Pages 11011106 
M. W. Dunlop^{1}, T. I. Woodward^{1}, D. J. Southwood^{1}, K.H. Glassmeier^{2}, and R. Elphic^{3}
^{1}Space Physics, Imperial College, London SW72BZ, Email:
m.dunlop@ic.ac.uk
^{2}Institut für Geophysik und Meteorologie (IGM), Technische
Universität Braunschweig, Mendelssohnstraße 3, D38106 Braunschweig,
Germany.
^{3}MS D466, Los Alamos National Laboratory, Los Alamos, NM 87545,
U. S. A.
ABSTRACT
Shocks and discontinuities in space plasmas have received much attention in both theoretical studies and spacecraft data analyses. The advent of multispacecraft missions, such as ISEE, have provided for a vast improvement of experimental information available on these structures. The ESA CLUSTER, multispacecraft mission presents the community with a unique opportunity to investigate these structures with considerably more information than in previous missions. The objective of the current study is to consider the balance of information on the topology and motion of these structures, gained from four probe (CLUSTER) array data (but also from less than four spacecraft arrays). In particular, this balance is reviewed with regard to the order (or complexity) of the information sought in relation to the uncertainties involved in the determination. Typical magnetic field signatures computed from simulated trajectories through simple planar and curved discontinuity models are used as input data to the tool. This allows us to grasp several subtleties arising from competing influences of the physical properties reflected in the data, for example curvature and acceleration. In this way we propose a sophisticated discontinuity analysis tool for CLUSTER.
INTRODUCTION
In this paper we discuss multispacecraft analysis of the structure and motion of discontinuities in the magnetic field. The techniques fall into the ‘macroscopic’ class of multispacecraft analysis methods (see Dunlop et al. 1988) by virtue of the assumption, which requires justification, that the spacecraft separation distances are much in excess of the scale lengths of the physical event. It is stressed that this paper concentrates on the analysis of magnetic field data alone, leaving the discussion of analysis methods which incorporate both particle and field data to a future paper. The specific technique described here, termed a ‘Discontinuity Analyser’ (a label originally coined, but not discussed in detail, by Dunlop et. al. 1988), is currently built on a method which seeks to find a planar discontinuity, or to determine the degree of nonplanarity, if present, based on knowledge of the boundary motion and evolution (planarDA). The method here can determine motional properties of a boundary which is demonstrably planar and give qualitative indications of curvature in the limit of small deviations from planarity on the spacecraft separation scale. An important aspect driving this cautious approach is that low noise levels on the data signature (arising from either instrument effects or additional signal) are required to show the significance of either nonconstant motion or curvature. The ability to distinguish between nonconstant motion and surface curvature is discussed conceptually. In the current report we introduce an established method which has been tested against models in simulated event situations. In a future paper we shall describe the use of the technique; and application to data. Below, we first discuss the methodology adopted (slightly generalised to explore related concepts) and then briefly present the planar technique, applied to both planar and nonplanar event models.
METHODOLOGY
The principle components of the procedure may be identified as follows:
1. Data Preselection.
This process serves to identify candidate data intervals which may possess the signature of a discontinuity detected on all spacecraft present. Routines such as variance or spectral analysis and crosscorrelations are useful here, but simple tests for planarity (Farrugia et. al. 1990) and loose forms of stationarity (Chapman and Dunlop, 1993), as reflected in the signature, best serve as precursors to the discontinuity analysis and have been adopted for the technique demonstrated below.
2. Normal Stability.
In the technique described below, the stability of boundary (surface) normals, determined using minimum variance (MVA) analysis on a set of nested data intervals, is assessed in each spacecraft data set separately; a single spacecraft analysis. Equivalent inspection of the boundary (eg. tanjential discontinuity normals) and the degree of planar ordering is contained within a diagnostic analysis of the variance. Failure to detect a stable normal requires a return to the data preselection process. Either further preselection or prefiltering is then carried out, or rejection of that particular interval of data. For example, a wave propagating along a boundary will affect the result obtained from normal determination, but may often be suppressed or the effect clarified by suitable high or low pass filtering (Dunlop et. al. 1996).
3. Crossing Times.
This procedure seeks the times at which the boundary, identified in step 2, crossed each spacecraft. In order to identify these times, the centre of the boundary signature for each spacecraft data set has to be located by curve fitting, or otherwise (eg. by inspection). Currently, inspection is used by default (this is found to be best during use of simulated data). Note that the success of MVA provides a canonical (natural) coordinate system which best represents the boundary in the data and this is taken advantage of in the current procedure.
4. Boundary Analysis.
In principle, the motion and/or topology of the boundary can be analysed. The core technique is currently based on an assumption of planarity, but where deviations from planarity can be identified. In general, any interpretation has to be made in the context of selfconsistency between the normal directions and the motion/topology of the boundary. The form of any check on consistency depends on the implied properties of the boundary (planar or nonplanar, constant motion): a planar boundary is consistent with colinear normals, but nonplanarity is not necessarily the only source for noncolinear normals. For instance, the differences between the spacecraft normals may arise from other than simple boundary structure and then further prefiltering must be attempted or there may exist no physical significance to the differences between the spacecraft normals and then the analysis is ambiguous. Consistency may also be obscured in step 3, where changes in relative timing can arise from a combined effect of nonconstant motion and curvature.
If the process is successful, the results will yield parameters characterising the motion and/or topology of the boundary. The procedural loop(s) implied by the checks on consistency in steps 3 and 4, however, assume some interaction between the interpretation of the normal analysis in step 2 (single spacecraft) and the choice of combined analysis. Clearly, good planarity is implied by strong, consistent and closely colinear normals and then allows more detail on the boundary motion to be determined in principle (noise permitting): for instance, an acceleration term. Nonplanar boundaries are implied by significantly noncolinear (stable) normals, but the significance needs to be established. Then, some assumption of motion (or structure), is required to quantify the curvature analysis. The method of choice for normal determination is MVA, although alternative methods for estimating the normals can also be used.
Use of MultiSpacecraft
Consider now the amount of information that can be learnt from arrays of spacecraft. Intuitively we would expect that with more spacecraft available, more can be gleaned on the macroscopic properties of the physical event. For a single spacecraft, of course, planarity is implicitly assumed in the application of MVA (although any deviations are reflected in the quality of the normals) and the determination of motional information requires multiinstrument analysis. We summarise below the relative pieces of information that can be calculated from spacecraft arrays of increasing size (up to 4, the size of CLUSTER) in the case of: firstly, planar discontinuities and secondly, in the presence of curved boundaries (it may help the visual interpretation to refer to Figure 1, which is described more fully later).
Planarity may be checked to within the projection of the spacecraft separation by making use of computed boundary normals, n (interpreted from MVA, for example). The boundary velocity, v_{}, relative to the spacecraft and parallel to n may also be directly determined from the time delay, but with assumption of constant velocity (simple convection) and quasistatic structure.
These give a better check on planarity (two spacecraft pairs are available, providing a mean, <n>). Although similar assumptions must be made as for two spacecraft, a more realistic velocity can be found since the two estimates may give some quantitative indication of acceleration. Again, the motion must be one dimensional (convective).
Here, a good planarity fit and velocity estimate can be obtained. In addition, a possible estimate of acceleration, a_{}, parallel to <n>, may be made if noise or other additional signal is sufficiently low and normal stability and colinearity is sufficiently high.
Unless planarity is assumed, special, multiple boundary crossings are required with 1 spacecraft for even qualitative analysis, or the curvature must be small (nearly planar). Quantitative information generally requires either limited motion or curvature with more than 1 spacecraft. Explicitly, with four spacecraft, this could be constant velocity of a convected structure, which would allow some check on the level of curvature to be made, via surface fitting procedures (the result cannot be fully decoupled from the determination of normals, however, and curvature must not be significant on the scale of the boundary layer, as discussed later). Alternatively, if some linear acceleration is present, simple curvature properties must be assumed; for instance, a single radius of curvature. Generally, with less than four spacecraft, the number of unknowns sought, even at lowest order in curvature and acceleration, are too many for direct estimation. An assumption of either rigid, constant motion, or planarity must be assumed. Furthermore, error and noise conflicts severely limit the ability to disentangle motional from topological factors and only a qualitative distinction between these systematic variations is generally possible with multiple spacecraft.
It is in view of the above considerations, that the technique described here first attempts to fit a planar discontinuity, and determine its motional properties. Initially here, curvature is considered only as a test for nonplanarity. Curvature introduces characteristic (to the sense of curvature) and systematic deviations from a planar fit, as in the discussion below (on Figure 2). Only in the case of very pure events can quantitative surface fitting be attempted.
Fig. 1. The basic planar discontinuity analyser.
THE BASIC ANALYSIS TECHNIQUE
We are principally concerned here with the boundary normal analysis section above. This is discussed from two viewpoints: firstly planar analysis; and secondly deviations from planarity.
Planar structures
For the moment, consider the planar case, demonstrated in Figure 1. The boundary is assumed to cross each spacecraft in the order shown once, with the projected crossing distances along r_{} being known from spacecraft position data and the computed n. These individual boundary normals, obtained at each spacecraft, provide an estimate of the mean normal, which is a better estimate for the common boundary normal. An error weighted mean is currently used. Although we can estimate the velocity parallel to this normal for each pair if spacecraft individually, we choose to fit a polynomial to the projected spacecraft separation vectors parallel to the normal of the form:
where t represents t_{i,j}, the relative crossing times, and r_{} represents , the projected spatial separation for each spacecraft pair. This allows a constant acceleration term only and some initial velocity at the first spacecraft crossing (there is a subtle problem of sorting crossing times against spacecraft number which is contained in the technique but is not dealt with here). Of course, for planar boundaries having constant motion, v_{} and the common normal can be obtained as in Russell et. al. 1983 with a minimum of four spacecraft. With less than four spacecraft this analysis is ambiguous, unless the plane of the discontinuity is known in one direction, or unless other conditions along the normal can be employed (for example, in the case of shock jump conditions).
The method has been tested, using simple magnetic field models for the discontinuity, by flying the spacecraft through the model. Various motions and evolution of the spacecraft configuration can be simulated and these are treated progressively in terms of: line of flight, orientation of the configuration and changes in configuration. The latter two are unlikely to occur in most actual orbital situations and only fixed, relative orientation of the configuration is considered here, as example. The top graph in Figure 1 shows a typical result of flying the spacecraft with constant velocity. There may be a component of v perpendicular to n, as shown, for which no information is revealed by the structure. The lower graph shows the result of an additional, constant acceleration. The relative times of the crossings are plotted against the relative, components of r, parallel to n. Note that one point (for spacecraft 1, here) lies at the origin. For constant acceleration, a quadratic can easily be fitted to the points as shown.
The presence of noise in the data signature will increase the scatter of the points about the fits and the corresponding uncertainty in v, since crossing times will be less well determined, but also it will degrade the normal analysis through uncertainties in n. Clearly, then, the quality of the estimate of the acceleration term is degraded by the addition of noise or instrument uncertainty. In fact, timing errors usually remain well below 1% and are relatively unimportant in this respect. Since the mean normal, used for determining r_{}, is affected to a reduced degree, noise levels associated with uncertainties in normal components of up to 20% only affect the estimate of Dr_{} (and hence in v), to below 10%. Of more practical concern are tracking errors, which are typically of order 10% in the projected r_{} (and hence in v). The acceleration will remain quantitatively significant, therefore, if deviations from a linear fit (corresponding to implied acceleration, as in Figure 1) are greater than the uncertainty in r_{,}. In simulation tests of noise, which result in scatter of order the affect of the typical acceleration indicated in Figure 1 (although then not systematic), the fit often remains significant, although the estimate is quantitatively poor.
NonPlanar structures
A curved discontinuity is drawn in Figure 2, with the corresponding graph of crossing times, plotted against the projected r. The curvature is assumed to be 2D and only three spacecraft are shown for clarity. In this case, the dashed line in the graph is drawn to correspond to the constant velocity motion of a planar discontinuity, as indicated by the dashed line in the sketch. The situation shown, represents a typical result of simulated flythroughs, using simple 2D magnetic boundary models. The sketch serves to indicate how the crossing times deviate asymmetrically from the planar line. For instance, spacecraft 2 will cross the boundary at t_{2} but crosses the dashed line at t_{2}'. This shift (to: t_{2}'t_{1}) is shown on the graph, and similarly for t_{3}. The sense of this systematic deviation depends only on the relative orientation of the spacecraft configuration with respect to the curvature and not on the sense of motion (down or up along the line shown in Figure 2). It is clear, however, that in actual encounters, where the relative orientations are not known, the affect of noise will more easily obscure the effect of acceleration, although the sense of the deviations may still be apparent. The asymmetry in the deviations shown in the plot shown in Figure 2 is also apparant in the individual data sets using simple tests for planar ordering in the field (Farrugia et al. 1990) and is present for actual dual spacecraft events (Dunlop et al. 1996).
Clearly, once the analysis indicates the presence of a curvature, direct fitting of a surface to the individual normals is in principle possible, given some knowledge or assumption of motion. For instance, with constant velocity, the crossings at each spacecraft can be mapped back to the positions at the time of the first crossing. Both components of velocity must be determined in this case, or be treated as parameters in the fit. There is, however, a need to ensure that the fit is achieved selfconsistently, although the normals might be significantly noncolinear. The whole analysis is further complicated by the fact that the normal determination is itself affected, at each crossing point, by the direction of motion through the boundary of each spacecraft, particularly if the curvature is large. This arises because of the need to identify a finite data interval (which maps to a distance along the motion) through the boundary in order to perform MVA and, unless this distance is small on the scale of the curvature, the field structure will not wholly represent the boundary orientation local to the crossing, thus affecting the implied normal direction.
Fig. 2. The analysis of a nonplanar boundary structure.
DISCUSSION AND CONCLUSIONS
This paper has briefly introduced a proposed methodology for a magnetic fieldbased discontinuity analyser technique, suitable for multispacecraft analysis, and has discussed conceptually the issues arising for study of different structure. A later paper will describe in more detail the application of the method to simulated and actual data (from 2+ spacecraft) as a discontinuity analyser (DA) technique. The success of the analysis depends on the degree of stationarity (here, also taken to include quasistatic or stable structure) and the planarity properties of events. Although these preselection methods have not been explicitly described here, they have been implied in the description of procedure. The restrictions introduced by limitations in the number of spacecraft has also been addressed.
Here, this short account has only briefly demonstrated a planar technique together with the associated motional analysis, using example output from simulated data. There is a natural progression from motional to topological parameters which can be extracted. Thus, for planar structures one can potentially determine v_{} , a_{} , and a unique n for the boundary. Uncertainties lead to a playoff between the determination of v_{} and a_{}, however, and events need to be particularly clean (in terms of noise, or the absence of complex properties) to analyse deviations from planarity. If the structure is nonplanar, less information can be determined about the motion. The nature of the obtained parameters of course also depend upon the degree of stationarity found. Topological and motional parameters therefore compete for their representation in the data.
The analysis of a model, curved discontinuity (2D) using the planar technique has also been demonstrated. Since the planar analysis can be used to order the data, deviations from planarity can yield unique information on the surface. Direct analysis of the surface topology, however, is limited by the need to ensure selfconsistency with the normal analysis, currently a single spacecraft analysis. For real events, determination of a nonplanar surface will depend upon the separation of noise and quality of each spacecraft normal, n, from the parameters sought. Additionally, the individual normals are dependent on spacecraft trajectory, with the dependence being related to the form of the surface. Hence, any surface analysis will affect the interpretation of the normals. Full curvature analysis, therefore, is generally possible only under conditions of constant motion or very limited curvature, as indicated in the table below.
Table: Indication of the ability of the analysis to determine the unknowns in different physical situations and to lowest order in curvature.
Dimension 
No of spacecraft: 
1 s/c 
2 s/c 
3 s/c 
4 s/c 
1D 
No acceleration 

c 


Acceleration 


c 


2D 
No acceleration 




Acceleration 





3D 
No acceleration 


c 

Acceleration 



c 
The table shows the quantitative information which is, in principle, obtainable from magnetic field analysis alone. To construct the table, it has been assumed that the normals are given and are not modified by the analysis (not always true). It is also assumed that the discontinuity is nondispersive and convecting. The ticks refer to situations where the number of identifiable unknowns is less than the number of equations, defined in terms of the motion through the structure. The crosses to the reverse situation where there are more unknowns than equations. The letter ‘c’ indicates situations where there are the same number of unknowns and equations (critically constrained). To deduce this only lowest order parameters have been allowed: a constant acceleration, a single constant radius of curvature (two principal values in the case of 3D). For the cases which are underconstrained, so that not all parameters can be determined, qualitative indications can still be obtained, such as the existence of curvature, or acceleration, as discussed above. The use of other instrument data, of course, can potentially add information on structure, such as independent determination of velocity in the case of electric field measurements.
REFERENCES
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Russell, C. T., M. M. Mellott, E. J. Smith, and J. H. King, Multiple observations of interplanetary shocks: four spacecraft determination of shock normals, J. Geophys. Res., 88, 47394748, 1983.