Indonesian J our nal of Electrical Engineering and Computer Science V ol. 42, No. 3, June 2026, pp. 865 ∼ 874 ISSN: 2502-4752, DOI: 10.11591/ijeecs.v42.i3.pp865-874 ❒ 865 A computational framew ork f or detection, classication, and visualization of magnetic nulls in multi-spacecraft obser v ations Sri Ekawati 1,2 , Dongsheng Cai 2,3 , Hir oyuki K udo 2 1 Research Center for Climate and Atmosphere, National Research and Inno v ation Agenc y (BRIN), Bandung, Indonesia 2 Department of Computer Science, Uni v ersity of Tsukuba, Tsukuba, Japan 3 Nago ya Uni v ersity of Commerce and Business (NUCB), Nago ya, Japan Article Inf o Article history: Recei v ed Oct 15, 2025 Re vised Mar 17, 2026 Accepted May 26, 2026 K eyw ords: Magnetic null detection MMS mission T opological classication Computational pipeline Plasma ph ysics ABSTRA CT Magnetic nulls, dened as locations where the magnetic eld magnitude be- comes zero, are theoretically well dened ho we v er practically dif c ult to lo- cate, v alidate, and interpret. T o address these challenges, this paper introduces a no v el, modular , and fully reproducible automated frame w ork for magnetic null detection, classication, and visualiza tion based on multi-spacecraft ob- serv ations. The proposed frame w ork consists of tw o open-source modules: an automated data ingestion and null detection module, and a topological classi- cation and three-dimensional visualization module. Magnetic nulls are de- tected by combining eigen v alue analysis of the magnetic eld gradient tensor with tetrahedron-based geometric v alidation, enabling both numerical stability assessment and ph ysical consistenc y che cks. Meanwhile, detected nulls are clas- sied into radial (T ype A, B) and spiral (T ype As, Bs) topologies, and their lo- cal magnetic structures are visualized through reconstructed three-dimensional magnetic eld lines. The main contrib ution of this w ork is the tight inte gra- tion of detection, numerical v alidation, classication, and visualization within a single end-to-end pipeline, ensuring consistenc y between computational output and ph ysic al interpretation. The frame w ork is v alidated using four pre viously reported ele ctron dif fusion re gion (EDR) e v ents and one storm-time substorm e v ent. The detected null times closely agree with the reported EDR interv als, with se v eral e v ents sho wing sub-second dif ferences. Among all detected can- didates, three nulls satisfy strict numerical v alidity criteria, including a T ype A null, a T ype As null, and a T ype Bs null. This is an open access article under the CC BY -SA license . Corresponding A uthor: Sri Eka w ati Department of Computer Science, Graduate School of Science and T echnology , Uni v ersity of Tsukuba 1-1-1 T ennodai, Tsukuba, Ibaraki 305-8573, Japan Email: sri.eka w ati@ca v elab .cs.tsukuba.ac.jp 1. INTR ODUCTION One of the important problems in space plasma ph ysics is ho w the solar wind interacts with the ear th’ s magnetosphere [1], as illustrated in Figure 1. This interact ion occurs at the magnetopause [2], [3], which is a comple x boundary as a transition re gion between the solar wind and the earth’ s magnetosphere. The magne- topause is the boundary re gion where the solar wind, a continuous stream of char ged particles from the Sun, interacts directly with the Earth’ s intrinsic magnetic eld [4]-[6]. At the magnetopause, changes in the solar J ournal homepage: http://ijeecs.iaescor e .com Evaluation Warning : The document was created with Spire.PDF for Python.
866 ❒ ISSN: 2502-4752 wind pressure and interplanetary magnetic eld orientation frequently trigger reconnection e v ents, enabling solar wind plasma and ener gy to enter the magnet o s phere [1], [6], [7]. These processes are closely link ed to magnetospheric dynamics, including the onset of storms and substorms, which ultimat ely af fect space weather near Earth. Magnetic reconnection is thought to originate and persist within a conned, electron-scale re gion called the el ectron dif fusion re gion (EDR) [2], [3], [8], [9] where electrons become demagnetized [7], [10] uncoupled from the magnetic eld [4]-[6]. In astroph ysical plasma, such as the solar corona [11], coronal mass ejections [12], solar are [13], stellar ares [14], and solar -terrestrial plasma, especially in magnetospheric sub- storms [15], [16], magnetic reconnection is a crucial process. Distincti v e topological features naturally arise in magnetic reconnection re gions, where the magnetic eld s trength v anishes [17], [18] the local magnetic topol- ogy becomes highly structured [19] commonly referred to as magnetic nulls [20]-[22]. As inte gral elements of reconnection geometry , magnetic null plays an important role i n ener gy con v ersion and particle acceleration f acilitating the breaking and rejoining of magnetic eld lines into a ne w conguration [23], [24]. Figure 1. Illustration of the solar wind interaction with the Earth’ s magnetosphere, sho wing magnetic reconnection re gions [4], [6] at the dayside magnetopause and nightside magnetotail Magnetic nulls are theoretically well-dened, ho we v er , the y remain practica lly c hallenging to l ocate, v alidate, and interpret within three-dimensional space plasma en vironments due to sensiti vity to spacecraft ge- ometry , numerical instability , and magnetic eld uctuations. Furthermore, magnetic nulls and the magnetic eld topology surrounding the null also need visualization, as ef fecti v e visualization plays a crucial role in in- terpreting comple x, multidimensional scientic data [25], particularly in space plasma ph ysics where magnetic topologies must be represented in three dimensions [26]. Accordingly , this study aims to de v elop an automated and reusable frame w ork for magnetic null detection and visualization. Therefore, the main contrib utions of this frame w ork are summarized as follo ws: − An adapti v e barycentric-based magnetic null detection algorithm that restricts localization to the nearest null within the multi-spacecraft tetrahedron,compared with e xisting approaches [1], [17], [27], [28], a v oiding distant or e xtrapolated null candidates, impro ving geometric reliability and reducing f alse positi v es. − A fully automated frame w ork that eliminates manual parameter tuning required in pre vious studies [3]. − Ph ysical v alidati on criteria, including magnetic eld re v ersal and e xplicit numerical criteria to ensure ph ys- ically consistent solutions, were not implemented in pre vious studies [3]. − Three-dimensional visualization of the local magnetic topology applied to v e e v ents (the pre vious study [3] included only one e v ent and no visualization). 2. PR OPOSED COMPUT A TION AL FRAMEW ORK In this study , we introduce an adapti v e null -detection method that automatically adjusts search pa- rameters and incorporates magnetic eld re v ersal v alidation and e xplicit numerical criteria, which ha v e not been addressed in pre vious studies [3], enabling reuse across dif ferent e v ents or datasets. In addition, three- dimensional visualization is inte grated, and the o v erall w orko w is sho wn in Figure 2. The frame w ork is or g anized into tw o modular MA TLAB components: Indonesian J Elec Eng & Comp Sci, V ol. 42, No. 3, June 2026: 865–874 Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian J Elec Eng & Comp Sci ISSN: 2502-4752 ❒ 867 1 Null detection module [29], which performs temporal synchronization of the four -spacecraft data, ingests magnetic eld and position measurements, detects magnetic null locations and times using barycentric interpolation with adapti v e thresholds, v alidates eld re v ersal, and stores the detected candidates. 2 Null classication and visualization module [30], which computes the magnetic gradient tensor ∇ B , clas- sies null topology via eigen v alue analysis, applies numerical v alidation criteria, and generates three- dimensional visualizations of the local magnetic structure around each null. Figure 2. Flo wchart of the proposed computational frame w ork for automatic magnetic null detection, classication, and visualization 3. METHOD 3.1. Multi-spacecraft data The frame w ork le v erages observ ations from N ASA ’ s magnetospheric multiscale (MMS) mission sat el- lite, launched on March 12, 2015, which consists of four identically instrumented spacecraft ying in a tetra- hedral formation. This conguration enables simultaneous measurements at four distinct spatial locations, pro viding the minim um requirements for reconstructing three-dimensional magnetic eld gradients and ac- curately determining the position and topology of magnetic nulls. The MMS data pro vide publicly a v ailable high-resolution plasma and magnetic eld measurements through the science data center (SDC) at https: //lasp.colorado.edu/mms/sdc/public/ . The MMS data are originally distrib uted in CDF format, ho we v er , the datasets used in this study were preprocessed and con v erted into MA TLAB-ready ASCII les for ef cient analysis. All processed datasets corresponding to the screened e v ents are openly a v ailable on Zenodo [31]. This s tudy analyzes v e e v ents: four pre viously reported EDR e v ents (E1–E4)—E1 (2015-09-19 07:43:30 UTC) [2], [3], [9], E2 (2015-10-16 13:07:02 UTC) [2], [9], E3 (2015-10-22 06:05:22 UTC) [2], [9], and E4 (2017-08-10 12:18:33 UTC) [32], and one geom agnetic storm/substorm e v ent (S1, 2017-05-28) [33]. In total, a substantially lar ger collection of MMS interv als w as processed and screened using the pro- posed automated pipeline; ho we v er , only these v e e v ents [29] were detected. 3.2. Null detection Before detecting magnetic null points, the con v erted ASCII les underwent additional preproce ssing to ensure consistenc y across the four spacecraft. T emporal synchronization w as applied to guarantee that all MMS data streams were aligned within a tolerance of ϵ = 0 . 001 s, while timestamps were further adjusted t o the UTC reference using cross-correlation with ϵ = 0 . 01 s tolerance, and an y non-o v erlapping interv als were A computational fr ame work for detection, classication, and visualization of ma gnetic ... (Sri Ekawati) Evaluation Warning : The document was created with Spire.PDF for Python.
868 ❒ ISSN: 2502-4752 discarded. A custom MA TLAB routine w as de v eloped to check synchronization, in which the detection time w as considered v alid only if all four spacecraft matched within 1 ms, with MMS1 serving as the reference clock. Data cleaning w as also performed by remo ving in v alid measurements, such that the cleaned dataset w as formally dened as D clean = { B i , r i | B i ̸ = NaN , r i ̸ = NaN } , ensuring that only v alid magnetic eld v ectors B i and spacecraft positions r i were retained for subsequent null detection analysis. The automatic detection of magnetic null points w as carried out using an adapti v e threshold e xpansion method. The search w as initialized with parameters a = 0 , b = 1 , and step size δ = 0 . 01 , and the boundaries were progressi v el y e xpanded up to max expand = 7, which allo wed the algorithm to search be yond the initial MMS tetrahedron. At each iteration, the magnetic eld v ectors B i and spacecraft positions r i were e xtracted from the four MMS spacecraft. These relati v e magnetic eld v ectors were then emplo yed to construct the matrix A , which serv es as the basis for estimating the null point l o c ation through a system of linear equations. The null is obtained by solving A [ s, t, u ] T = − B 1 , where A = [ B 2 − B 1 , B 3 − B 1 , B 4 − B 1 ] T , using the approach pre viously reported in [3]. By solving this system for the coef cients ( s, t, u ) , the fourth barycentric coef cient is obtained as d = 1 − s − t − u . When the conditions s, t, u ∈ [ a, b ] and s + t + u ≤ b are satised, the null position r null can be determined as a weighted combination of the four spacecraft posi tions. Once a candidate null w as found within the specied tolerance, its position and properties were recorded for further analysis. After a candidate null point is detected, it under goes a series of checks to conrm its ph ysical and geometric plausibility . The magnetic eld magnitudes ∥ B i ∥ are e v aluated for each MMS spacecraft and the signs of the B x , B y , and B z components are e xamined across all four spacecraft, as a v alid null gen- erally coincides with a re v ersal of the magnetic eld in at least one component, indicating a true change in eld direction. T o assess the spatial conguration, the inter -satellite distances d ij = ∥ r i − r j ∥ and the dis- tances from the null to each spacecraft d null-MMS = ∥ r i − r null ∥ are calculated, and the centroid of the tetra- hedron is computed as r centroid = 1 4 ( r 1 + r 2 + r 3 + r 4 ) to ensure that the null lies reasonably within the observ ation v olume. 3.3. Null classication The detected candidate null i s classied from the eigen v alues of the magnetic gradient tensor ∇ B [1], [3], [17], [19], [24], [34], enabling identication of radial or spiral three-dimensional topologies, as summarized in T able 1. At the detected candidate null, the local magnetic eld is linearly approximated as B ( r ) ≈ ∇ B ( r − r 0 ) , where the magnetic gradient tensor ∇ B ∈ R 3 × 3 is obtained by solving a full 9 × 9 multi-point linear system constructed from the four MMS spacecraft measurements. The eigen v alues λ i = eig ( ∇ B ) are then computed to characterize the local topology . Ph ysical consistenc y is enforced using the di v er gence-free constraint η = tr( ∇ B ) = λ 1 + λ 2 + λ 3 ≈ 0 together with the normalized metric p x = | η | / ( | λ 1 | + | λ 2 | + | λ 3 | ) , and only solutions satisfying | η | < ϵ 1 and p x < ϵ 2 are retai ned. The null type is subsequently determined from the eigen v alue signs: tw o ne g a ti v e and one positi v e eigen v alue indicate type A, tw o positi v e and one ne g ati v e indicate type B, while the presence of a comple x conjug ate pair corresponds to spiral types (As/Bs). T able 1. Classication of magnetic nulls based on eigen v alues of ∇ B [18], [34] λ 1 λ 2 λ 3 Null type Dimension/Structure Labelling 0 + λ − λ X 2-D X 0 + iλ − iλ O 2-D ⃝ − λ 1 − λ 2 +( λ 1 + λ 2 ) A 3-D (radial) △ + λ 1 + λ 2 − ( λ 1 + λ 2 ) B 3-D (radial) ▽ + λ 1 − λ 1 2 + iλ 2 − λ 1 2 − iλ 2 As 3-D (spiral) ▲ − λ 1 + λ 1 2 + iλ 2 + λ 1 2 − iλ 2 Bs 3-D (spiral) ▼ 3.4. Null visualization The visualization of the magnetic topology is achie v ed through a 3D streamline plot generated around the null point. T o characterize the local magnetic eld structure, a rst-order linear approximation is emplo yed. Gi v en the small spatial scales under consideration, t he eld is modeled as B model ( r ) = ∇ B · ( r − r null ) , where ∇ B is the gradient tensor deri v ed from the four MMS measurements and r denotes the position v ector relati v e to the null. This reconstructed eld is then e v aluated on a re gular 3D grid centered at r null , and the eld lines are numerically inte grated to produce the stre amlines. In MA TLAB, the streamlines are rendered together with the MMS tetrahedron and color -coded according to the null type, pro viding an intuiti v e representation of the local Indonesian J Elec Eng & Comp Sci, V ol. 42, No. 3, June 2026: 865–874 Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian J Elec Eng & Comp Sci ISSN: 2502-4752 ❒ 869 3D magnetic structure. A re gular 3D grid of seed points is constructed, typically forming a cube of side length L centered at r null , where L is proportional to the a v erage inter -satellite distance of the MMS tetrahedron. The model eld B model ( r ) is computed at each grid point, and eld lines are inte grated numerically using standard streamline inte gration. In MA TLAB, the visuali zation is performed by constructing a re gular 3D gri d centered at the null point, computi ng the local magnetic eld at each grid point as B local = ∇ B · ( r − r null ) , coloring the eld lines according to the null type, and o v erlaying the MMS tetrahedron to pro vide spatial conte xt. Streamlines are inte grated numerically to represent the local magnetic topology of fering an intuiti v e depiction of the 3D eld structure near each null. This visual ization pro vides an intuiti v e representation of magnetic eld lines near each null, f acilitating the interpretation of local 3D magnetic structures. 4. RESUL TS The detection frame w ork identied magnetic nulls in all v e analyzed e v ents. The computa tional characteristics for each e v ent are summarized in T able 2. The number of iterations required for con v er gence v aries across e v ents. Ev ent E1 (2015-09-19) con v er ged aft er a single iteration using the initial search parameters ( a, b ) = (0 . 00 , 1 . 00) . Ev ent E2 (2015-10-16) and Ev ent E3 (2015-10-22) required 71 and 150 iterations, respecti v ely . Ev ent E4 (2017-08-10) required 594 itera tions. The special case S1 (2017-05-28) con v er ged after 15 iterations wi th 365 synchronized time st eps. The dif ferences between the detected null times and the reported EDR interv als are less than one second for Ev ents E1, E2, and E4, while Ev ent E3 sho ws an of fset of approximately six seconds. The detected null times for all e v ents are listed in T able 2. T able 2. Computational performance and null detection time compared with reported EDR interv als Ev ent Date Iteration Final (a,b) T ime steps Null time (UTC) EDR time (UTC) Reference E1 2015-09-19 1 (0.00, 1.00) 1999 07:43:30.840 07:43:30 [2], [9] E2 2015-10-16 71 (-0.70, 1.70) 4741 13:07:02.490 13:07:02 [2], [9] E3 2015-10-22 150 (-1.49, 2.49) 374 06:05:16.779 06:05:22 [2], [9] E4 2017-08-10 594 (-5.93, 6.93) 1112 12:18:33.423 12:18:32 [32] S1 2017-05-28 15 (-0.15, 1.15) 365 05:21:25.583 – Not reported The geometric characteristics of the detected nulls are summarized in T able 3. The MMS separat ion represents the characteristic size of the spacecraft tetrahedron, while the null–centroid distance and the nor - malized ratio ( d/R ) describe the relati v e position of the nu l l with respect to the formation. Ev ent E1 e xhibits barycentric coordinates within the range 0 < s, t, u < 1 , indicating that the null lies inside the tetrahedron. In contrast, E2 sho ws one ne g ati v e parameter and a ratio close to unity , placing the null near the boundary . Ev ents E3 and E4 sho w lar ger normalized distances and ratios e xceeding one, indicating locations f arther from the tetrahedron. The special case S1 yields a relati v ely small ratio ( ≈ 0 . 37 ) with one ne g ati v e parameter , corresponding to a near -boundary position. T able 3. Geometric parameters and relati v e position of detected nulls with respect to the MMS tetrahedron Ev ent Date P arameter MMS separation Null–Centroid Ratio Location s t u (km) (km) (d/R) E1 2015-09-19 0.051 0.625 0.306 71.662 28.892 0.40 Inside E2 2015-10-16 1.448 0.253 -0.699 13.784 16.122 1.17 Near E3 2015-10-22 1.096 1.041 0.345 16.967 29.262 1.72 F ar E4 2017-08-10 3.810 -5.921 1.305 20.548 73.221 3.56 F ar S1 2017-05-28 0.151 0.313 -0.144 59.228 22.142 0.37 Near T able 4 summarizes the a v erage magnetic eld magnitude and the re v ersal check across the three components for each e v ent. Ev ent E1 (2015-09-19) sho ws a relati v ely strong mean eld of 25.249 nT with re v ersals detected in all three components. Ev ent E2 (2015-10-16) e xhibits a weak er mean eld of 5.502 nT ; no re v ersal is observ ed in B x , while B y and B z display sign changes. Ev ent E3 (2015-10-22) presents a stronger mean eld of 22.246 nT with no re v ersal in an y component. Ev ent E4 (2017-08-10) is characterized by a weak mean eld of 3.497 nT with a single re v ersal observ ed in B y , while B x and B z remain monotonic. Finally , the special case S1 (2017-05-28) sho ws the lo west mean eld of 0.859 nT with re v ersals present in all three components. A computational fr ame work for detection, classication, and visualization of ma gnetic ... (Sri Ekawati) Evaluation Warning : The document was created with Spire.PDF for Python.
870 ❒ ISSN: 2502-4752 T able 4. Magnetic eld statistics and re v ersal checks for each e v ent Ev ent Date Mean | B | (nT) B x Re v ersal B y Re v ersal B z Re v ersal E1 2015-09-19 25.249 Detected ([-1 1 -1 1]) Detected ([-1 -1 -1 1]) Detected ([-1 -1 -1 1]) E2 2015-10-16 5.502 No re v ersal ([-1 -1 -1 -1]) Detected ([1 -1 -1 -1]) Detected ([1 -1 -1 -1]) E3 2015-10-22 22.246 No re v ersal ([1 1 1 1]) No re v ersal ([1 1 1 1]) No re v ersal ([1 1 1 1]) E4 2017-08-10 3.497 No re v ersal ([1 1 1 1]) Detected ([-1 1 1 1]) No re v ersal ([-1 -1 -1 -1]) S1 2017-05-28 0.859 Detected ([-1 -1 1 -1]) Detected ([-1 -1 1 1]) Detected ([1 -1 -1 -1]) T able 5 summarizes the classication outcomes of the detected nulls for the v e analyzed e v ents (E1– E4 and S1). Three e v ents (E1, E2, and S1) satisfy the v alidation criteria and are classied as v alid nulls, while the remaining tw o e v ents (E3 and E4) e xceed the prescribed thresholds and are therefore classied as in v alid. Among the v alidated cases, both radial and spiral topologies are observ ed, indicating v ariability in the local magnetic eld structure across e v ents. T able 5. Eigen v alues of ∇ B at detected nulls, v alidated via the η = λ 1 + λ 2 + λ 3 and P x = η / max( | λ i | ) . Nulls are considered v alid if | η | < 0 . 1 and | P x | < 0 . 3 , and cla ssied according to T able 1 Ev ent Date λ 1 λ 2 λ 3 η P x V alidity T ype E1 2015-09-19 0.5444 -0.3736 -0.2040 -0.0338 -0.0621 V alid A E2 2015-10-16 -0.0415+0.4185i -0.0415-0.4185i 0.1782 0.0952 0.2265 V alid As E3 2015-12-05 -0.4953 0.1092 0.1906 -0.1956 -0.3948 In v alid - E4 2017-08-10 -0.0253 0.0246 0.0135 0.0128 0.5049 In v alid - S1 2017-05-28 -0.0173+0.0000i 0.0066+0.0124i 0.0066-0.0124i -0.0041 -0.2381 V alid Bs Figure 3 presents three-dimensional reconstructions of the magnetic eld topology surrounding the detected nulls for representati v e e v ents. Figure 3(a) sho ws Ev ent E1 (2015-09-19), a T ype A null, featuring a clear radial conguration with eld lines di v er ging along one principal direction and con v er ging along the others. Figure 3(b) illustrates Ev ent E2 (2015-10-16), a T ype As null, where the eld e xhibits a distinct spi- ral pattern indicati v e of rotational topology . Figure 3(c) depicts Ev ent S1 (2017-05-28), a T ype Bs null, also displaying a spiral structure consistent with its eigen v alue-based classication. These three-dimensional vi- sualizations f aithfully reproduce the geometrical dif ferences between radial and spir al nulls, pro viding direct, visual conrmation of the classications reported in T ables 3-5. The reconstructed eld lines , computed from the locally linearized magnetic eld, con vincingly highlight the topological si g na tures unique to each null type. Figure 3. Three-dimensional reconstructions of magnetic eld topology around detected nulls: (a) T ype A (radial, E1), (b) T ype As (spiral, E2), and (c) T ype Bs (spiral, S1). Field lines are computed from the locally linearized magnetic eld and illustrate the distinct topological structures associated with each null type Indonesian J Elec Eng & Comp Sci, V ol. 42, No. 3, June 2026: 865–874 Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian J Elec Eng & Comp Sci ISSN: 2502-4752 ❒ 871 5. DISCUSSION The detected null times clos ely match the reported EDR interv als, often within sub-second dif ference s (Ev ents E1 and E2 sho wn in T ables 2), demonstrating temporal accurac y and conrming that the identied nulls represent ph ysically meaningful reconnection structures rather than numerical artif acts. Dif ferences in iteration counts primarily reect the relati v e position of the null with respect to the spacecraft tetrahedron rather than numerical i nstability: interior nulls con v er ge rapi d l y (Ev ent E1), whereas more distant nulls require substantially more iterations (Ev ent E4), indicating a computationally more demanding search. Nulls located inside or near the tetrahedron are generally better constrained by the four -point measurements and therefore more rob ustly dened (e.g., E1, E2 and S1 sho wn in T ables 3), whereas distant nulls (e.g., E3 and E4 sho wn in T ables 3) are more sensiti v e to interpolation errors and gradient uncertainties. Magnetic eld re v ersals pro vide an important ph ysical v alidation of the detected nulls. Ev ents with re v ersals in all or at least tw o components (E1, E2, and S1 in T able 4) are consistent with true null structures, whereas partial or absent re v ersals (E3–E4) indicate weak er or less reliable candidates. Importantly , the pres- ence of magnetic eld re v ersals across multiple components pro vides a more reliable ph ysical signature of a true null than the absolute eld magnitude. The Eigen v alue analysis of the gradient tensor ∇ B further elucidates the local magnetic topology . Radial and spiral congurations observ ed in Ev ents E1, E2, and S1 agree with theoretical e xpectations for three-dimensional nulls associat ed with reconnection. In contrast, in v alid classications in Ev ents E3 and E4 lik ely arise from asymmetric gradients or increased uncertainty when the null is l o c ated f ar from the spacecraft, highlighting a practical limitation of linear interpolation methods, namely that the accurac y of ∇ B decreases with e xtrapolation distance. Combining geometric constraints with eigen v alue v alidation is therefore essential to minimize f alse positi v es. These geometric indicators, together with magnetic eld re v ersal and eigen v alue analyses, pro vide comprehensi v e v alidation of each detection. Figure 3 further conrms the results through three-dimensional topology reconstructions, where Ev ent E1 sho ws a radial T ype A null, while Ev ents E2 and S1 e xhibit spiral T ype As/Bs structures. The reconstructed eld lines, obtained from the locally linearized magnetic eld, clearly dif ferentiate radial and spiral t opo l ogies and visually conrm the classications reported in T ables 3–5. Ov erall, the agreement between geometric, ph ysical, eigen v alue, and visual analyses, together with the clear separation between radial and spiral patterns, visually supports the classications and reinforces the rob ustness of the proposed frame w ork. 6. CONCLUSION Combining adapti v e numerical search, geometric v alidation, and ph ysical topology analysis pro vides a reliable, computationally ef cient strate gy for automatic magnetic null detection in multi-spacecraft observ a- tions. The frame w ork consistently distinguishes well-constrained nulls from distant or ambiguous candidates, rob ustly characterizes local topology , and of fers practic al utility for reconnection studies. Across all in v esti- g ated e v ents, the frame w ork consistently detected candidate nulls wit h timings closely aligned with reported reconnection and EDR interv als, dem onstrating both numerical rob ustness and ph ysical consistenc y . Geometric and topological v alidations further distinguished well-constrained nulls located inside or near t he MMS tetrahe- dron from mar ginal candidates at lar ger dista nces, while eigen v alue analysis enabled systemat ic classication into radial and spiral types. These results sho w that the proposed approach pro vides not only accurate local- ization b ut also meaningful ph ysical interpretation of the three-dimensional magnetic topology . In summary , the frame w ork of fers an ef cient and scalable tool for aut o m ated magnetic null detection in multi-spacecraft observ ations. Future w ork wil l focus on inte grating direct ingestion of MMS CDF les into the MA TLAB pipeline, applying the frame w ork to additional datasets, implementing Earth Mo v er’ s Distance (EMD)-based methods, and e xtending the frame w ork to w ard multi-null detection using higher -order polynomial interpolation. A CKNO WLEDGMENTS The authors thank the MMS (Magnetospheric Multiscale) mission team for pro viding the data. The authors also thank anon ymous re vie wers for their constructi v e comments. A computational fr ame work for detection, classication, and visualization of ma gnetic ... (Sri Ekawati) Evaluation Warning : The document was created with Spire.PDF for Python.
872 ❒ ISSN: 2502-4752 FUNDING INFORMA TION Authors state no funding in v olv ed. A UTHOR CONTRIB UTIONS ST A TEMENT This journal uses the C o nt rib utor Roles T axonomy (CRediT) to recognize indi vidual author contrib u- tions, reduce authorship disputes, and f acilitate collaboration. Name of A uthor C M So V a F o I R D O E V i Su P Fu Sri Eka w ati ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ Dongsheng Cai ✓ ✓ ✓ ✓ ✓ ✓ Hiro yuki K udo ✓ ✓ ✓ C : C onceptualization I : I n v estig ation V i : V i sualization M : M ethodology R : R esources Su : Su pervision So : So ftw are D : D ata Curation P : P roject Administrati on V a : V a lidation O : Writing - O riginal Draft Fu : Fu nding Acquisition F o : F o rmal Analysis E : Writing - Re vie w & E diting CONFLICT OF INTEREST ST A TEMENT Authors state no conict of interest. D A T A A V AILABILITY The preprocessed datasets corresponding to the v e analyzed e v ents are openly a v ailable on Zenodo at https://doi.org/10.5281/zenodo.16070912 , and all preprocessed MMS datasets associated with the screened interv als used for magnetic null detection are openly a v ailable at https://zenodo.org/ records/18437216 . The original high-resolution data are publicly accessible through the Magnetospheric Multiscale (MMS) mission Science Data Center (SDC) at https://lasp.colorado.edu/mms/sdc/ public/ . REFERENCES [1] E. Eriksson, A. V ai v ads, Y . V . Khotyaintse v , V . M. Khotyayintse v , and M. Andr ´ e, “Statistics and accurac y of magnetic null identication in multispacecraft data, ” Geophysical Resear c h Letter s , v ol . 42, no. 17, pp. 6883–6889, Sep. 2015, doi: 10.1002/2015GL064959. [2] J . M. W ebster et al. , “Magnetospheric multiscale dayside reconnection electron dif fusion re gion e v ents, ” J ournal of Geophysical Resear c h: Space Physics , v ol. 123, no. 6, pp. 4858–4878, Jun. 2018, doi: 10.1029/2018J A025245. [3] S . Eka w ati and D. Cai, “In-situ observ ation of magnetic null on 19 September 2015 e v ent using magnetospheric multiscale mission, ” J ournal of Geophysical Resear c h: Space Physics , v ol. 128, no. 2, Feb . 2023, doi: 10.1029/2021J A029571. [4] J . L. Burch, T . E. Moore, R. B. T orbert, and B. L. Giles, “Magnetospheric multiscale o v ervie w and science objecti v es, ” Space Science Re vie ws , v ol. 199, no. 1–4, pp. 5–21, Mar . 2016, doi: 10.1007/s11214-015-0164-9. [5] J. L. Burch et al. , “Electron-scale measurements of magnetic reconnection in space, ” Science , v ol. 352, no. 6290, Jun. 2016, doi: 10.1126/science.aaf2939. [6] Q. Lenouv el et al. , “Identication of electron dif fusion re gions with a machine learning approach on MMS data at the earth’ s magnetopause, ” Earth and Space Science , v ol. 8, no. 5, May 2021, doi: 10.1029/2020EA001530. [7] J. L. Burch and T . D. Phan, “Magnetic reconnection at the dayside magnetopause: Adv ances with MMS, ” Geophysical Resear c h Letter s , v ol. 43, no. 16, pp. 8327–8338, Aug. 2016, doi: 10.1002/2016GL069787. [8] S. A. Fuselier and W . S. Le wi s, “Properties of near -earth magnetic reconnection from in-situ observ ations, ” Space Science Re vie ws , v ol. 160, no. 1–4, pp. 95–121, Oct. 2011, doi: 10.1007/s11214-011-9820-x. [9] S. A. Fuselier et al. , “Lar ge-scale characteristics of reconnection dif fusion re gions and associated magnetopause crossings observ ed by MMS, ” J ournal of Geophysical Resear c h: Space Physics , v ol. 122, no. 5, pp. 5466–5486, May 2017, doi: 10.1002/2017J A024024. [10] T . D. Phan et al. , “Electron magnetic reconnection without ion coupling in Earth’ s turb ulent magnetosheath, ” Natur e , v ol. 557, no. 7704, pp. 202–206, May 2018, doi: 10.1038/s41586-018-0091-5. [11] M. S. Freed, D. W . Longcope, and D. E. McK enzie, “Three-year global surv e y of coronal null points from potential-eld-source- surf ace (PFSS) modeling and solar dynamics observ atory (SDO) observ ations, ” Solar Physics , v ol. 290, no. 2, pp. 467–490, Feb . 2015, doi: 10.1007/s11207-014-0616-5. Indonesian J Elec Eng & Comp Sci, V ol. 42, No. 3, June 2026: 865–874 Evaluation Warning : The document was created with Spire.PDF for Python.
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BIOGRAPHIES OF A UTHORS Sri Ekawati is currently a researcher at the Research Center for Climate and Atmosphere, In- donesia’ s National Research and Inno v ation Agenc y (BRIN), and a Ph.D. student in the Department of Computer Science at the Uni v ersity of Tsukuba, Japan. She recei v ed her B.Sc. de gree in Ph ysics from Uni v ersitas P adjadjaran, Indonesia (2005), and her M.Sc. de gree from Institut T eknologi Ban- dung, Indonesia (2014). Since 2006, she has been with the Indonesian Nat ional Institute of Aero- nautics and Space (LAP AN), which w as later inte grated into BRIN in 2021, where her research has focused on space weather , particularly magnetospheric and ionospheric ph ysics. F or this paper , she led the resea rch design, de v eloped the computational programs, and prepared the manuscript. She can be contacted at email: sri.eka w ati@ca v elab .cs.tsukuba.ac.jp or sri.eka w ati@brin.go.id. A computational fr ame work for detection, classication, and visualization of ma gnetic ... (Sri Ekawati) Evaluation Warning : The document was created with Spire.PDF for Python.
874 ❒ ISSN: 2502-4752 Dongsheng Cai w as a professor in the Department of Computer Science at the Uni v ersity of Tsukuba, Japan. He is c urrently a Professor of Management at the Nago ya Uni v ersity of Commerce and Business (NUCB), Japan, and recei v ed his Ph.D. de gree from Stanford Uni v ersity , USA. His research interests include high-performance computing, articial intelligence, quantum computing, data science, and space ph ysics simulations. F or this paper , he contrib uted to the conceptualization of the study , pro vided guidance on the initial me thodology , supervised the research, and supplied the data used. He can be contacted at email: cai@cs.tsukuba.ac.jp. Hir oyuki K udo is a professor in the Department of Computer Science at the Uni v ersity of Tsukuba, Japan. He recei v ed his Doctor of Engineering de gree in Electrical and Communication Engineering from T ohoku Uni v ersity , Japan, in March 1990. His research interests include intelligent informatics, biomedical engineeri ng, biomaterials, and medical imaging systems. F or this paper , he pro vided supervision, access to essential resources throughout the study , and manuscript renement. He can be contacted at email: kudo@cs.tsukuba.ac.jp. Indonesian J Elec Eng & Comp Sci, V ol. 42, No. 3, June 2026: 865–874 Evaluation Warning : The document was created with Spire.PDF for Python.