The combined processing of geomagnetic intensity vector projections and absolute magnitude measurements

The combined processing of geomagnetic intensity vector projections and absolute magnitude measurements

The combined processing of geomagnetic intensity vector projections and absolute magnitude measurementsThe combined processing of geomagnetic intensity vector projections and absolute magnitude...Victor G. Getmanov et al.

Victor G. Getmanov^{1,2},Alexei D. Gvishiani^{1,2},and Roman V. Sidorov^{1}Victor G. Getmanov et al. Victor G. Getmanov^{1,2},Alexei D. Gvishiani^{1,2},and Roman V. Sidorov^{1}

^{1}Laboratory of Geoinformatics and Geomagnetic Studies, Geophysical Center of the Russian Academy of Sciences (GC RAS), 119296 Moscow, Russian Federation

^{2}Department V: Mathematical Geophysics and Geoinformatics, Laboratory of Geoinformatics, Schmidt Institute of Physics of the Earth of the Russian Academy of
Sciences (IPE RAS), 123242 Moscow, Russian Federation

^{1}Laboratory of Geoinformatics and Geomagnetic Studies, Geophysical Center of the Russian Academy of Sciences (GC RAS), 119296 Moscow, Russian Federation

^{2}Department V: Mathematical Geophysics and Geoinformatics, Laboratory of Geoinformatics, Schmidt Institute of Physics of the Earth of the Russian Academy of
Sciences (IPE RAS), 123242 Moscow, Russian Federation

Correspondence: Roman V. Sidorov (r.sidorov@gcras.ru)

A method for the combined processing measurements of the
projections and the absolute magnitudes of geomagnetic field intensity
vectors, based on mathematical technology of local approximation models, is
proposed. The approach realized in this paper, based on the proposed method,
provides an increase in quality of the measurements of the projections of
the geomagnetic intensity vector. An algorithm for the two-stage combined
processing of the measurements of projections and absolute magnitudes of
geomagnetic field intensity vectors is developed. The operation of the
combined processing algorithm was tested on models and observatory
measurements. The estimates of the combined processing algorithm errors were
obtained using statistical modelling. The reduction of the root-mean-square
error values was achieved for the estimates of the projections of
geomagnetic field intensity vectors.

Please read the corrigendum first before continuing.

In this article, a method and an algorithm for combined processing
measurements of the components (projections and the absolute magnitudes) of
geomagnetic field intensity vectors is proposed. The approach realized in
the paper, based on the proposed method, provides an increase in quality of
the measurements of the projections of the geomagnetic intensity vector. The
considered measurements are carried out by INTERMAGNET observatories
equipped with systems of vector and scalar magnetometers; the definitive
type data are used, containing systematic errors equal to zero (Mandea and Korte,
2011; INTERMAGNET, 2018). The measurement errors of vector and scalar
magnetometers are represented by random, normally distributed errors with
zero expectation and a predetermined variance. As usual, the measurement
errors of the projections of geomagnetic field intensity vectors are
significantly larger than the ones of the absolute vector magnitudes
performed by the mentioned measurement devices. The formulation of the
problem of reducing the noise root-mean-square (RMS) error values in the
geomagnetic field intensity projection measurements is due to combined
processing of the values of all its components.

In the following research, the following steps are outlined:

A method for combined processing of the measurements of projections and
absolute magnitudes of geomagnetic field intensity vectors is formulated,
based on formation of the sequences of piecewise-constant local models
followed by their weighted averaging;

A two-stage algorithm for combined processing of the measurements of
projections and absolute magnitudes of geomagnetic field intensity vectors
is developed;

Testing of the algorithm on model and observatory data is performed;

The estimates of the algorithm errors, calculated using statistical
modelling, are presented; the reduction of the RMS noise errors for the
estimates of the geomagnetic field vector projections is proved.

The material in this research paper is intended for specialists
(magnetologists) engaged in digital processing of geomagnetic field
measurements. The need to reduce noise errors in estimates of the
projections of the geomagnetic field intensity vectors measured by vector
magnetometers arises in a number of technical and scientific applications.
For instance, technogenic disturbances can affect the geomagnetic
observatory hardware, affecting vector and scalar magnetometers differently:
as a rule, the noise errors from vector magnetometers occurring due to such
interference are greater than the noise errors from scalar magnetometers.
The decrease in noise errors from vector magnetometers is necessary, e.g. for calculation of the gradients of the projections of the
geomagnetic field intensity vectors in the navigation problems.

Nowadays the reduction of errors for vector magnetometers (with certain
assumptions) is achieved using optimization of the calibration from scalar
magnetometers (Merrayo et al., 2000; Olsen et al., 2003) or refinement of
calibration characteristics (Soborov et al., 2008) based on special
mathematical processing. In the measurement systems considered in this
paper, in fact, parallel measurements are performed; a possible algorithm
providing the decrease in errors for such measurements can be formed based
on Kalman filters (Shakhtarin, 2008). However, due to the peculiarities of
this problem, the construction of the resulting non-linear filters is
associated with certain problems due to the inaccuracies of linearization
and the accepted hypothesis concerning the type of initial intensity vector
function. In other research (Soloviev et al., 2018), joint processing of
vector and scalar magnetometer measurements aimed at improving the
calibration accuracy of the so-called baseline, which only indirectly
provides the considered reduction in errors. The combined processing of
measurements of projections and absolute magnitudes of geomagnetic field
intensity vectors proposed in this paper is significantly free from the
mentioned problems.

2 A method for combined processing of measurements of projections and absolute magnitudes of geomagnetic field intensity vectors

Let H_{1}(Ti), H_{2}(Ti), H_{3}(Ti), and ${H}_{\mathrm{0}}\left(Ti\right)=\sqrt{{H}_{\mathrm{1}}^{\mathrm{2}}\left(Ti\right)+{H}_{\mathrm{2}}^{\mathrm{2}}\left(Ti\right)+{H}_{\mathrm{3}}^{\mathrm{2}}\left(Ti\right)}$ be the
initial functions for the projections and absolute magnitudes of the
geomagnetic field intensity vectors; we assume that Y_{1}(Ti), Y_{2}(Ti), Y_{3}(Ti), and Y_{0}(Ti) are their values registered by
vector and scalar magnetometers, $i=\mathrm{0},\mathrm{1},\mathrm{\dots},{N}_{\mathrm{f}}-\mathrm{1}$; the sampling interval T=1 s; 1 s
measurements from INTERMAGNET observatories are analysed in this study. For
$n=\mathrm{0},\mathrm{\dots},\mathrm{3}$, we represent the noise errors of the
measurement values W_{n}(Ti) in the form of uncorrelated,
normally distributed random values with zero mathematical expectations and
some variances. Such a representation of errors is, to a large extent, valid
for cases of large technogenic noises that can occur when geomagnetic
measurements are carried out. We consider, with some assumptions, that the
spectrum of random components for the functions of the geomagnetic field is
concentrated almost entirely in the low-frequency domain, and the spectrum
of random measurement errors is concentrated in the high-frequency domain.

We assume that the measurements, the initial functions, and the errors are
related by linear additive dependences:

Using the specified observation values Y_{n}(Ti), we demand
the determination of the estimates ${Y}_{\mathrm{1}}^{\circ}\left(Ti\right)$, ${Y}_{\mathrm{2}}^{\circ}\left(Ti\right)$, and ${Y}_{\mathrm{3}}^{\circ}\left(Ti\right)$, where $i=\mathrm{0},\mathrm{1},\mathrm{\dots},{N}_{\mathrm{f}}-\mathrm{1}$, that would be close to the
initial functions of the intensity vector projections. We perform the
combined processing for the projections and magnitudes of the geomagnetic
field intensity vectors in two stages.

At the first stage, on the main interval with the points $i=\mathrm{0},\mathrm{1},\mathrm{\dots},{N}_{\mathrm{f}}-\mathrm{1}$, we introduce the N-point sliding
local intervals with limiting points ${N}_{{\mathrm{1}}_{j}}$,
${N}_{{\mathrm{2}}_{j}}$, and the sliding step N_{d} as well as the quantity of
sliding intervals m_{0}.

To simplify the considerations, we require the relations of multiplicity
mN=N_{f} and N_{d}m_{d}=N; here, if m and m_{d} are
integers, then ${m}_{\mathrm{0}}={m}_{\mathrm{d}}(m-\mathrm{1})+\mathrm{1}$.

For a sliding interval with a number j, we define the model functions of a
form ${Y}_{{M}_{\mathrm{1}}}\left({c}_{{\mathrm{1}}_{j}},Ti\right)$,
${Y}_{{M}_{\mathrm{2}}}\left({c}_{{\mathrm{2}}_{j}},Ti\right)$, and ${Y}_{{M}_{\mathrm{3}}}\left({c}_{{\mathrm{3}}_{j}},Ti\right)$;
here, ${c}_{{n}_{j}}$ is the vectors of model parameters, n=1, 2, 3. These
model functions can be, in particular, polynomial, piecewise constant,
piecewise linear, etc. The size of local intervals determines the
approximation errors. At small N there will be large fluctuation errors, and
at large N there will be large systematic approximation errors.

Based on the above-defined measured values, models, and the maximum
likelihood method (Kramer, 1975), we define the local functional
S(c_{j}Y_{j}) which determines the measure of closeness
for local measurements and models, similar to Getmanov (2013) as the sum
of the four functionals:

Here,
${c}_{j}^{T}=({c}_{{\mathrm{1}}_{j}}^{T}{c}_{{\mathrm{2}}_{j}}^{T},{c}_{{\mathrm{3}}_{j}}^{T})$ and ${Y}_{j}^{T}=({Y}_{{\mathrm{1}}_{j}}^{T}{Y}_{{\mathrm{2}}_{j}}^{T},{Y}_{{\mathrm{3}}_{j}}^{T})$ are the parameter and value vectors
related to the jth local interval. Taking into account the
assumption of errors in measurements, we carry out the identification of the
optimal estimates of the model parameters ${c}_{j}^{\circ}$ using the solutions of the sequence of optimization
problems for local functionals:

At the second stage we introduce the unit functions E_{j}(Ti), equal to zero outside of local sliding intervals, add them
together to get E(Ti), and calculate the sequence of weighting
coefficients R(Ti):

We calculate the estimates ${Y}_{n}^{\circ}\left(Ti\right)$ and $i=\mathrm{0},\mathrm{1},\mathrm{\dots},{N}_{\mathrm{f}}-\mathrm{1}$, where n=1, 2, 3, based on linear operations in Eq. (6) corresponding to the second stage.

The method of combined two-step processing of the values of the projections
and the absolute magnitudes of geomagnetic field intensity vectors consists
of the sequential execution of the first and second stages in accordance
with Eqs. (2)–(6).

3 An algorithm for two-stage processing of measurements of projections and absolute magnitudes of geomagnetic field intensity vectors for piecewise-constant models

Let us build the local intervals using Eq. (1) and define the local models
on them as piecewise-constant functions ${Y}_{{M}_{n}}\left({c}_{{n}_{j}},Ti\right)={c}_{{n}_{j}}$, $j=\mathrm{1},\mathrm{\dots},{m}_{\mathrm{0}}$, and ${N}_{{\mathrm{1}}_{j}}\le i\le {N}_{{\mathrm{2}}_{j}}$. In this case it is obvious that the
initial functions for geomagnetic field vector projections must be
approximately constant at local intervals with duration NT. The local interval
can be expanded if we treat piecewise-constant functions as local models.
Let us formulate the equation for local functionals:

We differentiate Eq. (7) with respect to ${c}_{{n}_{j}}$, equate the derivatives
to zero, and get the necessary conditions for an extremum in the form of a
system of three non-linear algebraic equations as a result:

In this case, an exact analytical solution to this system is possible.
Omitting the calculations, we obtain expressions for local estimates, $j=\mathrm{1},\mathrm{\dots},{m}_{\mathrm{0}}$:

where n=1, 2, 3, and get the piecewise-constant functions for local
estimates ${Y}_{n}^{\circ}\left(Ti\right)={c}_{{n}_{j}}^{\circ}$, ${N}_{{\mathrm{1}}_{j}}\le i\le {N}_{{\mathrm{2}}_{j}}$,
${Y}_{n}^{\circ}\left(Ti\right)=\mathrm{0}$,
$\mathrm{0}\le i<{N}_{{\mathrm{1}}_{j}}$, ${N}_{{\mathrm{2}}_{j}}<i\le {N}_{\mathrm{f}}-\mathrm{1}$, and n=1, 2, 3,
using Eq. (8) according to Eq. (4); let us present them as the realization of
the first stage.

Weighted averaging of sequences of piecewise-constant local estimates and
the calculations of the estimate functions ${Y}_{n}^{\circ}\left(Ti\right)$ for the second stage are performed
using Eqs. (5) and (6).

4 Testing of the algorithm for combined processing on model and observatory measurements

The developed algorithm for combined processing was tested on model
measurements. Initial model functions for the projections of the geomagnetic
intensity vector ${H}_{M{G}_{n}}\left(Ti\right)$ were presented as
quadratic functions; the model function for the absolute vector magnitude
${H}_{M{G}_{\mathrm{0}}}\left(Ti\right)$ was calculated based on
${H}_{M{G}_{n}}\left(Ti\right)$, n=1, 2, 3.

and $i=\mathrm{0},\mathrm{1},\mathrm{\dots},{N}_{\mathrm{f}}-\mathrm{1}$. The noise error
functions W_{n}(Ti) and n=0, 1, 2, 3 were modelled using
normally distributed values with zero mathematical expectation and the
variances ${\mathit{\sigma}}_{\mathrm{1}}^{\mathrm{2}}={\mathit{\sigma}}_{\mathrm{2}}^{\mathrm{2}}={\mathit{\sigma}}_{\mathrm{3}}^{\mathrm{2}}={\mathit{\sigma}}^{\mathrm{2}}$ and ${\mathit{\sigma}}_{\mathrm{0}}^{\mathrm{2}}$. The model values of the projections the
absolute magnitude of the intensity vector were represented as sums:

where n=0, 1, 2, 3.
The following parameter values for the model functions were assigned:
${a}_{\mathrm{0},\mathrm{1}}=\mathrm{30}\phantom{\rule{0.125em}{0ex}}\mathrm{000}$, ${a}_{\mathrm{1},\mathrm{1}}=\mathrm{0.4206}$,
${a}_{\mathrm{2},\mathrm{1}}=-\mathrm{0.0040}$,
${a}_{\mathrm{0},\mathrm{2}}=\mathrm{4300.4}$,
${a}_{\mathrm{1},\mathrm{2}}={a}_{\mathrm{1},\mathrm{2}}=-\mathrm{0.4252}$, ${\mathit{\alpha}}_{\mathrm{2},\mathrm{2}}=\mathrm{0.0043}$, ${a}_{\mathrm{0},\mathrm{3}}=\mathrm{4089.5}$, ${a}_{\mathrm{1},\mathrm{3}}=\mathrm{0.5013}$, ${\mathit{\alpha}}_{\mathrm{2},\mathrm{3}}=-\mathrm{0.0056}$, and T=1 s. Model
measurement values ${H}_{{M}_{n}}\left(Ti\right)$ were the input of Eq. (8) of the algorithm for combined processing based on piecewise-constant
models.

The case of local intervals without sliding was considered, with the number
of points N, ${N}_{\mathrm{1}j}=N(j-\mathrm{1})$ and
${N}_{\mathrm{2}j}={N}_{\mathrm{1}j}+N-\mathrm{1}$, where
$j=\mathrm{1},\mathrm{\dots},m$ and mN=N_{f}. The local estimates
${H}_{{n}_{j}}^{\circ}={c}_{j}^{\circ}$, where
$j=\mathrm{1},\mathrm{\dots},m$, were calculated, and the sequences of piecewise
constant estimates ${H}_{{L}_{n}}^{\circ}\left(Ti\right)$,
n=1, 2, 3, corresponding to the first processing stage, were
formed.

For a sliding local intervals case with the number of points N, the
sliding step N_{d} was selected, as well as the number of sliding local
intervals m_{0}. Local estimates ${H}_{{n}_{j}}^{\circ}={c}_{j}^{\circ}$, $j=\mathrm{1},\mathrm{\dots},{m}_{\mathrm{0}}$, and
the estimates of functions ${H}_{{n}_{j}}^{\circ}\left(Ti\right)$, where n=1, 2, 3 and
$i=\mathrm{0},\mathrm{1},\mathrm{\dots},{N}_{\mathrm{f}}-\mathrm{1}$, corresponding to the first processing
stage, were calculated. Next, the second stage was performed where the
estimates ${H}_{{S}_{n}}^{\circ}\left(Ti\right)$ were found.

Figure 1Results of testing the processing algorithm on model
measurement data: 1 – initial model functions, 2 – noised model functions,
3 – first-stage result (piecewise-constant model estimates without
sliding), and 4 – second-stage result (estimates with weighted averaging).

For testing, the following values were selected: N_{f}=96,
N=12, m=8, N_{d}=1,
m_{0}=84, σ=1.0 nT, and σ_{0}=0.5 nT. In Fig. 1, the calculation results
are displayed for ${H}_{M{G}_{\mathrm{1}}}$ (Fig. 2a) and ${H}_{M{G}_{\mathrm{2}}}$ (Fig. 2b),
${H}_{M{G}_{\mathrm{3}}}$ (Fig. 2c); dashed lines with index 1 depict the initial
functions for the projections of the geomagnetic field intensity vector
${H}_{M{G}_{n}}\left(Ti\right)$; lines with index 2 represent the
noised measurements of the projections of the geomagnetic field intensity
${H}_{{M}_{n}}\left(Ti\right)$; piecewise-constant lines with index 3
represent the results of the first stage ${H}_{{L}_{n}}^{\circ}\left(Ti\right)$; solid lines with index 4 show the estimates
${H}_{{S}_{n}}^{\circ}\left(Ti\right)$ – the second-stage
results with weighted averaging.

Figure 2Results of testing the processing algorithm on observatory
measurement data: 1 – initial data, 2 – first-stage result ( piecewise
constant model estimates without sliding), and 3 – second-stage result
(estimates with weighted averaging with sliding).

The performed testing of the processing algorithm on model data for a number
of parameters led to the conclusion that the second stage of processing
reduces the RMS of the errors of the first stage by 60 %–80 % on average.

4.2 Testing on real observatory geomagnetic data

The developed algorithm was tested using combined processing of 1 s
sampled geomagnetic measurements from the INTERMAGNET observatory MBO
(Mbour, Senegal). The measurements were recorded on 2 January 2014, they began
at 01:16:37 UT, and the length of a test fragment was 96 s (N_{f}=96). For the processing algorithm, N=12 and the
sliding step N_{d}=1 were assigned.

In Fig. 2 the test results are shown for H_{1} (Fig. 2a), H_{2} (Fig. 2b), and H_{3} (Fig. 2c). Dashed lines with index 1 depict the observatory
measurements of the geomagnetic vector projections
H_{n}(Ti); piecewise-constant lines with index 2 are
related to the first processing stage – the functions for piecewise
constant estimates ${H}_{{L}_{n}}^{\circ}\left(Ti\right)$
without sliding are displayed; index 3 stands for the line corresponding to
the result of the second processing stage – the estimate with weighted
averaging ${H}_{{S}_{n}}^{\circ}\left(Ti\right)$ with sliding.

The results of testing the algorithm for combined processing of
measurements of projections and absolute magnitudes of geomagnetic field
intensity, displayed on Figs. 1 and 2, proved their satisfactory performance.

5 Error estimation for the algorithm for combined processing of measurements

The estimates of errors of the proposed algorithm for combined processing
were found using statistical modelling. The first stage of combined
processing was analysed.

The initial functions for the intensity vector projections were assumed to
be constant on a local interval. The values
${H}_{n}(k,m,{H}_{\mathrm{0}},Ti)={H}_{n}(k,m,{H}_{\mathrm{0}})$,
$i=\mathrm{0},\mathrm{1},\mathrm{\dots},N-\mathrm{1}$, n=1, 2, 3 were found using the
following equations:

Here, H_{0} is an assigned absolute magnitude value, φ_{k}ϑ_{m} are the azimuthal and zenithal angles, θ_{m}=Δϑm, $\mathrm{\Delta}\mathit{\vartheta}=\mathrm{2}\mathit{\pi}/{M}_{\mathrm{0}}$, m and k are integer parameters,
$m=\mathrm{0},\mathrm{1},\mathrm{\dots},{M}_{\mathrm{0}}-\mathrm{1}$, φ_{k}=Δφk, $\mathrm{\Delta}\mathit{\phi}=\mathrm{2}\mathit{\pi}/{K}_{\mathrm{0}}$, and $k=\mathrm{0},\mathrm{1},\mathrm{\dots},{K}_{\mathrm{0}}-\mathrm{1}$.

For all possible values of indices k and m, the realizations of sequences of
model normally distributed random numbers with zero mathematical expectation
${W}_{n,s}(k,m,Ti)$ were formed,
where $i=\mathrm{0},\mathrm{1},\mathrm{\dots},N-\mathrm{1}$, and n=0, 1, 2, 3, and
$s=\mathrm{1},\mathrm{\dots},{S}_{\mathrm{0}}$ is the number of realization for
statistical modelling. For n=0, the variance determining the noise
error level for a scalar magnetometer assumed the value ${\mathit{\sigma}}_{\mathrm{0}}^{\mathrm{2}}$; for n=1, 2, 3, the noise error
variances for a vector magnetometer assumed the value σ^{2}. For ${H}_{n}(k,m,{H}_{\mathrm{0}})$,
${W}_{n,s}(k,m,Ti)$, and S_{0}, random
realizations were constructed:

The results of the algorithm implementation for the first stage – the estimates
${H}_{n,s}^{\circ}(k,m,{H}_{\mathrm{0}},N,\mathit{\sigma},{\mathit{\sigma}}_{\mathrm{0}})$, n=1, 2, 3, $k=\mathrm{0},\mathrm{1},\mathrm{\dots},{K}_{\mathrm{0}}-\mathrm{1}$, $m=\mathrm{0},\mathrm{1},\mathrm{\dots},{M}_{\mathrm{0}}-\mathrm{1}$, $s=\mathrm{1},\mathrm{\dots},and{S}_{\mathrm{0}}$ – were
calculated depending on the parameters ${H}_{\mathrm{0}},N,\mathit{\sigma},{\mathit{\sigma}}_{\mathrm{0}}$. The error of the processing algorithm ${\mathit{\epsilon}}_{n}^{\mathrm{2}}(k,m,{H}_{\mathrm{0}},N,\mathit{\sigma},{\mathit{\sigma}}_{\mathrm{0}})$ was found using averaging over the number of
realizations for fixed $n,k,$, and m:

The error described by Eq. (10) was averaged over the number of the
absolute geomagnetic vector projections n=1, 2, 3 and then over
different k and m. The final formula for estimating the error was the
following:

The results of the combined processing algorithm for the first stage were
compared with the results of the operation of a possible linear filtering
algorithm that was separately applied to the recordings of vector
magnetometer channels. The linear filtering algorithm in this case was
represented by standard equations:

The error estimate ${\mathit{\epsilon}}_{\mathrm{1}\mathrm{f}}^{\mathrm{2}}\left({H}_{\mathrm{0}}N\mathit{\sigma}\right)$ for the linear filtering algorithm was
calculated similar to Eqs. (10) and (11).

The efficiency of the proposed algorithm for combined processing of
measurements was estimated using the introduction of a relative decrease
factor for the RMS error values $\mathit{\rho}({H}_{\mathrm{0}},N,\mathit{\sigma},{\mathit{\sigma}}_{\mathrm{0}})$:

For statistical modelling, the following values have been assumed:
H_{0}=10 000 nT, N=5, M_{0}=50, K_{0}=50, S_{0}=100, σ_{0}=0.5, 0.3, 0.1, 0.03, and σ= 0.156, 0.312, 0.625, 1.25, 2.50, 5.00, 10.0. Figure 3 displays the results of the $\mathit{\rho}({H}_{\mathrm{0}},N,\mathit{\sigma},{\mathit{\sigma}}_{\mathrm{0}})$ calculation depending on
${\mathrm{log}}_{\mathrm{2}}(\mathit{\sigma}/\stackrel{\mathrm{\u203e}}{\mathit{\sigma}})$ and $\stackrel{\mathrm{\u203e}}{\mathit{\sigma}}=\mathrm{0.156}$. Numbers 1, 2, 3, and 4 indicate
the estimates for σ_{0}=0.5, 0.3, 0.1, and 0.03,
respectively. The estimates of the introduced factor made it possible to get
an idea of the effectiveness of the proposed processing.

Figure 3Results of calculation of the relative decrease factors for the
errors: 1 – σ_{0}=0.5; 2 – σ_{0}=0.3; 3 –
σ_{0}=0.1; 4 – σ_{0}=0.03.

Analysis of the graphs shows that, for a fixed value σ,
the introduced factor ρ decreases when σ_{0} decreases, which
is physically understandable. It is also seen that this factor tends to
limit values with increasing σ. For a fixed
H_{0} , an increase in N leads to a decrease in the factor
ρ. The performed statistical modelling for a wide range of parameters
shows that the estimates for ρ are about 0.15–0.3, which indicates the
efficiency of the proposed combined processing.

The proposed method for combined processing of the measurements of
projections and absolute magnitudes of geomagnetic field intensity vectors
and the corresponding two-stage algorithm developed appear to be
satisfactorily workable. The testing of the developed combined processing
algorithm on model and observatory measurement data proved its efficiency.

The approach realized in this paper, based on the proposed method, provides
an increase in quality of the measurements of the projections of the
geomagnetic intensity vector, allowing us to eliminate possible unwanted
disturbances of artificial (anthropogenic) origin.

Statistical modelling for the developed algorithm for combined processing of
measurements shows that at the first stage the relative decrease factor of
RMS errors can reach values of 0.15–0.3, and at the second stage the
decrease in the first-stage RMS errors can reach approximately 60 %–80 %.

Further reduction of the RMS noise errors can be implemented based on
combined processing using local piecewise linear models for the values of
projections and absolute magnitudes of geomagnetic field intensity vectors.
The proposed combined processing algorithm can be implemented for many
practically important tasks, in particular, when optimizing the operation of
three-component accelerometer systems, three-component angular velocity
sensors, or other three-component data arranged in a similar way. The
technique allows for processing of the data with sampling intervals smaller than 1 s.

The observatory geomagnetic data used for the tests in our study are
available at the INTERMAGNET website (http://www.intermagnet.org; INTERMAGNET, 2018) as magnetograms or as digital data files.

VGG is the author and the main developer of the method presented
in this research. Also, the basic work on programming was done by him. ADG performed the optimization and upgrade of the algorithm. RVS collected, analysed, and prepared the geomagnetic data for the
computational tests.

The results presented in this paper rely on data collected at the
INTERMAGNET magnetic observatories. We thank the national institutes that
support them and INTERMAGNET for promoting high standards of magnetic
observatory practice (http://www.intermagnet.org).

We thank Igor M. Aleshin, as well as the anonymous reviewers, for their
valuable comments that helped us to improve our article and clarify some
significant details.

This research has been supported by the Budgetary funding of the
Geophysical Center of RAS, adopted by the Ministry of Science and Higher
Education of the Russian Federation (grant no. 0145-2018-0002).

Mandea, M. and Korte, M. (Eds.): Geomagnetic Observations and Models, Springer, -XV, 343 pp., available at: https://www.springer.com/gp/book/9789048198573 (last access: 3 August 2019),
2011.

Getmanov, V. G.: Nonlinear filtering of observations of a system of vector and
scalar magnetometers, Meas. Tech.+, 56, 683–690,
https://doi.org/10.1007/s11018-013-0265-3, 2013.

Getmanov, V. G., Sidorov, R. V., and Dabagyan, R. A.: The method of filtering
signals using local models and weighted averaging functions, Meas. Tech.+,
58, 1029–1039, https://doi.org/10.1007/s11018-015-0837-5, 2015.

Kramer, G.: Matematicheskie metody statistiki (Mathematical Methods of
Statistics), Mir Publ., Moscow, Russia, 1975.

Merayo, J. M. G., Brauer, P., Primdahl, F., Petersen, J. R., and Nielsen, O. V.:
Scalar calibration of vector magnetometers, Meas. Sci. Technol., 14,
120–132, 2000.

Olsen, N., Tøffner-Clausen, L., Sabaka, T. J., Brauer, P., Merayo, J. M. G., Jørgensen, J. L., Leger, J. M., Nielsen, O. V., Primdahl, F., and Risbo, T.: Calibration of the Ørsted vector magnetometer, Earth Planets Space, 55, 11–18, 2003.

Shakhtarin, B. I.: Fil'try Vinera i Kalmana (Wiener and Kalman filters), Gelios ARV Publ., Moscow, Russia, 2008.

Soborov, G. I., Skhomenko A. N., and Linko Y. R.: Sposob opredeleniya
graduirovochnoy harakteristiki magnitometra (The method for determining the
calibration characteristics of the magnetometer), Patent RF, no. 2481593, Federal Service for Intellectual Property of Russian Federation, Moscow, Russia,
2008.

Soloviev, A., Lesur, V., and Kudin, D.: On the feasibility of routine baseline
improvement in processing of geomagnetic observatory data, Earth Planets
Space, 70, 16, https://doi.org/10.1186/s40623-018-0786-8, 2018.

Search articles

Download

Notice on corrigendum

The requested paper has a corresponding corrigendum published.
Please read the corrigendum first before downloading the article.

The material in this research paper is intended for specialists engaged in digital processing of geomagnetic field measurements. A technique is discussed that can help to reduce the errors in measurements and can be applied in various tasks of digital processing of geomagnetic data from vector magnetometers and other three-component data. The results of the tests on model and real geomagnetic data are provided for the algorithm along with the conclusions about its possibilities.

The material in this research paper is intended for specialists engaged in digital processing of...