Minimum variance projection is widely used in geophysical and space plasma measurements to identify the wave propagation direction and the wavenumber of the wave fields. The advantage of the minimum variance projection is its ability to estimate the energy spectra directly in the wavenumber domain using only a limited number of spatial samplings. While the minimum variance projection is constructed for discrete signals in the data, we find that the minimum variance projection can reasonably reproduce the spectral slope of the power-law spectrum if the data represent continuous power-law signals. The spectral slope study using the minimum variance projection is tested against synthetic random data with a power-law spectrum. The method is applicable even for a small number of spatial samplings. Conversely, the spatial aliasing causes a flattening of the spectrum.

Minimum variance estimator proposed by

The wave energy in Capon's method is evaluated from the covariance matrix of
the measured data and the fluctuation model (referred to as the steering
vector) as

Here we study the applicability of Capon's minimum variance projection in turbulence data analysis. Turbulence occurs in many geophysical flows and space plasmas, and the inertial range fluctuations are characterized by power-law spectra. There are many signals in turbulence in the form of waves or eddies, and the number of sensors is by far smaller than that of signals. Our approach is numerical; we generate synthetic turbulence data, sample using a sensor array, and find the wave energy using Capon's method. We then compare Capon's minimum variance estimator spectra with that used in the synthetic data.

Taylor's frozen inflow hypothesis

The synthetic data are generated in the following steps. We set a mesh of
one-dimensional spatial coordinates

A power-law spectrum is defined at the mesh points of the wavenumbers as

The spectral amplitudes are computed in the wavenumber domain (

The fluctuations are obtained in the spatial coordinates using the inverse
fast Fourier transform (FFT):

To double check the computation, the forward Fourier transform is applied to
the spatial fluctuation data,

Synthetic data (left panels) and the energy spectra used to generate the synthetic data (right panels).

The synthetic data are sampled at the positions of the sensor array, and the
energy spectra are evaluated from the sampled data as follows. An array of
four sensors is set with a sensor distance of

The dimension of the averaged energy spectrum is converted from square
amplitudes (nT

The minimum variance estimator is tested under different spatial samplings:

Minimum variance spectra for different numbers of spatial samplings
for a spectral slope of

The major discovery from the numerical test for the minimum variance estimator is that the power-law indices or spectral slopes can reasonably be reproduced in a limited range of wavenumbers if the true spectrum follows a power law. Moreover, this ability to detect the true spectral slope is not lost, even using only a few spatial sampling points.

For comparison, we present two case studies for 4-point measurements
(Fig.

For the four-point measurements (Fig.

Minimum variance spectra for different slopes using four sampling points. Gray lines indicate the true spectral slope.

For the 20-point measurements (Fig.

Since the spatial samplings are made at discrete points, the measurement of
the wavenumber spectra is influenced by the spatial aliasing. That is,
signals or fluctuations with a shorter wavelength than the sensor separation
can affect the spectral measurements. In the case of the power-law spectra,
the spatial aliasing affects the measurement through a flattening of the
spectrum around the Nyquist wavenumber,

Examples of the regular and irregular aliasing-affected spectra are displayed
in Figs.

In Fig.

Minimum variance spectra for different slopes using 20 sampling points. Gray lines indicate the true spectral slope.

In Fig.

It is encouraging that the minimum variance estimator is capable of not only identifying the wavenumbers in the discrete wave signals but also evaluating the spectral slope of the power-law spectrum in the continuous turbulence signals. The capability of the power-law estimation is still valid even using a small number of spatial samplings (typically below five spatial sampling points) at the cost of a limited wavenumber range and a smoothed shape of the spectral curve. Another finding from the presented numerical studies is that the spatial aliasing contributes in different ways. For regularly spaced (or equidistant) samplings, the minimum variance spectra show resonance peaks in the high wavenumber domain beyond the Nyquist wavenumber. For irregularly spaced samplings, the minimum variance spectra show no more resonance peak.

Minimum variance spectra with aliasing for regular spatial samplings
(

One does not need a frequency resolution since the technique is applied in the time domain. The strongest limitation is the fact that one has to assume that the length scales of the sensor array (both the separation distance and the total length) should be in the power-law spectrum range of the wavenumber domain. In the mean flow direction, one may use the time series data and Taylor's hypothesis to diagnose the existence of the power law in the corresponding wavenumber range (even though Taylor's hypothesis is not strictly valid). In the directions perpendicular to the mean flow, there is no means of diagnosing the existence of the power-law spectrum in the wavenumber domain.

The minimum variance estimator can be extended to a vectorial data set such as flow velocities or magnetic fields. In that case, the spectrum projected onto the wavevector domain is obtained as a matrix with the elements consisting of the covariance (represented in the wavevector domain) between different components of the measured vectorial field.

The minimum variance projection can be used for a variety of applications in space plasma observations. Examples are proposed below as a conclusion:

one-dimensional application to magnetospheric plasma turbulence along the
magnetic field line in the magnetotail using five THEMIS spacecraft

one-dimensional application to plasma turbulence in the inner heliosphere
using multi-point measurements either along the magnetic field line or along
the streamline by finding a suitable orbit configuration for Solar Orbiter

three-dimensional applications to local structures of plasma turbulence in
space, e.g., shock-upstream and shock-downstream turbulence of the Earth
magnetosphere, using tetrahedral spacecraft configuration of the Cluster
mission

Minimum variance spectra with aliasing for irregular spatial
samplings (

No experimental data are used in this article. The numerical data are reproducible using the algorithm presented in the article.

The authors declare that they have no conflict of interest.

Y. Narita thanks Karl-Heinz Glassmeier and Uwe Motschmann for scientific
discussions on the adaptive filter theory. The work conducted in Braunschweig
is financially supported by the German Science Foundation under contract MO
539/20-1