Mathematical expressions of the drag-based models for predicting the arrival time of coronal mass ejection and their development and evolutionary processes

Mitigating the lethal threats caused by coronal mass ejections (CMEs) on human and space operations can be accomplished with a fast and accurate forecast of Earth-directed CME transit times. The current paper presents a robust Cascade Forward Neural Network (CFNN) framework to predict the transit times of Earth-directed halo CMEs using a total of 290 CME/Interplanetary coronal mass ejections (ICME) pairs of datasets for the past two and half decades (solar cycle; SC 23, 24, 25). It is the first time incorporating deprojected speeds into a machine learning framework to mitigate uncertainties due to projection effects during CME transit time prediction. The CFNN model forecasted the transit times of 87 Earth-directed CMEs and recorded a mean absolute error (MAE) of 7.3 h. In addition, 5 selected fast-moving (energetic) halo CME episodes during the deep phases (solar minima) of SC 25 were reconstructed using the Graduated Cylindrical Shell (GCS) forward-modeling technique. The events were predicted using the CFNN model (MAE = 4.5 h) in comparison with the Drag-Based model (DBM; MAE = 6.2 h) and Empirical Shock Arrival model (ESA; MAE = 13.5 h) to evaluate the robustness and the flexibility of the CFNN framework as well as the reliability of the CME 3D speed as a proxy for the space speed. The CFNN framework demonstrated satisfactory performance in contrast to previously used models by minimizing CME arrival time prediction errors due to projection effects. Hence, the study validated the efficiency of the GCS model for studying the 3D kinematics of CMEs and emphasized the essence of utilizing deprojected speeds in machine learning frameworks as better alternatives for fast, reliable, and accurate CME arrival (transit) time predictions.

Forecasting the arrival time of coronal mass ejections (CMEs) and their associated shocks is one of the key aspects of space weather research. One of the commonly used models is the analytical drag-based model (DBM) for heliospheric propagation of CMEs due to its simplicity and calculation speed. The DBM relies on the observational fact that slow CMEs accelerate whereas fast CMEs decelerate and is based on the concept of magnetohydrodynamic (MHD) drag, which acts to adjust the CME speed to the ambient solar wind. Although physically DBM is applicable only to the CME magnetic structure, it is often used as a proxy for shock arrival. In recent years, the DBM equation has been used in many studies to describe the propagation of CMEs and shocks with different geometries and assumptions. In this study, we provide an overview of the five DBM versions currently available and their respective tools, developed at Hvar Observatory and frequently used by researchers and forecasters (1) basic 1D DBM, a 1D model describing the propagation of a single point (i.e., the apex of the CME) or a concentric arc (where all points propagate identically); (2) advanced 2D self-similar cone DBM, a 2D model which combines basic DBM and cone geometry describing the propagation of the CME leading edge which evolves in a self-similar manner; (3) 2D flattening cone DBM, a 2D model which combines basic DBM and cone geometry describing the propagation of the CME leading edge which does not evolve in a self-similar manner; (4) DBEM, an ensemble version of the 2D flattening cone DBM which uses CME ensembles as an input; and (5) DBEMv3, an ensemble version of the 2D flattening cone DBM which creates CME ensembles based on the input uncertainties. All five versions have been tested and published in recent years and are available online or upon request. We provide an overview of these five tools, as well as of their similarities and differences, and discuss and demonstrate their application.

The so-called drag-based model (DBM) simulates analytically the propagation of coronal mass ejections (CMEs) in interplanetary space and allows the prediction of their arrival times and impact speeds at any point in the heliosphere ("target"). The DBM is based on the assumption that beyond a distance of about 20 solar radii from the Sun, the dominant force acting on CMEs is the "aerodynamic" drag force. In the standard form of DBM, the user provisionally chooses values for the model input parameters, by which the kinematics of the CME over the entire Sun--"target" distance range is defined. The choice of model input parameters is usually based on several previously undertaken statistical studies. In other words, the model is used by ad hoc implementation of statistics-based values of the input parameters, which are not necessarily appropriate for the CME under study. Furthermore, such a procedure lacks quantitative information on how well the simulation reproduces the coronagraphically observed kinematics of the CME, and thus does not provide an estimate of the reliability of the arrival prediction. In this paper we advance the DBM by adopting it in a form that employs the CME observations over a given distance range to evaluate the most suitable model input parameters for a given CME by means of the least-squares fitting. Furthermore, the new version of the model automatically responds to any significant change of the conditions in the ambient medium (solar wind speed, density, CME--CME interactions, etc.) by changing the model input parameters according to changes in the CME kinematics. The advanced DBM is shaped in a form that can be readily employed in an operational system for real-time space-weather forecasting by promptly adjusting to a successively expanding observational dataset, thus providing a successively improving prediction of the CME arrival.

One of the goals of Space Weather studies is to achieve a better understanding of impulsive phenomena, such as Coronal Mass Ejections (CMEs), to improve our ability to forecast their propagation characteristics and mitigate the risks to our technologically driven society. The essential part of achieving this goal is to assess the performance of forecasting models. To this end, the quality and availability of suitable data are of paramount importance. In this work, we merged publicly available data of CMEs from both in-situ and remote observations in order to build a dataset of CME properties. To evaluate the accuracy of the dataset and confirm the relationship between in-situ and remote observations, we have employed the Drag-Based Model (DBM) due to its simplicity and modest consumption of computational resources. In this study, we have also explored the parameter space for the drag parameter and solar wind speed using a Monte Carlo approach to evaluate how efficiently the DBM determines the propagation of CMEs for the events in the dataset. The geoeffective CMEs selected as a result of this work are compliant with the hypothesis of DBM (isolated CME, constant solar wind speed beyond 20 R⊙) and also yield further insight into CME features such as arrival time and arrival speed at L1 point, lift-off time, speed at 20 R⊙ and other similar quantities. Our analysis based on the acceptance rate in the DBM inversion procedure shows that almost 50% of the CME events in the dataset are well described by DBM as they propagate in the heliosphere. The dataset includes statistical metrics for the DBM model parameters. The probability distribution functions (PDFs) for the free parameters of DBM have been derived through a Monte Carlo-like inversion procedure. Probability distribution functions obtained from this work are comparable to PDFs employed in previous works. The analysis showed that there exist two different most probable values (median values) of solar wind speed for DBM input based on slow (wslow ≈ 386 km/s) and fast (wfast ≈ 547 km/s) solar wind type. The most probable value for the drag parameter (γ ≈ 0.687 × 10−7 km−1) in our study is somewhat higher than the values reported in previous studies. Using a data-driven approach, this procedure allows us to present a homogeneous, reliable, and robust dataset for the investigation of CME propagation. Additionally, possible CME events are identified where the DBM prediction is not valid due to model limitations and higher uncertainties in the input parameters. These events require further thorough investigation in the future.

Coronal mass ejections: a driver of severe space weather

Coronal Mass Ejections (CMEs) are explosive large-scale outbursts of the Sun’s coronal
\nplasma and magnetic field. They can induce strong geomagnetic storms at Earth, which
\npose serious threats to space systems, communications and navigation. Hence, arrival predictions
\nof CMEs are of special interest to the humane society. Such predictions require a
\nmeticulous analysis of CME properties in the earliest possible stage. Coronagraph observations
\ncan provide important insights into the CME kinematics, morphology and mass at
\nCME distances of only a few solar radii away from the Sun. However, the 3-dimensional
\nstructure of CMEs can only by analysed, based on their 2-dimensional projection in coronagraph
\nimages, which means that they are affected by projection effects.
\nThis thesis has the goal to present the state-of-the-art methods of CME parameterisation
\nderived from coronagraph observations and to discuss arising issues resulting from projection
\neffects. A focus is laid on the measurements of the CME mass and morphology as well
\nas the question under which conditions they can be determined with highest accuracy. Further,
\nthe solar mass loss caused by CMEs is investigated. Also, CME mass determination
\nis currently not feasible in real-time and therefore not applicable in actual terrestrial CME
\narrival predictions. Thus, it is discussed how the CME mass and the CME morphology
\ncan be empirically estimated from the CME speed.
\nThe thesis presents a new combined method which enables the measurement of relevant
\nCME kinematics, morphology and mass in a consistent and comparable manner. The two
\nvantage points of the COR2 coronagraphs onboard of the twin NASA STEREO spacecraft
\nare used to apply the method to a set of 122 events with intense brightness. The modelling
\nresults are analysed to derive empirical correlations with the CME speed. Further, a CME
\npropagation model – the Drag-Based Model (DBM) – is combined with the GCS model to
\npredict the CME arrival of a sample event at Earth.
\nIt is shown that the largest CME parameterisation uncertainties arise for events emerging
\nfrom close to the disk centre towards or away from the observer. For these events the term
\n”disk events“ is adopted. If an event is seen as disk event in both coronagraphs, the CME
\nmorphology can be overestimated by up to a factor of two from stereoscopical modelling.
\nEqually the CME mass of disk events can be overestimated by a factor of 10 and more in
\nthe case of overlapping coronal streamers. Therefore, stereoscopical measurements of disk
\nevents are not always reliable, at least under a very active background corona. Though, the
\nCME mass M can be estimated from the initial apex velocity vapex with the empirically
\nderived equation
\nlog10(M) = 3.4* 10^(-4) v_apex + 15.479.
\nThis result is used to predict the terrestrial CME arrival of a CME with an Earth-directed
\ninitial speed of 1172 km/s with the GCS plus DBM model. The CME arrival time and the
\narrival speed are both strongly affected by the solar wind density and CME mass. For the
\npresented case the arrival prediction limits spread to DeltaT = 59 h and Deltav = 748 km/s for
\ntypical CME mass and solar wind values. It is demonstrated that the derived empirical
\nequation can be very valuable to improve the arrival prediction accuracy.

<p>Coronal mass ejections (CMEs) typically cause the strongest geomagnetic storms so a major focus of space weather research has been predicting the arrival time of CMEs. Most arrival time models fall into two categories: (1) drag-based models that integrate the drag force between a simplified CME structure and the background solar wind and (2) full magnetohydrodynamic (MHD) models. Drag-based models typically are much more computationally efficient than MHD models, allowing for ensemble modeling. While arrival time predictions have improved since the earliest attempts,both types of models currently have difficulty achieving mean absolute errors below 10 hours. Here we use a drag-based model ANTEATR to explore the sensitivity of arrival times to various input parameters. We consider CMEs of different strengths from average to extreme size, speed, and mass (kinetic energies between 9x10^29 and 6x10^32 erg). For each scale CME we vary the input parameters to reflect the current observational uncertainty in each and determine how accurately each must be known to achieve predictions that are accurate within 5 hours. We find that different scale CMEs are the most sensitive to different parameters. The transit time of average strength CMEs depends most strongly on the CME speed whereas an extreme strength CME is the most sensitive to the angular width. A precise CME direction is critical for impacts near the flanks, but not near the CME nose. We also show that the Drag Based Model has similar sensitivities, suggesting that these results are representative for all drag-based models.</p><p> </p>

The Effective Acceleration Model (EAM) predicts the Time-of-Arrival (ToA) of the Coronal Mass Ejection (CME) driven shock and the average speed within the sheath at 1 AU. The model is based on the assumption that the ambient solar wind interacts with the interplanetary CME (ICME) resulting in constant acceleration or deceleration. The upgraded version of the model (EAMv3), presented here, incorporates two basic improvements: (a) a new technique for the calculation of the acceleration (or deceleration) of the ICME from the Sun to 1 AU and (b) a correction for the CME plane-of-sky speed. A validation of the upgraded EAM model is performed via comparisons to predictions from the ensemble version of the Drag-Based model (DBEM) and the WSA-ENLIL+Cone ensemble model. A common sample of 16 CMEs/ICMEs, in 2013-2014, is used for the comparison. Basic performance metrics such as the mean absolute error (MAE), mean error (ME) and root mean squared error (RMSE) between observed and predicted values of ToA are presented. MAE for EAM model was 8.7$\pm$1.6 hours while for DBEM and ENLIL was 14.3$\pm$2.2 and 12.8$\pm$1.7 hours, respectively. ME for EAM was -1.4$\pm$2.7 hours in contrast with -9.7$\pm$3.4 and -6.1$\pm$3.3 hours from DBEM and ENLIL. We also study the hypothesis of stronger deceleration in the interplanetary (IP) space utilizing the EAMv3 and DBEM models. In particularly, the DBEM model perform better when a greater value of drag parameter, of order of a factor of 3, is used in contrast to previous studies. EAMv3 model shows a deceleration of ICMEs at greater distances, with a mean value of 0.72 AU.

<p>Halo coronal mass ejections (CMEs) are one of the most effective drivers of intense geomagnetic storms. Despite the recent advances in space weather forecasting, the accurate arrival prediction of halo CMEs remains a challenge.  This is because in general CMEs interact with the background solar wind during their propagation in the interplanetary space. In addition, in the case of halo CMEs, the accurate estimation of their kinematics is difficult due to projection effects in the plane-of-sky.</p><p>In this study, we are revisiting the arrival of twelve geoeffective Earth-directed fast halo CMEs using an empirical and a numerical approaches. For this purpose we refine the input to the Drag-based Model (DBM) and to the EUropean Heliospheric Forecasting Information Asset (EUHFORIA), which are recently available for users from the ESA Space Situational Awareness Portal (http://swe.ssa.esa.int).</p><p>The DBM model has been tested using different values for the input drag parameter.  On average, the predicted arrival times are confined in the range of ± 10 h. The closest arrival to the observed one has been achieved with a drag value higher than the recommended for fast CMEs. Setting a higher drag also helped to obtain a closer to the observed CME arrival speed prediction. These results suggest that the exerted solar wind drag was higher than expected. Further, we are searching for clues about the CME propagation by performing EUHFORIA runs using the same CME kinematics. Preliminary results show that both models perform poorly for CMEs that have possibly undergone CME-CME interaction, underlying again the importance of taking into account the state of the interplanetary space in the CME forecast.</p>

<p><span>Drag-Based Ensemble Model (DBEM) is a probabilistic model for heliospheric propagation of Coronal Mass Ejections (CMEs) that predicts the CME hit chance, most probable arrival times and speeds, quantify the prediction uncertainties and calculate the confidence intervals. DBEM is based on the 2D analytical Drag-based Model (DBM) with very short computational time. By using CME cone geometry with flattening DBM calculates the CME arrival time and speed at Earth or any other given target in the solar system. DBEM considers the variability of model input parameters by making an ensemble of n different input parameters to obtain the distribution and significance of the DBM results. As an important tool for space weather forecasters, DBM/DBEM web application is integrated as one of the ESA Space Situational Awareness portal services (</span><span>https://swe.ssa.esa.int/current-space-weather</span>). Important requirement to perform DBM calculations is to assume that two input parameters namely background solar wind speed and the drag parameter γ are constant in order to have the analytical solution and fast computational times. However, this assumption is not always valid in more complex heliospheric conditions. Thus, to further increase the accuracy of CME propagation forecast we developed the new DBEMv4 version that calculates CME propagation in more steps with variable solar wind speeds. <span>This allows also to employ as DBEMv4 input the dynamic solar wind data in real-time taken from simple persistence model under consideration of the CME propagation direction.</span></p>

The propagation of geoeffective fast halo coronal mass ejections (CMEs) from solar cycle 24 has been investigated using the European Heliospheric Forecasting Information Asset (EUHFORIA), ENLIL, Drag‐Based Model (DBM) and Effective Acceleration Model (EAM) models. For an objective comparison, a unified set of a small sample of CME events with similar characteristics has been selected. The same CME kinematic parameters have been used as input in the propagation models to compare their predicted arrival times and the speed of the interplanetary (IP) shocks associated with the CMEs. The performance assessment has been based on the application of an identical set of metrics. First, the modeling of the events has been done with default input concerning the background solar wind, as would be used in operations. The obtained CME arrival forecast deviates from the observations at L1, with a general underestimation of the arrival time and overestimation of the impact speed (mean absolute error [MAE]: 9.8 ± 1.8–14.6 ± 2.3 hr and 178 ± 22–376 ± 54 km/s). To address this discrepancy, we refine the models by simple changes of the density ratio (dcld) between the CME and IP space in the numerical, and the IP drag (γ) in the analytical models. This approach resulted in a reduced MAE in the forecast for the arrival time of 8.6 ± 2.2–13.5 ± 2.2 hr and the impact speed of 51 ± 6–243 ± 45 km/s. In addition, we performed multi‐CME runs to simulate potential interactions. This leads, to even larger uncertainties in the forecast. Based on this study we suggest simple adjustments in the operational settings for improving the forecast of fast halo CMEs.

Context. One of the most widely used space weather forecast models to simulate the propagation of coronal mass ejections (CMEs) is the analytical drag-based model (DBM). It predicts the CME arrival time and speed at Earth or at a specific target (planets, spacecraft) in the Solar System. The corresponding drag-based ensemble model (DBEM) additionally takes into account the uncertainty of the input parameters by making n ensembles and provides the most probable arrival time and speed as well as their uncertainty intervals. An important input parameter for DBM and DBEM is the drag parameter γ, which depends on the CME cross-section and mass, as well as the solar-wind density. Aims. The reverse-modeling technique applied to the DBM allows us to derive γ values that minimize transit time (TT) and arrival-speed (vtar) errors. The present study highlights the limitations and constraints of such a procedure. Methods. We searched for optimal γ values that would yield the perfect TT within one hour of the actual observed CME transit time as well as perfect vtar within ±75 km s−1. This optimal window for vtar was found by increasing vtar from ±10 to ±100 km s−1, where the ±75 km s−1 window gave the perfect TT and vtar in the case of 87% of CMEs compared to the ±10 km s−1 window, which was used in some previous reverse-modeling studies and gave optimal results for only 45% of the events from our CME list. For our analysis, a 31 CME-ICME pair sample is used from the period spanning 1997–2018. The reverse-modeling method is applied using the DBEMv3 tool for different γ ranges from 0.01 to 10 × 10−7 km−1. We tested whether and how the obtained optimal γ depends on the chosen γ range. Results. By increasing the γ range, we find that the optimal γ converges to a certain value for two thirds of the analyzed events. The highly constrained γ ranges resulted in shifted and skewed γ distributions. By using the largest γ range (0.01–10 × 10−7 km−1), the medians of the optimal γ distributions are obtained for two thirds of the events in the common operational DBEMv3 range of 0.01–0.5 × 10−7 km−1. We also found that the important quantity in determining the range of γ distribution and ability to find an optimal γ is the difference between the CME launch speed and the solar-wind speed (v0 − w), which together with γ define the drag acceleration in the DBM. For small v0 − w differences (e.g., < 200 km s−1), the reverse modeling may not be the appropriate method to find the optimal γ due to large divergence of γ values found, which may additionally be caused by larger input uncertainties and physical model limitations in turn leading to inappropriate γ values.

Forecasting the arrival of coronal mass ejections (CMEs) is vital for protecting satellites, power systems, and human spaceflight. We present the Heliospheric Observer for Predicting CME Arrival via Nonlinear Drag Assimilation (HELIOPANDA), a framework that integrates the drag-based model (DBM) with spacecraft observations using iterative parameter estimation and Kalman filter assimilation. We introduce a method for estimating the solar wind speed w and drag parameter γ , two key but usually unknown quantities controlling CME propagation, through direct solutions of the DBM equations. We tested the method on 4480 synthetic CME profiles spanning CME speeds of 200–3500 km s −1 , solar wind speeds of 250–800 km s −1 , and drag parameters of 0.1–1.0 × 10 −7 km −1 . The results demonstrate that the framework provides accurate reconstructions of the DBM input parameters, providing a solid basis for in situ and remote-sensing applications. By testing a single virtual spacecraft positioned at nine distances along the Sun–Earth line, HELIOPANDA achieved arrival-time errors as low as 0.6 hr for a 600 km s −1 CME and 1 hr for a 2500 km s −1 CME when the spacecraft was located 30 million km from the Sun. We developed a Kalman filter framework to assimilate noisy heliospheric data into the DBM, enabling recursive updates of CME kinematics and robust estimates of w and γ , and yielding Earth and Mars arrival-time predictions within 1–2 hr using 160 simulated hourly measurements. By combining DBM, parameter recovery, and data assimilation, HELIOPANDA provides a pathway to real-time, multipoint CME forecasts, suited to observations from Solar Orbiter, Parker Solar Probe, PUNCH, and planned L4/L5 missions.

We analyze the interaction between an interplanetary coronal mass ejection (ICME) detected in situ at the L1 Lagrange point on 2016 October 12 with a trailing high-speed stream (HSS). We aim to estimate the region in the interplanetary (IP) space where the interaction happened/started using a combined observational-modeling approach. We use minimum variance analysis (MVA) and the Walen test to analyze possible reconnection exhaust at the interface of ICME and HSS. We perform a graduated cylindrical shell reconstruction of the CME to estimate the geometry and source location of the CME. Finally, we use a two-step drag-based model (DBM) model to estimate the region in IP space where the interaction took place. The magnetic obstacle observed in situ shows a fairly symmetric and undisturbed structure and shows the magnetic flux, helicity, and expansion profile/speed of a typical ICME. The MVA together with the Walen test, however, confirms reconnection exhaust at the ICME–HSS boundary. Thus, in situ signatures are in favor of a scenario where the interaction is fairly recent. The trailing HSS shows a distinct velocity profile which first reaches a semi-saturated plateau with an average velocity of 500 km s−1 and then saturates at a maximum speed of 710 km s−1. We find that the HSS's interaction with the ICME is influenced only by this initial plateau. The results of the two-step DBM suggest that the ICME has started interacting with the HSS close to Earth (∼0.81 au), which compares well with the deductions from in situ signatures.

<p>ICME (Interplanetary Coronal Mass Ejection) are violent phenomena of solar activity that affect the whole heliosphere and the prediction of their impact on different solar system bodies is one of the primary goals of the planetary space weather forecasting. The travel time of an ICME from the Sun to the Earth can be computed through the Drag-Based Model (DBM), which is based on a simple equation of motion for the ICME defining its acceleration as a=-Γ(v-w)v-w, where a and v are the CME acceleration and speed, w is the ambient solar-wind speed and Γ is the so-called drag parameter (Vršnak et al., 2013).<br>In this framework, Γ depends on the ICME mass and cross-section, on the solar-wind density and, to a lesser degree, on other parameters. The typical working hypothesis for DBM implies that both Γ and w are constant far from the Sun. To run the codes, forecasters use empirical<br>input values for Γ and w, derived by pre-existent knowledge of solar-wind condition and by solving the “inverted problem” (where the ICME travel time is known and the unknowns are Γ and/or w). In<br>the 'Ensemble' approaches (Dumbovich et al., 2018; Napoletano et al. 2018), the uncertainty about the actual values of such inputs are rendered by Probability Distribution Functions (PDFs), accounting for the values variability and our lack of knowledge. Among those PDFs, that of Γ is poorly defined due to the relatively scarce statistics of recorded values. </p><p>Employing a list of past ICME events, for which initial conditions when leaving the Sun and arrival conditions at the Earth are known, we employ a statistical approach to the Drag-Based Model to determine a measure of Γ and w for each case. This allows to obtain distributions for the model parameters on experimental basis and, more importantly, to test whether different conditions of relative velocity to the solar wind influence the value of the drag efficiency, as it must be expected for solid objects moving into an external fluid. In addition, we perform numerical simulations of a solid ICME-shaped structure moving into the solar-wind modelled as an external fluid. Outcomes from these simulations are compared with our experimental results, and thus employed to interpret them on physical basis.</p>