Eur. Phys. J. D (2022) 76 :4 THE EUROPEAN
https://doi.org/10.1140/epjd/s10053-021-00328-9
PHYSICAL JOURNAL D
Regular Article – Atomic Physics
Assessing the quality of a network of vector-field sensors
Joseph A. Smiga1,2,a
1 Helmholtz-Institut Mainz, GSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, Germany
2 Johannes Gutenberg-Universität Mainz, 55128 Mainz, Germany
Received 8 October 2021 / Accepted 9 December 2021 / Published online 9 January 2022
© The Author(s) 2022
Abstract. An experiment consisting of a network of sensors can endow several advantages over an exper-
iment with a single sensor: improved sensitivity, error corrections, spatial resolution, etc. However, there
is often a question of how to optimally set up the network to yield the best results. Here, we consider a
network of devices that measure a vector field along a given axis; namely for magnetometers in the Global
Network of Optical Magnetometers for Exotic physics searches (GNOME). We quantify how well the net-
work is arranged, explore characteristics and examples of ideal networks, and characterize the optimal
configuration for GNOME. We find that by re-orienting the sensitive axes of existing magnetometers, the
sensitivity of the network can be improved relative to the past science runs.
1 Introduction For this work, the Global Network of Optical Magne-
tometers for Exotic physics searches (GNOME) [15–18]
Various experiments make use of a network of sensors is of particular interest. GNOME consists of shielded
in lieu of a single, centralized device. A network of sen- magnetometers around the Earth and has the goal
sors can come with several advantages such as having of finding new, exotic (vector) fields that couple to
better sensitivity than a single device, being able to fermionic spin. For example, GNOME searches for
catch and correct errors, and achieving superior spatial axion-like particle (ALP) domain walls [15,17,18] via
resolution. However, there are also a few challenges and the coupling of the ALP field gradient to nucleon spin.
complexities that arise when involving many devices. In The gradient, in this case, is in effect a vector field;
addition to the logistical challenges of managing mul- albeit with typical constraints, such as having a van-
tiple devices at once and making sense of several data ishing curl.
streams, there is the foundational question of how to This paper is organized as follows: a method of calcu-
best arrange the network. This question is explored for lating the network sensitivity is described in Sect. 2, a
a network consisting of a specific class of sensors: ones quantification of network quality is described in Sect. 3,
that measure a vector field. ideal and optimized networks are described in Sect. 4,
A common motivational interest in designing these and concluding remarks are given in Sect. 5. Through-
network experiments is in measuring spatially extended out this paper, the network under consideration will
phenomena. For example, interferometer networks used consist of the GNOME magnetometers. However, the
to measure gravitational waves [1–3], gravimeters used principles explored here can be extended to other net-
for geodesy [4], and magnetometers used for geophysics work experiments.
[5–7]; all of which measure vector-field phenomena1 on
scales the size of the Earth or larger. Networks have also
been used to search for direct evidence of dark matter,
which is believed to dominate the mass of the galaxy 2 Sensitivity
but only weakly interacts with visible matter; see, e.g.,
reviews in Refs. [8–10]. This includes both gravimeters The magnetometers in the network each possess a “sen-
[11–14] that search for the motion of dark matter cap- sitive axis” that results in the attenuation of a signal
tured by the Earth as well as magnetometers [15–20] when the vector field is not parallel or anti-parallel to
that search for coupling between dark and visible mat- the sensitive axis. Denote the sensitive axis of magne-
ter. tometer i with di . The magnitude of this vector reflects
the strength of the coupling such that a vector (field)
1 The interferometer networks can be understood as mea- m will induce a signal si = di · m in the ith magne-
suring something closer to a tensorial deformation in space- tometer. Consider the case in which only one vector m
time. However, similar to vector-field sensors that measure describes the signal. For a domain wall, this could be
a vector only along one axis, these interferometers are only the gradient at the center of the wall with the timing of
sensitive to certain polarizations of gravitational waves.
a e-mail: jsmiga@uni-mainz.de (corresponding author) the signal adjusted to account for delays as the domain
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wall crosses the network. The signals observed by the this distribution. If one is ambivalent about the direc-
network can be simplified into the linear equation, tion of the signal, the worst-case direction indicates a
bound on sensitivity.
Dm = s , (1) Ideally, the magnetometers in the network will be
oriented to evenly cover all directions. If there is a
{ } preferred and unpreferred direction, one could improvewhere D is a matrix whose rows are di and similarly
{ } the sensitivity in the unpreferred direction by rotatings = si . To avoid a trivial case, it is assumed that the sensitive axis of magnetometers toward this direc-
the network consists of at least one operating sensor;
 tion. Under practical conditions, it is not possible toso there exists a vector m such that Dm = 0. have GNOME always operating under optimal condi-
In practice, one will measure a set of signals s with tions because the noise in individual sensors varies over
some error. Let Σs be the covariance matrix in the mea- time and magnetometers will occasionally activate and
surements s that characterizes this error; the matrix deactivate.
will generally be diagonal as noise between the GNOME To judge how well the GNOME network is perform-
magnetometers is uncorrelated. An approximate solu- ing, it helps to define some quantitative “quality fac-
tion to Eq. (1) given the error in measurement is
2 − T −1 − tor.” This factor would ideally reflect how optimallyobtained by minimizing χ = (Dm s) Σs (Dm s). the network is set up with the magnetometers available
This solution is given by and not the absolute sensitivity of the network. That is,
− the quality of the network refers to how well the magne-m = Σ DT 1m Σs s (2a) tometers are oriented and is not affected by improving
for Σ−1 = DTΣ−1D , (2b) all magnetometers by a constant factor. One possibilitym s is the quotient of the best and worst sensitivity,
where Σm is the covariance matrix for m. As long as Σs
is a positive-definite matrix, as is the case for a realis- q0 := β1/β0 . (4)
tic network, Σ−1s is well-defined. However, Σm will not
be well-defined if D has a non-trivial kernel, Dm = 0 This factor will be zero if network has a blind spot
for m = 0; in other words, the network has a “blind (β0 → ∞) and one if the network has no preferred direc-
spot.” In this pathological case, one cannot uniquely tion. Generally, a more optimally oriented network has
reconstruct m, though one can still salvage a meaning- a larger q0. The quality factor for GNOME during the
ful definition of sensitivity. Science Runs is given in Fig. 1b.
The network sensitivity is defined as the magni- A few terms are defined here based on the quality of
tude √necessary to induce a |m| signal-to-noise ratio a network. Namely, an “ideal” network is one for which
= TΣ−1 of one. Thus, the sensitivity in the q0 = 1, while an “optimal” network is one in which theζ m m m
direction is sensitivity β0 cannot be improved by re-orienting them̂ sensors in the network.
√ Given a set of magnetometers with covariance matrix
β(m̂) = 1/ m̂TΣ−1m m̂ . (3) Σs and known coupling, it should be possible to deter-
mine a theoretical best sensitivity. For this, we will
This will vary by direction. However, one can find the consider a network with n independent magnetome-
range of sensitivities over different directions by solv- ters that are all described by a single sensitive axis di
ing for the eigenvalues of the symmetric, positive semi- with coupling strength κi := |di|. Consider the case( )
definite matrix Σ−1 = DTΣ−1D — the smallest in which the angle between any vector signal and anym s
−2 sensitive axis is random (this would be the case foreigenvalue λmin = β0 giving the “worst-case” sensitiv- ( )2
ity as a large signal β0 would be needed to induce a sig- many randomly oriented sensors). Because d̂i · m̂ ≈
nificant signal. Likewise the largest eigenvalue λ 〈 〉
− max
= 2
2 cos θ = 1/d, for d = 3 spatial dimensions, Eq. (3) inβ1 gives the “best-case,” and the corresponding eigen- this case becomes,
vectors are the directions that induce such signals. If
the network has a blind spot, then λmin = 0 so β → ∞ √√ /
along the corresponding direction. n−1
β ≈ √
√ ∑
opt d (κ2i /σ
2
i ) (5)
i=0
3 Quality factor This may not be the optimal sensitivity, but it pro-
vides a heuristic for an optimal network. The quo-
Given the sensitivity defined by Eq. (3), there remains tient between the observed and optimal sensitivity for
the question of how to optimize the network; in partic- GNOME over time is given in Fig. 1c.
ular, how to optimize the directions {di} for the best With the quality factor in mind, it helps to consider
network. If there is distribution of directions of interest exactly how sensitivity varies with direction. The net-
for the vector field, one could define the optimization work has been fairly stable with many active sensors
by performing some weighted average of Eq. (3) over during the recent Science Run 5. A map of the aver-
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Eur. Phys. J. D (2022) 76 :4 Page 3 of 8 4
(a)
(b)
(c)
Fig. 1 The quality factor over time for Science Runs 1–5. Solid lines represent 1 day rolling averages. a The number
of active sensors over time. b The quality factors over time. The color indicates the number of active sensors. c The
approximate factor by which the network could be improved if the sensors were optimally oriented using the approximate
optimized sensitivity, Eq. (5)
〈 〉−1 T
age sensitivity2, ( )−1β m̂ , in different directions is and Σ1 = PΣ0P ; that is, they are the same up to
ordering. Additionally, a network can be decomposed
shown in Fig. 2. The network quality could be improved into two complementary subnetworks N ∼= N ⊕ N
by improving the quality/reliability of stations sensi- A Bif
tive to insensitive directions (e.g., Moxa or Daejeon) or
rotating/adding additional sensor(s) toward the worst {[ ] [ ]}
direction. ∼ DN = A Σ, A 0 .DB 0 ΣB
4 Ideal and optimized networks Observe that the subnetworks NA and NB are indepen-dent/uncorrelated.
In addition to the basic equivalence relation described
With the quantitative definition of network quality above, there are some additional symmetries for a
given in the previous section, various optimized and network. First, the sensitivity β(m̂) is invariant with
ideal networks are given here. In particular, we consider respect to parity reversal of a subnetwork; i.e., DA →
properties and explicit arrangements of ideal networks −DA for N ∼= NA ⊕ NB . Further, though the sensi-
as well as numerical optimizations of more realistic net- tivity β(m̂) of a (non-ideal) network can change under
works. arbitrary rotation of the whole network DT → RDT ,
To better understand the characteristics of networks, the value of the worst sensitivity, best sensitivity, and
it helps to define some additional formalism. Define a quality factor do not.
network with a given set of orientations as a pair of
(n×d) directional matrix and (n×n) covariance matrix
N = {D,Σ}; for n sensitive axes and d = 3 spatial
{ } 4.1 Ideal networksdimensions. Two networks N0 = D0,Σ0 and N1 =
{D1,Σ1} can be considered equivalent N ∼0 = N1 if there There are a few useful characteristics of ideal net-
exists a permutation matrix P such that D1 = PD0 works that are worth considering. For an ideal net-
work, q0 = 1 so the smallest and largest eigenvalues of
2 Using the average of the inverse sensitivity accounts for DTΣ−1D are the same which implies that this matrix
blind spots β−1 → 0. is proportional to the identity matrix. In particular,
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Fig. 2 Average sensitivity β of GNOME during Science Run 5 (23 August–31 October 2021). A position on the Earth
represents the direction perpendicular to that point on the Earth. The direction of the sensitive axes of the stations is
represented by  (parallel) and ⊗ (anti-parallel). That is, if a sensor was moved to the corresponding geographical location,
then its sensitive axis would be vertical. The  markers are labeled with name of the station, given by the city in which
the sensor is physically located. The marker color is a visual aid to associate the pair of markers on opposite sides of the
Earth representing the same station
DTΣ−1D = β−21. It also follows that the sensitiv- then the resulting network will be ideal with sensitiv-
ity of an ideal network is independent of global rota- ity β = σ/κ. Heuristically, one would like to orient the
tions. magnetometers to evenly cover all directions. One way
Consider, now, how ideal networks are combined. Let to do this is to take some inspiration from the Platonic
NA and NB be two complementary, ideal subnetworks solids by designing a network in which the sensitive axes
of N , then of the sensors are oriented from the center of the solid to
each of the vertices; see Table 1. Most of these solids will
[ ] [ ]−1 [ ]Σ 0 D ( − − ) generate a network with two ideal subnetworks havingD T D T A A 2 2A B 0 Σ = βA + βB 1 .D opposite sensitive axes. Thus, one can obtain ideal net-B B works with three (octahedron), four (tetrahedron and
That is, the network ∼ ⊕ is also ideal with cube), six (icosahedron), and ten (dodecahedron) sen-N N N
( ) = A B− − −1/2 sors through this method; denoted N3, N4, N6, andsensitivity β 2 2A + βB . Further, because an ideal N . These arrangements have the sensitivity β = √σ/κ
network remains ideal under rotations, one can rotate 10 n/3
either ideal subnetwork without affecting the sensitiv- and are irreducible. With these networks alone, it is evi-
ity of the network N . Further, if N = NA ⊕ NB is an dent that there are multiple unique ways of orienting a
ideal network and NA is an ideal subnetwork, then NB given number of sensors that do not rely on using the
is also an ideal subnetwork. An ideal network that can- same set of ideal subnetworks; for example, six sensors
not be separated into ideal subnetworks is “irreducible.” can be arranged as N6 or N3 ⊕N3.
Because an ideal network needs at least d sensitive axes
(for d = 3 spatial dimensions), an ideal network with Combining N3 and N4 subnetworks, one can design
n < 2d is irreducible, because it cannot be split into ideal networks with three, four, six, or more sensors.
two ideal networks. What seems to remain is a way to orient five identi-cal, independent sensors into an ideal network. One can
One can consider certain explicit cases of ideal net- show that two such network arrangements N5a and N5b
works with some simplified conditions. In particular, let are given by
N = {D,Σ} be composed of n identical, independent,
single-axis sensors — that is, Σ = σ21 and |di| = κ
where d is the ithi row of D. If there are d = 3 sen-
sors oriented such that their sensitive axes are orthog-
onal,
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√ √
Table 1 Examples of optimal networks based on the pla- each with sensitivity β = σ/κ 5/3. These networks
tonic solids. The orientations of sensitive axes in an ideal are unique, even when considering parity reversal of
network are given as lines (dashed in one direction, solid in individual sensors, reordering, and global rotations; this
the other). The number of vertices are separated by ideal is evident becauseD5b has orthogonal sensors whileD5a
network. For example, the cube has eight vertices, and the does not, and orthogonality of two vectors is invari-
corresponding ideal network consists of two ideal networks ant under these operations. Further, these are irre-
with four sensors each; hence, “4+ 4” is listed. In each case ducible ideal networks because n < 6. Along with N3
here, X + X vertices describe two ideal networks with X and N4, one can generate any ideal network with at
sensitive axes in opposite directions least three identical, independent sensors using these
arrangements.
Though one can always arrange three or more iden-
tical, independent sensors into an ideal network, it is
not always the case that a given set of sensors can be
arranged into an ideal network. For example, consider
a set of n ≥ d = 3 sensors all with the same coupling
κ = 1, but n − 1 sensors h√ave noise σ0 and the last
sensor has noise σ1 < σ0/ (n− 1)/2. An optimized
network would be arranged with the first n− 1 sensors
evenly oriented around the plane or√thogonal to the last
sensor and have a quality q0 = σ1 n−1σ 2 < 1 and sen-0
sitivity β0 = √ σ0
(n− . The least-sensitive direction is1)/2
orthogonal to the last sensor’s sensitive axis, and any
adjustments to the sensors’ orientations would worsen
the sensitivity in this plane. The problem in this sce-
nario is that one sensor is much more sensitive than the
others, so they cannot compensate, even collectively.
The extreme case of this would be if the first n− 1 sen-
sors were so noisy that they were effectively inoperable;
this arrangement would not be much different than an
n = 1 network, which cannot be ideal.
4.2 Optimizing networks
Regardless of whether a given set of magnetometers
can be arranged into an ideal network, their arrange-
ment can always be optimized; at least at a given time.
In practice, the noise in each GNOME magnetometer
varies over time and magnetometers turn on and off
throughout the experiment, so an optimal arrangement
at one time may not be optimal at another.
A non-optimized network can still be improved via
an algorithm. An example of a “greedy” algorithm
⎡ √ √ ⎤ would be one in which, each step, a sensor is randomly
5 1
⎢ 0 √6 √ 6⎥ selected, removed from the network, and re-inserted in
⎢⎢ ⎥0 − 5 1⎥ the least-sensitive direction for the network without the⎢ √ 6 √ 6⎥ removed sensor. This step can then be repeated many
D ⎢ ⎥5a = κ⎢ 5 0 1⎥ times until reaching some optimization condition. For⎢⎢ √
6 √ 6⎥⎥ multi-axis sensors that always have the same relative
⎣− 5 0 1⎦6 6 angle between the sensitive axes, the orientation by
0 0 1 which to re-insert the sensor is a bit more complicated.
⎡ ⎤
1 0 0 Roughly, one would apply a rotation to the sensor to√ √
⎢ ⎥ align the best- and worst-directions for the multi-axis⎢− 13 − 2 0⎢ √ √ 3 ⎥⎥ sensor and the rest of the network (see Appendix A).⎢− 1 2 ⎥ This algorithm will also work regardless of whetherand D ⎢ 0 ⎥5b = κ⎢ 3 √ 3 √ ⎥ , (6) an ideal network arrangement exists, though the result-⎢⎢ 0 1 5⎥⎥ ing network may not be optimal. Subsequent steps of
⎣ √6 √ 6⎦ the algorithm may converge to a non-optimal network
0 − 1 56 6 or alternate between network configurations.
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Table 2 Optimizing GNOME for each run. In particular, a network is constructed usi〈ng al〉l magnetometers active for at−1
least 25% of the run. Noise is calculated as the average standard deviation of the data σ−1 after applying a 1.67mHz
high-pass filter, 20 s averaging, and notch filters to remove powerline frequencies. This choice in how the average noise
is calculated suppresses the influence of brief periods in which there was a spike in noise. The columns contain the run
number, number of sensors used in the optimization, network characteristics, optimized network characteristics, theoretical
optimized sensitivity from Eq. (5), and factor by which the optimization algorithm improved sensitivity
During run Optimized
Run Size β0 (pT) q0 β0 (pT) q0 βopt (pT) Improved
1 6 0.74 0.19 0.27 0.62 0.22 2.72
2 9 0.48 0.49 0.30 1 0.30 1.59
3 7 0.51 0.27 0.26 0.61 0.21 1.97
4 9 1.13 0.22 0.42 0.79 0.37 2.67
5 11 0.25 0.55 0.18 0.86 0.17 1.35
The optimization can be applied to the GNOME net- how the sensor is built, it may not be possible/practical
work to better understand the potential of the experi- to reorient it, or it may only be possible to orient its
ment. In particular, this was performed for GNOME’s sensitive axis vertically or horizontally. Additionally,
official Science Runs: what may be an optimal network under some set of
conditions may not be optimal under later conditions
• Science Run 1: 6 June–5 July 2017. depending on how the sensitivities of the sensors change
• Science Run 2: 29 November–22 December 2017. and which ones are active. However, if there are many
• Science Run 3: 1 June 2018–10 May 2019. decent/reliable sensors, the network as a whole will not
• Science Run 4: 30 January–30 April 2020. usually deviate too much from its optimal arrangement.
• Science Run 5: 23 August–31 October 2021. This work only considered the manner in which thesensors were oriented, not their position. The relative
positions and distances between the sensors are not rel-
The results of the optimization are given in Table 2 in evant to understanding the sensitivity of the network
which the coupling was assumed to be one. For most of as a whole. Briefly, the optimal placement of the sen-
the Science Runs, the improvement was less than a fac- sors in a network differs depending on the goal of the
tor of two with the largest improvement being by a fac- experiment. It is generally simpler to observe a signal
tor of 2.7 from Science Run 4. Additionally, the Science that crosses a network of nearby sensors; because the
Run 2 network could be made ideal with the same sen- potential crossing time is shorter, less data need to be
sitivity as predicted in Eq. (5). It should be noted that compared between the sensors. However, measuring the
the networks used in the optimization included sensors direction and speed of some phenomena crossing the
available at least 25% of the run, while it is not typi- network can be done more accurately when the sensors
cal for all these sensors to be active at the same time. are more distant from one another.
Another example of network optimization is given in The work described in this paper can be applied to
Appendix B wherein only a couple sensors are reori- any experiment involving a network of directionally sen-
ented and an additional sensor is added. sitive devices. These networks are useful in detecting
features in a vector field or gradient that traverse a spa-
tial region. Both the design and improvement of these
networks can be meaningfully improved through careful
5 Conclusions consideration of how sensors are oriented.
Moving forward, this study can have a direct influ-
In this paper, we have considered a network of sensors ence on Advanced GNOME; a planned general upgrade
with directional sensitivity and how the collective sensi- to the GNOME experiment. In particular, this study
tivity of the network is affected by the choice in how the provides the tools to understand how to orient new
sensors are oriented. To do this, a “quality factor” was sensors and re-orient existing sensors as to optimize
introduced to quantify how efficiently the network sen- the network sensitivity—taking advantage of the major
sors are oriented regardless of their underlying sensitiv- upgrade period to make improvements to the collective
ity. Various properties and examples of ideal networks network. The Advanced GNOME upgrade includes the
were presented, along with a means of optimizing an addition of SERF comagnetometers [21] with the option
existing network. By optimizing GNOME, we can show to operate in two-axis mode. The use of multi-axis sen-
some modest improvement in the network sensitivity sors can be incorporated into the work presented here
without requiring additional sensors or improvements by treating them as multiple sensors with correlated
in existing sensors. noise. These sensors also have the constraint that the
In practice, there are still some limitations in how sensitive axes must remain orthogonal. Optimization
sensors can be oriented. For example, depending on
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Eur. Phys. J. D (2022) 76 :4 Page 7 of 8 4
with multi-axis sensors is explored in Appendix A con- sensitive) for the respective network: x̂ → the worst-
sidering this constraint. direction and ẑ → the best-direction. Finally, define Ũi
as Ui with its columns reversed; this also reverses which
Acknowledgements Work for this paper was made possi- coordinate axis will be rotated to which direction.
ble through discussions with the GNOME collaboration as When orienting the multi-axis sensor to be included
well as characterization data from the experiment. Discus- in the rest of the network, it is optimal to orient the
sions with Dr. Derek Jackson Kimball, Dr. Dmitry Budker, respective best-direction of the multi-axis sensor with
Dr. Szymon Pustelny, Dr. Ibrahim Sulai, and Hector Masia- the worst-direction of the rest of the network and vice-
Roig greatly aided in the completion of this work. JAS has versa. This is accomplished by rotating the multi-axis
no funding sources to declare. sensor as follows:
( )T
Da → D Ũ UTa b a . (A1)
Author contributions
This rotation maps:
All authors have equally contributed to this work.
T
Funding Information Open Access funding enabled and ŨbUa :worst for a → x̂ → best for b
organized by Projekt DEAL. best for a → ẑ → worst for b .
Data Availability Statement This manuscript has no For the diagonal matrix with the eigenvalues in descend-
associated data or the data will not be deposited. [Authors’
comment: GNOME data can be made available upon rea- ing order Λ̃i (i ∈ {a, b}), the combined network will
T −1
sonable request. Data are displayed at https://budker. have DabΣab Dab = UbΛ
T
abUb for Λab = Λ̃a+Λb, a diag-
uni-mainz.de/gnome/.] onal matrix of eigenvalues (not necessarily ordered).
The quality factor is given by th√e ratio of the small-
Open Access est and largest eigenvalue q =
λmin . The eigenval-
This article is licensed under a Creative Com- 0 λmax
mons Attribution 4.0 International License, which permits ues for the combined network consist of the sum of the
use, sharing, adaptation, distribution and reproduction in smallest eigenvalues for one subnetwork and the largest
any medium or format, as long as you give appropriate credit eigenvalues for the other. This decreases the range of
to the original author(s) and the source, provide a link to eigenvalues and optimize the network.
the Creative Commons licence, and indicate if changes were For example, the Hayward, Krakow, and Mainz sen-
made. The images or other third party material in this arti- sors could operate as two-orthogonal-axis magnetome-
cle are included in the article’s Creative Commons licence, ters during Science Run 5; though at the cost of worse
unless indicated otherwise in a credit line to the material. If sensitivity, roughly doubling the variance. Replacing
material is not included in the article’s Creative Commons these three sensors during the Science Run with (uncor-
licence and your intended use is not permitted by statu- related) two-axis sensors and including them with opti-
tory regulation or exceeds the permitted use, you will need mal orientations improves the sensitivity by a factor
to obtain permission directly from the copyright holder. of 1.35 to β = 0.18 pT. This is slightly better than
To view a copy of this licence, visit http://creativecomm 0the 1.21 factor of improvement for optimizing the three
ons.org/licenses/by/4.0/. sensors in single-axis mode.
Appendix A: Multi-axis sensors Appendix B: Applied optimization
Consider a multi-axis magnetometer that can only be
reoriented by rotating the entire sensor—one cannot It is useful to consider an explicit examples of optimiz-
adjust individual sensitive axes. Some effort is needed ing parts of the network. Specifically, consider reorient-
to understand how to orient such a sensor with respect ing the Mainz and Krakow magnetometers and adding
to a larger network. another magnetometer with 0.2 pT standard deviation
Let = { noise after filtering/averaging. The new magnetometerNa Da,Σa} describe the orientation of the
multi-axis sensor and Nb = {Db,Σb} describe the rest will be located in Berkeley, CA, USA. This optimizationwill use the Science Run 5 characteristics.
of the network. The matrices DTi Σ
−1
i Di (for i ∈ {a, b}) Locally, the orientation is expressed in horizontal
can be diagonalized as U TiΛiUi where the Λi is a diag- coordinates; using altitude (alt) and azimuthal (az)
onal matrix whose jth element along the diagonal is angles expressed as the pair (alt, az). The altitude is
the jth-smallest eigenvalue λij , and Ui is an orthogonal the angle relative to the horizon, while the azimuth is
matrix (U−1 = UT ) whose jthi i column is the respective the angle relative to noise (measured clockwise). The
(normalized) eigenvector uij . The orthogonal matri- Mainz sensor is oriented with alt = −90◦, while the
ces have the effect of rotating the coordinate axes to Krakow sensor is oriented as alt = +90◦ during Science
the best- and worst-directions (i.e., most- and least- Run 5.
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Optimizing the direction of the three magnetome- 17. H. Masia-Roig et al., Analysis method for detecting
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q0 = 0.89. Using Eq. (5), the optimal sensitivity would netometer network. Phys. Dark Universe 28, 100494
be βopt = 0.16 pT. When optimized, the Mainz mag- (2020)
netometer is oriented as (74◦, 99◦), the Krakow mag- 18. S. Afach et al., Search for topological defect dark matter
netometer is oriented as (6◦,−19◦), and the Berkeley with a global network of optical magnetometers. Nat.
magnetometer is oriented as (10◦, 123◦). Phys. 1–6, 2102.13379 (2021)
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