Multi-spacecraft analysis methods

Reciprocal vectors

Paschmann and Daly [1], Chapter 4, Paschmann and Daly [2], Chapter 14,

SPEDAS.position_tensor — Function

\[𝐑 = ∑_α (𝐫_α-𝐫_b) (𝐫_α-𝐫_b)' = ∑_α 𝐫_α 𝐫_α'-𝐫_b 𝐫_b'\]

with $𝐫_b = ∑_α 𝐫_α / N$ and N is the number of positions.

References

Paschmann and Daly [1] Paschmann & Daly, 2008. Section 4.7

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SPEDAS.reciprocal_vector — Function

reciprocal_vector(r_βα, r_βγ, r_βλ)

Compute the reciprocal vector $𝒌_α$ for a vertex of a tetrahedron given the relative position vectors.

\[𝒌_α = \frac{𝐫_{βγ} × 𝐫_{βλ}}{𝐫_{βα} ⋅ (𝐫_{βγ} × 𝐫_{βλ})}\]

where $𝐫_{αβ} = r_β - r_α$ are relative position vectors.

References

Multi-spacecraft analysis methods revisited : 4.3 Properties of reciprocal vectors

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reciprocal_vector(rα, rβ, rγ, rλ)

Compute the reciprocal vector $𝒌_α$ for a vertex of a tetrahedron given the position vectors of all vertices.

The vertices (α, β, γ, λ) must form a cyclic permutation of (1, 2, 3, 4).

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reciprocal_vector(rα, r0s::AbstractVector{<:AbstractVector})

Generalised reciprocal vector for N != 4

\[𝐪_α = 𝐑^{-1} 𝐫_α\]

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SPEDAS.reciprocal_vectors — Function

Compute the set of reciprocal vectors {$𝒌_α$}, which is also called the reciprocal base of the tetrahedron.

Estimation of spatial gradients

SPEDAS.lingradest — Function

lingradest(B1, B2, B3, B4, R1, R2, R3, R4)

Compute spatial derivatives such as grad, div, curl and curvature using reciprocal vector technique (linear interpolation).

Arguments

B1, B2, B3, B4: 3-element vectors giving magnetic field measurements at each probe
R1, R2, R3, R4: 3-element vectors giving the probe positions

Returns

A named tuple containing: • Rbary: Barycenter position • Bbc: Magnetic field at the barycenter • Bmag: Magnetic field magnitude at the barycenter • LGBx, LGBy, LGBz: Linear gradient estimators for each component • LD: Linear divergence estimator • LCB: Linear curl estimator • curvature: Field‐line curvature vector • R_c: Field‐line curvature radius

References

Based on the method of Chanteur (ISSI, 1998, Ch. 11).

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lingradest(B1::MatrixLike, args...)

Vectorized method for simplified usage. Returns a StructArray containing the results.

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lingradest(B1::AbstractDimArray, args...)

Method for handling dimensional arrays. Takes AbstractDimArray inputs with a time dimension and returns a DimStack containing all computed quantities.

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lingradest(
    Bx1, Bx2, Bx3, Bx4,
    By1, By2, By3, By4,
    Bz1, Bz2, Bz3, Bz4,
    R1, R2, R3, R4
)

SPEDAS-argument-compatible version of lingradest.

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Since $\tilde{g}$ and $\tilde{\boldsymbol{V}}$ are linear functions, the calculation of spatial derivatives, such as the gradient of some scalar function or the divergence or curl of a vector function, can be done quite easily. The results are:

\[\begin{aligned} \nabla g \simeq \nabla \tilde{g} & =\sum_{\alpha=0}^3 \boldsymbol{k}_\alpha g_\alpha \\ \hat{\boldsymbol{e}} \cdot \nabla g \simeq \hat{\boldsymbol{e}} \cdot \nabla \tilde{g} & =\sum_{\alpha=0}^3\left(\hat{\boldsymbol{e}} \cdot \boldsymbol{k}_\alpha\right) g_\alpha \\ \nabla \cdot \boldsymbol{V} \simeq \nabla \cdot \tilde{\boldsymbol{V}} & =\sum_{\alpha=0}^3 \boldsymbol{k}_\alpha \cdot \boldsymbol{V}_\alpha \\ \nabla \times \boldsymbol{V} \simeq \nabla \times \tilde{\boldsymbol{V}} & =\sum_{\alpha=0}^3 \boldsymbol{k}_\alpha \times \boldsymbol{V}_\alpha \end{aligned}\]

Multi-spacecraft timing

SPEDAS.ConstantVelocityApproach — Function

CVA(positions, times)

Constant Velocity Approach (CVA) for determining boundary normal and velocity. Solve timing equation: $D * m = Δts$

Parameters:

positions: Positions of 4 spacecraft (4×3 array)
times: Times of boundary crossing for each spacecraft

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ConstantVelocityApproach(positions, times, durations)

Given durations of the boundary crossings, calculate the thickness of the boundary

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[1]: G. Paschmann and P. W. Daly. Multi-Spacecraft Analysis Methods Revisited (ESA Communications, 2008).
[2]: G. Paschmann and P. W. Daly. Analysis Methods for Multi-Spacecraft Data (ESA Publications Division, 2000).
[3]: J. C. Samson and J. V. Olson. Some Comments on the Descriptions of the Polarization States of Waves. Geophysical Journal International 61, 115–129 (1980).
[4]: J. D. Means. Use of the Three-Dimensional Covariance Matrix in Analyzing the Polarization Properties of Plane Waves. Journal of Geophysical Research (1896-1977) 77, 5551–5559 (1972).