Spatial Interpolation Between Satellites: Development of a Data Fusion Implementation for Multi-Spacecraft Missions, 15-9306

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Principal Investigators
Jrg-Micha Jahn
Sandee Jeffers
Martin West

Inclusive Dates: 04/01/02 - 09/30/03

Background - Satellite constellation missions give for the first time full access to the true three-dimensionality of the space environment. They allow the detection and measurement of vector and scalar field gradients and discontinuities. Determining spatial gradients is a challenging endeavor, since spacecraft constellations not only move with respect to the region of interest, but the constellation members themselves move constantly with respect to each other. Recent literature has provided a theoretical framework for this analysis and the errors affecting the gradient calculation. In this project we implemented a data fusion system based on the barycentric coordinate and the least squares minimization formalisms, allowing us to calculate spatial gradients for constellations of four or more spacecraft. For development we were primarily using data from the Cluster Plasma Electron And Current Experiment (PEACE), which is part of the joint European-US Cluster mission launched in 2000.

Approach - In order to address the objectives formulated above, we built a software package which, based on a set of input data, calculates spatial gradients of desired quantities. The following main procedures are implemented in the package:

  1. Time-alignment of input data. All data need to be brought to a common timeline using time series resampling methods before any kind of meaningful analysis or fusion can be performed.
  2. Evaluation of the tetrahedron geometry. By calculating the "volumetric tensor" we quantify and classify the geometry of a tetrahedron at any given time step.
  3. Calculation of spatial gradients and curls. Spatial gradients of all physical quantities are calculated under inclusion of the satellite orbital positions at each measurement instance. In addition, the curl of vectors is calculated, if appropriate.

Accomplishments - We produced an operational software package that represents an implementation of two different algorithms to calculate spatial gradients and spacecraft tetrahedron geometrical quantities and quality factors. The package is written in object-oriented C++, it is highly modular, generic, portable, and adaptable. A User's and Programmer's Guide is available for this software package.
The software reads ASCII or Southwest Research Institute® IDFS™ (Instrument Data File System™) input data, then it performs (1) a multi-satellite orbit interpolation to calculate the exact tetrahedron shape at the time of science measurements, (2) the gradient calculation, (3) a calculation of the tetrahedron characteristic quantities (size, elongation, planarity), and (4) data output into ASCII files for easy post-processing analysis. The system is capable of processing arbitrarily long data sets. It is also capable of calculating the curl of a vector and the divergence of a quantity. The software package has been extensively tested with synthetic ASCII data and data in IDFS™ format from the Cluster PEACE instrument. Without changes, the software can be applied to any constellation of four or more spacecraft, as was demonstrated when we analyzed data form four spacecraft located in the solar wind.

Figure 1. Example Cluster pass through the Cusp from 1215 to 1250 UT. The bottom three panels show the total pressure for two spacecraft each, clearly showing differences in pressure from spacecraft to spacecraft. These differences are larger than the instrument error bar and thus represent an ideal test case for calculating the pressure gradient.

Figure 2. Pressure gradient for example Cluster pass through the Cusp. Red lines indicate orbits of individual Cluster spacecraft, the green line indicates the orbital trace of the meso-center of the spacecraft constellation. Blue dots indicate the positions of the four cluster spacecraft at the start of the time interval which is 1215 UT to 1250 UT. Blue lines emanating from the meso-center trace indicate the value (= length) and direction of the calculated pressure gradient (using the total scalar pressure as input).

Figure 3. Solar wind speed gradient for a two-day time period in May 1998. Displayed are, from the top, the three GSE components of the pressure gradient and its absolute value. Data from SOHO, Wind, ACE and IMP8 were used for this plot. At approximately 21 UT a CME with associated shock passes through the spacecraft cluster. Note the increased spatial gradients in solar wind speed after this event. (Data courtesy Dr. R. Goldstein, Space Science and Engineering Division).

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