, e.g., the aforementioned magnetopause models. While a correlation between SHAP value and Bz value is visible, the performed investigation indicates that the Bz component is not as important as widely believed, altering the standoff distance marginally. Also, the Bx components ranks higher than the Bz component for multiple different XGBoost models, further justifying the importance of the cone angle. The clock angle ranks as the lowest of all investigated parameters, supporting the conclusion of .

On top of the discussed parameters, additional quantities have been investigated. They are not shown here since they are either not relevant (e.g. magnetosonic Mach number, plasma beta), or too closely connected to already investigated parameters (e.g. the proton density SHAP values carry the same information as the ones for the dynamic pressure).

The obtained results are somewhat difficult to compare to the ones by , because of a different underlying dataset (including the spatial coverage of MPCs), machine learning model, and parameter choice. Both studies conclude that the dynamic pressure and the IMF magnitude rank highest and the cone angle ranks higher than the clock angle. Additionally, the dependencies between SHAP value and feature value overlap with our results for most parameters. Still, finds that the Bz component is more important than the Bx component and ranks high in general (S(Bz)=0.11RE). The differences in the results are possibly due to the mentioned differences in the analysis.

Based on the SHAP value investigation, the six most important parameters are selected for further use as inputs for a magnetopause model. The parameters are the dynamic pressure, the AE and SYMH indices, the IMF magnitude, Earth's dipole tilt angle and the IMF cone angle. Despite the F10.7 solar flux being ranked as fourth important parameter, it is not considered in the following. That is because later analysis shows that the inclusion of F10.7 does not significantly reduce the RMSE of the models. Therefore, the influence of F10.7 is believed to be already covered by other parameters, thus not contributing much additional information. As a possible seventh parameter, the alpha particle to proton ratio was considered. Given that including this parameter would significantly reduce the available training data, we ultimately chose not to incorporate it in magnetopause modeling. All other parameters ranked less important than the cone angle (S(i)<0.05RE) are also disregarded from further analysis.

SHAP values provide an overview of the influence on the magnetopause distance of individual parameters. Alternative visualization methods also enable a degree of quantitative analysis. However, to better understand the impact of individual parameters on the subsolar standoff distance, a second ML approach is utilized. A Support Vector Regression (SVR) model is trained using the scikit-learn Python module with the following nine input parameters: six parameters from the previous section identified as the most important, namely the solar wind dynamic pressure, the IMF cone angle and magnitude, the AE and SYMH indices, the dipole tilt angle, and a three-component n-vector, representing the position of each MPC. The vector is normalized and points from the center of the Earth to the respective MPC position. The target variable is the observed magnetopause distance. The maximal zenith angle is again limited to 30°, leaving a total of 32 667 MPCs. 80 % of the crossings are used to train the model, the remaining 20 % are the test set. SVR is a supervised machine learning algorithm that extends the principles of Support Vector Machines (SVM) to regression problems. SVR aims to find a function that approximates the relationship between input features and a continuous target variable while maintaining a margin of tolerance defined by the hyperparameter ε. Only data points with prediction errors larger than ε contribute to the cost function, promoting sparsity in the solution. The hyperparameter C controls the trade-off between model complexity and the degree to which deviations larger than ε are penalized; higher values of C allow the model to fit the training data more closely, at the risk of overfitting. The kernel coefficient γSVR defines the influence of individual training samples: lower values imply a smoother model, while higher values lead to more localized influence and greater model flexibility. utilize a SVR model to establish a global 3D magnetopause model, functioning as a proof of concept for the method of SVR. The authors used the solar wind dynamic pressure, the IMF Bz component and Earth's dipole tilt angle in combination with a n-vector as input parameters. The hyperparameter ε is set to 0, while C=20 and γSVR=1 are found by visual inspection of the resulting models. In this work, the optimal model parameters are found by hyperparmeter tuning to be ε=0.1, C=20 and γSVR=0.01. For SVR, it is important to standardize the input parameters and the target value by removing the mean and scaling to unit variance. The respective values used for scaling each parameter can be found in Table . After the prediction, the resulting r0 values and the input features need to be scaled inversely. The trained SVR model has an RMSE of 0.68RE, calculated for the 32 667 subsolar MPCs. Therewith, the error is decreased by approx. 17 % with respect to the Lin10 model, the one with the smallest error of the compared models (refer Fig. ). Note that the model is not able to make accurate predictions for parameter inputs outside the ranges specified in Table .

The dependency of r0 on different parameters is depicted in Fig.  in form of Ceteris Paribus plots. In a Ceteris Paribus plot, all parameters but the one on the x-axis is kept constant at a given value. In this case, the mean value (see Table ) of the respective parameters is chosen. Doing so allows to investigate the influence of each parameter independently. The n-vector is set to (nx,ny,nz)=(1,0,0), representing the subsolar point. The plot range of the abscissae is defined by the 1 % (99 %) quantile of the training data for the lower (upper) bound. All general trends of the individual parameters agree with the color-coding of the SHAP values from Fig. . Additionally, the ranges of the ordinates largely reflect the relative importance of the features, with pdyn showing the widest range and less important parameters smaller ranges. The fact that two different ML approaches result in qualitatively similar results underscores the validity of the observed patterns and increases confidence in the interpretability of the models. It further suggests that the identified relationships are not model-dependent artifacts.

Figure  panel (a) shows the influence of the solar wind dynamic pressure on r0. The typical non-linear decrease of r0 with increasing pdyn is present. A log-log representation of the same plot (not shown) reveals that the dependency is not linear, indicating that the relationship does not follow a power-law of the form r0∝pdyn-1/k with constant k, as assumed in all previously compared models. In fact, k is larger for low and high dynamic pressures (k>10 for pdyn<1.4 nPa and pdyn>9.6 nPa) and smaller for the intermediate dynamic pressure range (k<6 for 3nPa<pdyn<7.2 nPa). chose a similar approach to find the response of r0 to different parameters and found k varying for different values of pdyn too, likewise did .

The dependence of the subsolar standoff distance on the AE index is shown in Fig. b. Specifically, r0 decreases as AE increases. This trend aligns with established solar wind–magnetosphere coupling physics: elevated AE values correspond to increased substorm-related reconnection, which enhances magnetospheric current systems and erodes the dayside magnetopause . Therefore, less magnetic flux is present on the dayside during high AE intervals. The AE index may therefore capture magnetospheric processes that are not reflected by other parameters directly.

A high (low, meaning strongly negative) SYMH index is the result of a quiet (enhanced) ring current. The subsolar standoff distance decreases for high SYMH values, as shown in Fig. c. observe a similar relationship, but with the Dst* rather than the SYMH index, which reflects a related underlying principle, but do not discus possible reasons. We suggest that a stronger ring current is due to more particles being injected into the near–Earth magnetosphere. Therefore, the pressure just inside the magnetopause increases. To restore the pressure balance, the magnetopause has to move outward, explaining the identified relationship between SYMH and r0.

High IMF magnitude corresponds to high magnetic pressure, pushing the magnetopause earthward. Despite the magnetic pressure being small in comparison to the dynamic pressure, it is included in more recent magnetopause models. The SHAP value analysis shows that instead of the magnetic pressure, the IMF magnitude has a more severe impact on the magnetopause standoff distance, visible in Fig. d. As expected, r0 decreases with increasing IMF magnitude, agreeing with results from, e.g., .

The influence of the dipole tilt angle is investigated by , who performed MHD simulations for typical solar wind configurations. The authors conclude that the dipole tilt angle has little to no effect on the magnetopause position and shape for the equatorial magnetopause. The same conclusion is drawn by , stating that the influence of the tilt angle is small and can therefore be neglected. Our analysis (see Figure e) shows that the typical influence of the dipole tilt angle is about 0.4RE, up to 1RE for some cases (Fig. ), which is too much to be ignored. The relationship between r0 and γ is similar to the two aforementioned studies. They find a symmetric dependency with the maximum value of r0 at γ=0° and a cosine relationship . In our case, the curve is also symmetric but peaks at γ≈-4°. Note that the exact functional form of the dependency in our case is influenced by the values of the other parameters. One observation is that with increasing dynamic pressure, the impact of other parameters reduces. This is due to the fact that the dynamic pressure is the main driver of the magnetopause location. If it is weak, other parameters have a greater impact and vice versa. Therefore, the impact of the dipole tilt angle varies with other parameters - not only in magnitude, but also in functional form .

Figure  panel (f) depicts the influence of the cone angle. Values near 0° correspond to radial IMF, Parker spiral configuration is at θcone≈45° and larger values reflect quasi perpendicular IMF at the subsolar point. The results indicate that the magnetopause is located farther from Earth under quasi-radial IMF conditions, while it moves closer under perpendicular IMF orientations. This is in good agreement with former studies .

Despite being classified as not important, the solar wind speed v shall be discussed briefly. SHAP values indicate increased r0 with faster solar wind (ref. Fig. ). According to pdyn∝v2 and r0 decreasing with increasing dynamic pressure, the relation seems contradicting at first. However, faster solar wind usually is less dense, resulting in moderate dynamic pressure . Also, high solar wind speed increases the convection electric field and dayside reconnection and therewith the particle intake into the magnetosphere. The increased ring current results in an increase of the subsolar standoff distance, as discussed for the SYMH index. Lastly, fast solar wind often occurs at times of radial IMF, which in turn coincides with an extended magnetosphere. find the same relationship between solar wind speed and magnetopause standoff distance, but do not discuss the potential reasons.

Our analysis shows that a machine learning approach is suitable to investigate the influence of certain drivers on the position of the subsolar magnetopause. On the MPC dataset at hand, the SVR model outperforms the established magnetopause models from literature regarding the accuracy of the prediction. Additionally, the influence of certain parameters reflects results from previous studies, showing that the approach has a potential to improve the understanding of magnetopause dynamics under varying solar wind conditions. This raises the question of whether a similar methodology could be employed to develop a global 3D magnetopause model, following examples set by, e.g., .

To do so, a nine parameter SVR model is trained with in total 128 232 dayside MPCs (θ<90°). The input parameters and target value are the same as for the model discussed in the previous subsection. The trained model has an RMSE of 0.75RE, which is an improvement of more than 25 % compared to the best of the four compared models (ref. Fig. ) and only slightly worse than the model for the subsolar region. A comparison between the SVR-predicted magnetopause shape and that of the Lin10 model (see Fig. ) reveals significant discrepancies, especially in the y=0 plane. This is most likely due to the fact that the training dataset has only a few high latitude MPCs, covering, e.g., the cusps or polar regions. Consequently, the SVR algorithm struggles to accurately reproduce the magnetopause shape in those specific regions. Machine learning models are particularly effective at making predictions for parameters that lie within the parameter space seen during training. However, for uncovered or sparsely populated parameter space volumes, the model cannot predict the target variable accurately. This applies both to certain regions in space and to parameter combinations that are underrepresented in the training dataset. Since the majority of the MPCs lie close to the equatorial plane (with full longitudinal coverage on the dayside), this region is likely the most accurate region of the model. Comparing the equatorial plane with the Lin10 model shows a strong dawn-dusk asymmetry, where the dawn flank is up to 3RE closer to Earth than the dusk flank at the terminator. While the dawn–dusk asymmetry of Earth's magnetopause is an active area of research , the extent of the asymmetry continues to be a subject of debate. It therefore remains unclear whether the SVR model accurately represents the real magnetopause configuration, or whether the pronounced differences arise from artifacts in the data, such as orbital biases of the satellite missions.

Summarizing the findings, the discrepancies between our model and the Lin10 model are highest in the high latitude regions and on the dawn side, where a strong asymmetry is present. While discrepancies from established models are expected in the development of new magnetopause models, it is challenging to determine which model is more accurate based on visual inspection alone. Extensive hyperparameter tuning was performed on the SVR model to achieve an optimal trade-off between accuracy and overfitting. Doing so, the model is evaluated on unseen test data that has not been used for training the model. When plotting the residuals over the predicted value or individual parameters, no correlation is noticeable. The analysis also shows that the model performs equally well for different areas in space and over certain parameter ranges. Therewith, the SVR model is validated to the best of our abilities. The small RMSE of 0.75RE suggest good prediction capabilities. Subsequently, the ML approach, utilizing a SVR model, has the potential to model the dayside magnetopause more accurate than ever before. Nevertheless, the model can potentially lead to false predictions for certain cases, e.g., for uncommon (and therefore unseen during training) input parameter combinations (or parameters out of the bounds in Table ), especially for the high latitude regions. We suggest that ML models based on the given dataset can be used to predict the equatorial region accurately, but shall be handled with care when investigating other regions and extreme inputs, as ML models often provide unphysical results when used outside of the bounds of the training data set .