remote sensing

Communication
An Observation Density Based Method for Independent
Baseline Searching in GNSS Network Solution
Tong Liu 1 , Yujun Du 2, *, Wenfeng Nie 3,4 , Jian Liu 1 , Yongchao Ma 1 and Guochang Xu 1,5

1 Laboratory of Navigation and Remote Sensing, Institute of Space Science and Applied Technology (ISSAT),
Harbin Institute of Technology, Shenzhen 518055, China
2 The Industrial Sciences Group, Sydney, NSW 2060, Australia
3 Institute of Space Sciences, Shandong University, Weihai 264209, China
4 State Key Laboratory of Geo-Information Engineering, Xi’an 710000, China
5 Key Laboratory of Urban Land Resources Monitoring and Simulation, Ministry of Natural Resources,
Shenzhen 518000, China
* Correspondence: eugene@industrialsciences.com.au; Tel.: +61-411-644-071

Abstract: With applications such as precise geodetic product generation and reference frame main-
tenance, the global GNSS network solution is a fundamental problem that has constantly been a
focus of concern. Independent baseline search is a prerequisite step of the double-differenced (DD)
GNSS network. In this process, only empirical methods are usually used, i.e., the observation-max
(OBS-MAX), which allows for obtaining more redundant DD observations, and the shortest-path
(SHORTEST), which helps to better eliminate tropospheric and ionospheric errors between stations.
Given the possible limitations that neither of the methods can always guarantee baselines of the
highest accuracy to be selected, a strategy based on the ‘density’ of common satellites (OBS-DEN)
is proposed. It takes the number of co-viewing satellites per unit distance between stations as the
criterion. This method ensures that the independent baseline network has both sufficient observa-
Citation: Liu, T.; Du, Y.; Nie, W.; Liu, tions and short baselines. With single-day solutions and annual statistics computed with parallel
J.; Ma, Y.; Xu, G. An Observation processing, the method demonstrates that it has the ability to obtain comparable or even higher
Density Based Method for positioning accuracy than the conventional methods. With a clearer meaning, OBS-DEN can be an
Independent Baseline Searching in option alongside the previous methods in the independent baseline search.
GNSS Network Solution. Remote Sens.
2022, 14, 4717. https://doi.org/ Keywords: GNSS; independent baseline; GNSS network solution; observation-max; shortest;
10.3390/rs14194717 observation density; minimum spanning tree
Academic Editors: Zhetao Zhang,
Wenkun Yu and Giuseppe Casula

Received: 16 August 2022

1. Introduction
Accepted: 16 September 2022
Published: 21 September 2022
The global GNSS network solution plays an important role in geodesy, especially
geodetic parameter estimation [1], high-precision product generation [2,3], datum main-
Publisher’s Note: MDPI stays neutral tenance [4], and geodynamics applications [5,6]. As a well-developed method, double
with regard to jurisdictional claims in differencing (DD) [7] is widely used in well-known GNSS data processing software such as
published maps and institutional affil-
Bernese 5.2 developed by Rolf Dach et al. at the Astronomical Institute of the University of
iations.
Bern (AIUB), Switzerland [8] and GAIMIT/GLOBK 10.7 developed by T. A. Herring et al.
from MIT, Scripps Institution of Oceanography and Harvard University in America [9].
How to improve the accuracy of the GNSS DD network is a topic that has been continuously
Copyright: © 2022 by the authors.
explored.
Licensee MDPI, Basel, Switzerland.
In the implementation of the GNSS network solution, in order to reduce the computa-
This article is an open access article
tional load while not affecting the overall positioning accuracy, the independent baseline
distributed under the terms and solution of multiple stations should be used before the entire network adjustment [7]. The
conditions of the Creative Commons principle of independent baseline selection is that only one path exists between any two
Attribution (CC BY) license (https:// stations, while all the stations should be connected. For a network with n stations, a total
creativecommons.org/licenses/by/ of n(n + 1)/2 baselines exist, only n − 1 of which are independent. The objective of the
4.0/). independent baseline selection is to optimize the overall accuracy of the baseline solutions

Remote Sens. 2022, 14, 4717. https://doi.org/10.3390/rs14194717 https://www.mdpi.com/journal/remotesensing

Remote Sens. 2022, 14, 4717 2 of 12

in order to facilitate the subsequent network adjustment. Mathematically, this process can
be described by the minimum spanning tree (MST) [10,11] problem.
In the process of MST, the criteria for selecting baselines can be defined according to
the user’s needs. One of the most easily conceived solutions is to make the total length of
n − 1 baselines the shortest, which is known as the shortest path (SHORTEST) method [7,8].
This is because the shorter the distance between stations, the greater the number of co-
viewing satellites, thus more redundant observations are involved to facilitate the network
adjustment. More importantly, the tropospheric and ionospheric delays of neighboring
stations are similar. The shorter baselines help to better eliminate these errors. Since
the original intention of SHORTEST is to improve positioning accuracy by increasing the
number of co-viewing satellites, a more straightforward solution is to use the maximum
common satellites as the criterion. This method is known as the maximum observation
method (OBS-MAX) [8].
Both methods mentioned above have been investigated; for example, SHORTEST
was used in a massive GNSS network of more than 2000 globally distributed stations [12],
while OBS-MAX is shown to be beneficial in the tropospheric delay estimation [13]. In
the ideal situation, the shorter the baseline, the more common observations there are.
Then, SHORTEST and OBS-MAX should be fully equivalent. However, the statistics show
that they are not consistent [14], i.e., on various days, the baselines generated by different
methods could ultimately lead to different solution precision, which violates the assumption
that the two methods are equivalent. This means that the number of observations does
not necessarily increase as the baseline becomes shorter. This is due to the fact that the
satellites are usually not evenly distributed across the sky, e.g., sparse observations in local
areas and sufficient co-viewing satellites for some long baselines. In a word, the search for
optimal independent baselines is still an open question to be further investigated.
A scheme of setting up weights (WEIGHT) between SHORTEST and OBS-MAX has
been proposed [14]. The WEIGHT method was demonstrated to be of higher positioning
precision than that of SHORTEST and OBS-MAX. However, how to set up weights lacks
theoretical support and can only be empirical. For instance, the weights can be determined
based on the posterior accuracy of the final baseline solutions using each of the two methods;
on an a priori basis, the Bernese software could use a weight of 30% for the SHORTEST in
addition to OBS-MAX as an option [8].
To avoid setting up empirical weights or doubling the computational load brought
by a posteriori precision-based weighting, a new method called “observation-density”
(OBS-DEN) is proposed here. It takes the ratio of baseline length and the number of
observations between two stations as the criterion of the MST. The physical interpretation
of this criterion is the number of common satellites per unit distance, which overcomes the
degradation of baseline accuracy by seeking only maximum observations or the shortest
baselines. The advantage of OBS-DEN is that it provides an explainable weighting scheme
that can overcome the downsides of SHORTEST and OBS-MAX. This method can be used
in various types of GNSS network solution-related software, alongside existing options for
users to choose from.
The rest of the paper is organized as follows. In Section 2, the datasets and products,
the principle of MST, and the criterion with OBS-DEN are introduced. Then the flowchart
for generating independent baselines using various methods and the parallelization of
network processing is presented in Section 3. Afterward, the results of single-day solutions
and annual statistics are analyzed and discussed in Section 4. Finally, the paper is concluded
in Section 5.

2. Data and Method

2.1. Data
Observation data from about 100 IGS stations distributed worldwide, was used to test
the proposed method. The data was in Receiver Independent Exchange Format (RINEX)
and can be accessed through (https://cddis.nasa.gov/archive/gnss/data/daily/2012/,
 2.1. Data
Observation data from about 100 IGS stations distributed worldwide, was used to
test the proposed method. The data was in Receiver Independent Exchange Format
(RINEX) and can be accessed through
Remote Sens. 2022, 14, 4717 (https://cddis.nasa.gov/archive/gnss/data/daily/2012/, Weihai, China, 1 June 2018).3 First, of 12

data from 13 January 2012, was used to show the accuracy of a single‐day solution. After
that, data from the whole year 2012 was used for statistical analysis. We chose a data span
over the
Weihai, year1 2012,
China, as it was
June 2018). First,the
datafirst
from peak of the last
13 January 2012,solar cycleto[15],
was used which
show helps to
the accuracy
ofinvestigate
a single-daythe solution. After that,
performance of data
the from the whole
proposed year 2012
method under wasvarious
used forionospheric
statistical
analysis.
situations. We chose a data span over the year 2012, as it was the first peak of the last
solar cycle
Both [15],
GPSwhichand helps
GLONASSto investigate the performance
observations of the proposed
were included method under
in the processing. The
various ionospheric situations.
sampling interval of the data was 30 s. Precise products, including the precise orbit and
Both(*.SP3
clocks GPS and
andGLONASS observations
*.CLK), Earth rotationwere included(*.ERP),
parameters in the processing.
ionosphericTheparameters
sampling
interval
(*.ION),ofdifferential
the data was 30 s.biases
code Precise(*.DCB),
products, including
reference the precise(*.CRD),
coordinates orbit andantenna
clocks (*.SP3
phase
and *.CLK),
center Earth rotation
corrections, receiver parameters
types, and(*.ERP), ionospheric
tidal loading parameters
corrections (*.BLQ(*.ION), differen-
and *.ATL) were
tial code biases
adopted (*.DCB),
to enable reference coordinates
high‐precision GNSS network (*.CRD), antenna
solution. Thephase center
data was corrections,
processed with
receiver types, and tidal loading corrections (*.BLQ and *.ATL) were
the Bernese software developed at the Astronomical Institute of the University of Bern adopted to enable
high-precision
(AIUB). Detailed GNSS network solution.
descriptions The data
or flow charts was processing
of data processed with
withthe Bernese
Bernese cansoftware
be found
developed at the Astronomical Institute of the University of Bern
in the Bernese manual [8] or other publications [12,14,16–18]. The products (AIUB). Detailed descrip-
were
tions or flow
downloaded charts of data
and processing with
used Bernese can be found
following in the Bernese
the manual [8]
website
or(https://cddis.nasa.gov/Data_and_Derived_Products/GNSS/GNSS_product_holdings.ht
other publications [12,14,16–18]. The products were downloaded and used following
the
ml,website
Weihai,(https://cddis.nasa.gov/Data_and_Derived_Products/GNSS/GNSS_product_
China, 1 June 2018) and the instruction of Bernese [8].
holdings.html, Weihai, China, 1 June 2018) and the instruction of Bernese [8].
2.2. MST
2.2. MST
MST is the mathematical basis for GNSS DD independent baseline solutions given
MST is the mathematical basis for GNSS DD independent baseline solutions given
the baseline length and the number of observations that are calculated and counted. Let V
the baseline length and the number of observations that are calculated and counted. Let V
denote all possible Vertexes and E denote all possible edges. Let e denote an edge and w(e)
denote all possible Vertexes and E denote all possible edges. Let e denote an edge and w(e)
denote the criterion of that edge. This criterion w(e) can be distance or any other weighting
denote the criterion of that edge. This criterion w(e) can be distance or any other weighting
factors. MST is defined as follows. In an undirected graph G = (V, E), if there exists a subset
factors. MST is defined as follows. In an undirected graph G = (V, E), if there exists a subset
T ofofE,E,such
T suchthat
thatthethesum
sumofofw(e)w(e)ofofe ethat
thatcan
canconnect
connectallallnodes
nodes(V)
(V)isisminimal,
minimal,then
thenthat
that
subset T is called the minimum spanning tree of E, or the minimum
subset T is called the minimum spanning tree of E, or the minimum weight spanning tree. weight spanning tree.
InIngeometry,
geometry,the thebrief
briefexplanation
explanationofofMST MSTisisthat
thatititisisthe
theshortest
shortestpath
paththat
thatconnects
connects
all n nodes. It is easy to understand that there are always 𝑛 1edges
all n nodes. It is easy to understand that there are always n − 1 edges in an MST, and these in an MST, and these
edgesare
edges areindependent,
independent,i.e., i.e.,the
thepath
pathbetween
betweenany anyofoftwo
twonodes nodesisisunique.
unique.TheTheschematic
schematic
diagram is shown in Figure 1. Usually, the solution of MST
diagram is shown in Figure 1. Usually, the solution of MST can be obtained using can be obtained usingthethe
Kruskal method [19,20] or the Prim
Kruskal method [19,20] or the Prim method [21,22].method [21,22].

Figure1.1.The
Figure Theschematic
schematicdiagram
diagramfor
forselecting
selectingthe
theminimum
minimumspanning
spanningtree
treefrom
fromallallpossibilities.
possibilities.

2.3.
2.3.The
TheCriteria—Distance,
Criteria—Distance,Observation,
Observation,and
andOthers
Others
The goal of the baseline selection is to optimize the overall accuracy of baseline
solutions. For the SHORTEST method, only the length of the path is used as the criterion.
Mathematically, the weight w_SHORTEST (e) in MST is proportional to the baseline length,
i.e., the smaller the baseline length, the smaller w_SHORTEST (e) is. This is based on the
assumption that the closer two stations, the more thoroughly common errors can be
eliminated or reduced by DD, thus the higher overall accuracies can be achieved for
baseline solutions. For the OBS-MAX method, on the other hand, only the number of
DD observations is used as the criterion, i.e., w_OBS-MAX (e) is considered to be inversely
proportional to the number of observations. The more the DD observations, the smaller
w_OBS-MAX (e) is. This strategy is based on the fact that more observations can bring higher
Remote Sens. 2022, 14, 4717 4 of 12

redundancy in parameter estimation. WEIGHT is a synthesis of these two strategies. In this

process, a normalization is introduced since the dimensions of OBS-MAX (number) and
SHORTEST (meter) are not consistent [14]. Unfortunately, WEIGHT is still an empirical
operation lacking theoretical support.
Since both shorter baselines and more observations can result in higher accuracies
for baseline solutions, it is reasonable to adopt a special weight for MST which is both
proportional to the length of the path and inversely proportional to the number of observa-
tions. The proposed criterion, i.e., the number of DD observations per unit distance, can be
interpreted as the density of the observations over baselines. Thus, the proposed method is
named observation-density (OBS-DEN).
Equation (1) represents the definition of w(e) in different baseline selection methods.


 wSHORTEST (e) = msho
wOBS−MAX (e) = 1/mobs

(1)
w (e) = x1 /norm(mobs ) + x2 × norm(msho ), ( x1 ≥ 0, x2 ≥ 0, x1 + x2 = 1)
 WEIGHT


wOBS−DEN (e) = msho /mobs
where mobs denotes the number of co-viewing satellites observed by every two stations,
msho denotes the geodetic distance of each two stations; x1 and x2 denote the weights
applied to the mobs and msho factors, respectively, which can be obtained empirically or
based on the a posteriori accuracies of the solutions of the two methods. Note that in this
paper, the number of observations is counted on a daily basis.

2.4. The Calculation Process of the Independent Baseline

The calculation process is illustrated in Figure 2. First, the observation files of all
stations are loaded. After that, the co-observations between every two stations are retrieved
per epoch. The total co-observations of each station pair of a day are aggregated, respec-
tively. In this way, the common observation matrix Mobs is formed, and each element mobs
represents the number of common observations between every station pair. At the same
time, the geodetic distance between every station pair is calculated to form the distance
matrix Msho . The unit of element msho is meter. Mobs and Msho are shown in Equation (2).
Then, the MST is applied to the matrix Msho , which chooses the solution that lets the
sum of msho be the smallest. The main diagonal elements of the M matrix represent all
available observations of individual stations or 0 distances, which are not involved in the
MST generation.

m1,1 m1,2 m1,3 m1,n m1,1 m1,2 m1,3 m1,n

   
obs obs obs ··· obs sho sho sho ··· sho
m2,2 m2,3 ··· m2,n m2,2 m2,3 ··· m2,n
   
 obs obs obs   sho sho sho 

.. .. ..  
.. .. .. 
Mobs = . . . , Msho = . . . (2)
   

.. ..
   

. n−1,n  
. n−1,n 
 mobs   msho 
mn,n
obs mn,n
sho
For the matrix Mobs , the reciprocal of each element or the maximum spanning tree
should be used, since the largest observation needs to be chosen instead of the smallest. Cor-
respondingly, the WEIGHT matrix Mwei and the OBS-DEN matrix Mden can be computed
as follows:

Mwei = x1 × Msho + x2 /Mobs , ( x1 ≥ 0, x2 ≥ 0, x1 + x2 = 1)
(3)
Mden = Msho /Mbos

In the subsequent experiment and data analysis, both x1 and x2 of the WEIGHT were
set to 0.5. In order to apply the above methods with Bernese, the generated baseline file can
be used to replace the baseline file generated by Bernese’s default scheme. Except for the
independent baseline option, all other processing sessions and parameter settings follow
Bernese’s default options [8].
 tions are loaded. After that, the co-observations between every two stations are retrieved
per epoch. The total co-observations of each station pair of a day are aggregated, respec-
tively. In this way, the common observation matrix Mobs is formed, and each element mobs
represents the number of common observations between every station pair. At the same
Remote Sens. 2022, 14, 4717 time, the geodetic distance between every station pair is calculated to form the distance
5 of 12
matrix Msho. The unit of element msho is meter. Mobs and Msho are shown in Equation (2).

Remote Sens. 2022, 14, x FOR PEER REVIEW 6 of 14

 m1,1
obs m1,2
obs m1,3
obs  m1,n
obs
  m1,1
sho m1,2
sho m1,3
sho  m1,n
sho

 2,2 2,3 2,n   2,2 2,3 2,n 
 m obs m obs  m obs   msho msho  msho 
=M obs =     , M sho      (2)
 n −1,n
  n −1,n

  m obs    msho 
 m n,n   n,n 
msho
 obs   
For the matrix Mobs, the reciprocal of each element or the maximum spanning tree
should be used, since the largest observation needs to be chosen instead of the smallest.
Correspondingly, the WEIGHT matrix Mwei and the OBS-DEN matrix Mden can be com-
puted as follows:

M=wei x1 × M sho + x2 / M obs , ( x1 ≥ 0, x2 ≥ 0, x1 =

+ x2 1)
 (3)
 M den = M sho / M bos
In the subsequent experiment and data analysis, both x1 and x2 of the WEIGHT were
set to 0.5. In order to apply the above methods with Bernese, the generated baseline file
Figure
Figure 2.
2. Flow
can be used Flow chart forfor
chart
to replace independent
the baseline filebaseline
independent selection,
baseline
generated starting
selection,
by Bernese’s fromfrom
starting
default reading the
reading
scheme. RINEX
the for
Except files offiles
RINEX all
stations,
theall
of to generate
independent
stations, different
tobaseline
generate independent
option, all other
different baselinebaseline
processing
independent files according
sessions to different
and parameter
files according baseline
settings fol- selection
to different baseline
strategies.
low Bernese’s
selection default options [8].
strategies.

2.5. Parallel Computation

Then, Computation
2.5. Parallel the MST is applied to the matrix Msho, which chooses the solution that lets the
sum Since
of msho
Since the
thebe theofsmallest.
data
data aofglobal The GNSS
GNSS
a global main diagonal
network may beelements
network very be
may of parallel
large,
very thelarge,
M computation
matrix
parallelrepresent
computa-all
available
[12,16–18,23]
tion observations
is recommended
[12,16–18,23] of individual
for suchfor
is recommended stations
data or 0processing.
processing.
such data distances,
This can be which are
donecan
This not
by be involved
utilizing
done the in the
by utilizing
Bernese
MST
the Processing
generation.
Bernese EngineEngine
Processing (BPE). As shown
(BPE). Asinshown
Figure 3,
infirst,
Figurethe 3,
CPU filethe
first, thatCPU
comes with
file that comes
Bernese
with needs to
Bernese be defined.
needs After that,After
to be defined. commands are submitted
that, commands aretosubmitted
the supercomputing
to the supercom-
platform,
puting the main the
platform, server of which
main serveraccepts the commands
of which accepts the and performs parallel
commands compu- parallel
and performs
tation according
computation to the settings
according in the
to the CPU file.
settings In this
in the CPU experiment,
file. In thisthe parallel computa-
experiment, the parallel
computation was performed in two different layers, one was the parallelization ofa the BPEs
tion was performed in two different layers, one was the parallelization of the BPEs for
single-day solution of independent baselines, and the other was the parallelization of mul-
for a single-day solution of independent baselines, and the other was the parallelization of
tiple daily solutions.
multiple daily solutions.

Figure 3. Flowchart of parallel computing.

Figure 3. Flowchart of parallel computing.
3. Results
This section first shows the results of a single-day solution, including a comparison
of the precision of the different methods, and the generated baseline map for our proposed
method. The number of observations versus baseline length is also analyzed. After that,
Remote Sens. 2022, 14, 4717 6 of 12

3. Results
This section first shows the results of a single-day solution, including a comparison of
the precision of the different methods, and the generated baseline map for our proposed
method. The number of observations versus baseline length is also analyzed. After that,
statistical results for one year using different methods are shown.

3.1. Single-Day Solution

At first, an experiment was conducted using globally distributed stations on 13 January
2012. The GNSS network solution was performed in the ITRF08 (International Terrestrial
Reference Frame 2008). After the network adjustment, the results were converted to the
local ENU (East-North-Up) coordinate system using the final coordinate products in SINEX
format (Solution Independent Exchange Format) provided by CODE (Center for Orbit
Determination in Europe) as a reference.
As is shown in Table 1, OBS-DEN has the smallest 3D RMS error of 7.30 mm, followed
by SHORTEST. For SHORTEST, the large error in the E direction drags down its RMS.
Compared with the commonly used OBS-MAX and SHORTEST, OBS-DEN has mainly
improved the East and North accuracies.

Table 1. Accuracy comparison of single-day solutions of different methods. The statistics of station
accuracies are calculated in the local coordinate system. The three axes of the local coordinate
frame are East (E), North (N), and Up (U). The left, middle and right columns show the mean, the
standard deviation (STD), and the root mean square (RMS) of the station coordinate errors of each
method, respectively.

MEAN (mm) STD (mm) RMS (mm)

E N U E N U E N U 3D
SHORTEST −0.99 0.89 −0.53 3.66 3.15 5.97 3.79 3.28 6.00 7.81
OBS-MAX −0.70 −1.09 0.13 2.83 3.53 7.28 2.92 3.69 7.28 8.67
WEIGHT −1.45 0.29 0.00 2.96 3.07 6.95 3.30 3.08 6.95 8.29
OBS-DEN −0.70 0.38 0.19 2.78 2.41 6.25 2.86 2.44 6.25 7.30

STD represents the degree of dispersion of the error for all stations. Although OBS-
DEN has the lowest 3D RMS error, it has a larger STD compared to SHORTEST in the U
direction. The STD of N and U components of OBS-MAX are the largest, which indicates
that some individual stations may have large errors with OBS-MAX. Generally, it can be
seen that the positioning errors of OBS-DEN are less discrete compared to other approaches.
Histograms of the single-day solution showing the coordinate error distributions in the
E, N, and U directions of different methods are presented in Figure 4. In the East direction,
the error distribution of OBS-DEN is closest to 0 and has rare discrete bars (> 10 mm),
followed by OBS-MAX; the center of the error distribution of WEIGHT deviates from 0 at
about −1.4 mm, and there are agminated bars around −5 mm, which makes its precision
worse than OBS-DEN and OBS-MAX in East. In the N direction, the errors of OBS-DEN are
more concentrated, while OBS-MAX has the most discrete values. For the U component,
the results of the various methods are broadly similar, with the errors of OBS-MAX being
slightly dispersed.
Figure 5 shows the baseline map of OBS-DEN. Since OBS-MAX emphasizes more
DD observations, a ‘STAR’-like shape, i.e., a central station with plenty of observations
connected with multiple nearby stations [24,25], will exist in many regions. For example,
some stations in South America, Australia, and Europe shown in [14] could connect more
than 5 baselines. SHORTEST, on the other hand, has fewer baselines clustered towards the
central stations, i.e., most stations are connected to only two or three baselines. As a result
of a combination of the above two methods, some of the central stations of OBS-DEN, such
as POVE in South America and ALIC in Australia, are connected to four baselines, while
other stations are mostly connected to two or three baselines.
 RemoteSens.
Remote Sens.2022,
2022,14,
14,4717
x FOR PEER REVIEW 78 of
of1214

Figure 4. Histograms of single‐day solutions. The x‐axis of each subplot is the final station coordi‐
Figure 4. Histograms
nate accuracy of single-day
in millimeters, and solutions. The
the interval of x-axis of each
each bin subplot
of North andisEast
the final
is 1 mmstation coordinate
(3 mm for Up).
accuracy in millimeters, and the interval of each bin of North and East is 1 mm
The y‐axis of each subplot represents the number of stations accommodated in each bin. (3 mm for Up).
TheThecol‐
y-axis
umnsoffromeachleft
subplot represents
to right the East
denote the number
(E), of stations
North (N), accommodated in each bin.
and Up (U) component, The columns
respectively. The
Remote Sens. 2022, 14, x FOR PEER REVIEW 9 of 14
from
four left to right
methods denote
from thebottom
top to East (E),
areNorth (N), andOBS‐MAX,
SHORTEST, Up (U) component,
WEIGHT, respectively.
and OBS‐DEN, Therespec‐
four
tively. from top to bottom are SHORTEST, OBS-MAX, WEIGHT, and OBS-DEN, respectively.
methods

Figure 5 shows the baseline map of OBS‐DEN. Since OBS‐MAX emphasizes more DD
observations, a ‘STAR’‐like shape, i.e., a central station with plenty of observations con‐
nected with multiple nearby stations [24,25], will exist in many regions. For example,
some stations in South America, Australia, and Europe shown in [14] could connect more
than 5 baselines. SHORTEST, on the other hand, has fewer baselines clustered towards
the central stations, i.e., most stations are connected to only two or three baselines. As a
result of a combination of the above two methods, some of the central stations of OBS‐
DEN, such as POVE in South America and ALIC in Australia, are connected to four base‐
lines, while other stations are mostly connected to two or three baselines.

Figure 5.
Figure 5. Independent
Independent baseline network diagram
baseline network diagram of
of about
about 100
100 stations
stations generated
generated using
using OBS-DEN.
OBS‐DEN.

The number
The number ofof DD
DD observations versus the
observations versus the baseline length of
baseline length of each
each baseline
baseline isis plotted
plotted
in Figure 6. It can be seen that the co‐viewing satellites decrease roughly
in Figure 6. It can be seen that the co-viewing satellites decrease roughly linearly withlinearly with
increasing distance. When the
When the station spacing
spacing is greater than 17,000 km, the co‐viewing
co-viewing
are almost
satellites are almostabsent.
absent.However,
However,atatdistances
distancesof of several
several thousand
thousand kilometers,
kilometers, there
there are
are still
still largelarge numbers
numbers of common
of common observations
observations betweenbetween stations.
stations. It is obvious
It is obvious that the
that the baseline
baseline selection dominated by the number of DD observations, which is applied in OBS‐
MAX, is no longer applicable at this point. This is due to the long distances resulting in
different tropospheric and ionospheric conditions, especially the baselines from mid‐lati‐
tude to low‐latitude/equatorial regions.
 Figure 5. Independent baseline network diagram of about 100 stations generated using OBS‐DEN.

The number of DD observations versus the baseline length of each baseline is plotted
in Figure 6. It can be seen that the co‐viewing satellites decrease roughly linearly with
Remote Sens. 2022, 14, 4717 increasing distance. When the station spacing is greater than 17,000 km, the co‐viewing 8 of 12
satellites are almost absent. However, at distances of several thousand kilometers, there
are still large numbers of common observations between stations. It is obvious that the
baseline selection dominated by the number of DD observations, which is applied in OBS‐
selection dominated by the number of DD observations, which is applied in OBS-MAX, is
MAX,
no is no
longer longer applicable
applicable at this
at this point. point.
This This
is due to is
thedue to distances
long the long distances
resulting resulting
in differentin
different tropospheric and ionospheric conditions, especially the baselines from
tropospheric and ionospheric conditions, especially the baselines from mid-latitude to mid‐lati‐
tude to low‐latitude/equatorial
low-latitude/equatorial regions.regions.

6. Variation
Figure 6. Variationofofthe
thenumber
number ofof
DD DDobservations
observations between every
between twotwo
every stations withwith
stations distance. This
distance.
This
is is based
based on a single‐day
on a single-day solution.
solution. SomeSome
of theofstations
the stations
have have
only only GPS observations
GPS observations whilewhile
othersothers
have
have GPS
both bothand
GPSGLONASS
and GLONASS observations,
observations, leadingleading
to twotolinear
two linear patterns
patterns in thein the plot.
plot.

In the
the data
dataanalysis,
analysis,wewefound
foundthat
that some
some stations
stations have
have both
both GPS GPS
andand GLONASS
GLONASS ob‐
observations
servations while others can receive only GPS signals. That is why there are two linear
while others can receive only GPS signals. That is why there are
aggregations presented in Figure 6. In In addition
addition to
to the two obvious linear aggregations,
aggregations,
one can see some scattered dots to the the lower
lower left.
left. These dots indicate that although the
stations are close to each other,
other, there
there are
are not
not many
many common
common observations.
observations. This may be
due
due to a long‐time loss of signal lock or bad observations being excluded.
to a long-time loss of signal lock or bad observations being excluded. In this case,
In this case, the
the
baselines are short but with fewer observations. Thus, the SHORTEST method can possibly
degrade the accuracy of the baseline solutions due to the introduction of these stations with
a small number of observations, while OBS-DEN would avoid such baselines and instead
choose baselines with sufficient satellites, but which are slightly longer, i.e., those from the
upper-left region of Figure 6.

3.2. One-Year Statistical Results

To better evaluate the performance of various methods, we have tabulated the statisti-
cal results for a year. The RMS errors in each direction and the distribution are summarized
in Table 2 and Figure 7. Generally, the RMS and the distribution of the methods are compa-
rable. In more detail, the probabilities that 3D errors exceed ε, 2ε, and 3ε mm are presented,
respectively, in the right column of Table 2. The threshold ε is set as 9.67 mm, which is
the average 3D RMS value of the four methods. From the statistical results, we can see
that OBS-DEN has the most stations with accuracies within one ε, and WEIGHT the least.
However, the probability that WEIGHT is larger than 2ε and 3ε is the smallest. That is, the
coordinate errors of WEIGHT lie more in the interval from ε to 3ε. OBS-DEN and OBS-MAX
have more 3D errors larger than 3ε, which pulls down the performance of OBS-DEN and
OBS-MAX somewhat.
In addition to the tails of the distributions explored on the right side of Table 2, the
histograms showing the coordinate error distributions can be seen in Figure 7. Overall,
the distribution of the four methods is similar. However, SHORTEST has fewer burrs for
errors greater than 30 mm, especially in the North direction. The distributions of the four
methods in the East direction seem to be a little fatter than that in the North, which shows
that the STD is minimal in N. For the Up direction, there are large discrete errors around or
larger than 50 mm for all four methods.
Remote Sens. 2022, 14, 4717 9 of 12

Table 2. Statistics of one-year solutions. The left side represents the RMS, and the right side represents
the probability that the 3D errors for each method exceed certain thresholds. The threshold ε is set as
9.67 mm, which is the average 3D RMS value of the four methods.

RMS Probability
E (mm) N (mm) U (mm) 3D <ε <2ε <3ε
SHORTEST 4.38 4.21 7.63 9.75 71.89% 96.17% 99.38%
OBS-MAX 3.92 3.94 7.79 9.57 71.96% 96.54% 99.16%
Remote Sens. 2022, 14, x FOR PEER REVIEW 11 of 14
WEIGHT 4.14 3.92 7.78 9.64 71.82% 96.77% 99.41%
OBS-DEN 4.31 4.15 7.68 9.73 72.49% 96.45% 99.33%

Figure 7. Histograms of one-year

one‐year solutions. The x‐axis
x-axis of each subplot is the final station coordinate
accuracy in millimeters, and the interval of each bin is 11 mm.mm. The y-axis
y‐axis of each
each subplot,
subplot, which is
on aa logarithmic
on logarithmic scale,
scale, represents
represents the
the quotient
quotient of
of the
the number
number of
of stations
stations accommodated
accommodated in in each
each bin
bin
and the total number. The columns from left to right denote the East (E), North (N),
and the total number. The columns from left to right denote the East (E), North (N), and Up (U)and Up (U)
component, respectively. The four methods from top to bottom are SHORTEST, OBS‐MAX,
component, respectively. The four methods from top to bottom are SHORTEST, OBS-MAX, WEIGHT,
WEIGHT, and OBS‐DEN, respectively.
and OBS-DEN, respectively.

In addition to the tails of the distributions explored on the right side of Table 2, the
4. Discussion
histograms
Overall,showing
OBS-DEN theachieves
coordinate
theerror distributions
desired precision in can be seen
terms inRMS
of the Figure
3D7.ofOverall,
station
the distribution of the four methods is similar. However, SHORTEST
coordinates and shows its capability to get comparable or even better precision thanhas fewer burrs for
other
errors greater than 30 mm, especially in the North direction. The distributions
methods. OBS-MAX is overly focused on the number of observations, but it may include of the four
methods
some longinbaselines
the East direction
with low seem to be while
precision, a littleSHORTEST
fatter than that in the North,
is excessively which
focused onshows
base-
that the
lines’ STDand
length is minimal
may have in incorporated
N. For the Upsome
direction, there arewith
short baselines largeless
discrete errorssatellites.
co-viewed around
or larger than
OBS-DEN 50 mm
excludes for all
these twofour methods.
extreme conditions by both pursuing high observation num-
bers and also emphasizing short baselines. When compared with WEIGHT, although the
4. Discussion
accuracy improvement of OBS-DEN is limited, it provides a rational option rather than
determining
Overall,weights
OBS‐DEN empirically.
achieves the desired precision in terms of the RMS 3D of station
coordinates and shows its capability to get comparable or even better precision than other
methods. OBS‐MAX is overly focused on the number of observations, but it may include
some long baselines with low precision, while SHORTEST is excessively focused on base‐
lines’ length and may have incorporated some short baselines with less co‐viewed satel‐
Remote Sens. 2022, 14, 4717 10 of 12

In theory, with the same information obtained, the final results should be equivalent,
but the different ways of data processing led to inconsistent information or data involved.
The advantage of OBS-MAX is that it absorbs more redundant observations involved in
the adjustment. However, from the above results, especially in Figure 6, there are still a
considerable number of observations at a certain range with baselines getting too long. In
this case, OBS-MAX may pick some long baselines and make the results worse. In addition
to the impacts of the tropospheric and ionospheric delays, the DD ambiguity is more
difficult to deal with when the baseline becomes longer [26]. The advantage of SHORTEST
is that it uses stations from short baselines whose atmospheric delays are basically the same.
However, the shortest baseline could not necessarily exclude the baselines with few co-
viewing satellites. As a synthesis of the above two methods, the ratio of the baseline length
to the number of observations can be used to overcome the respective shortcomings of the
previous individual methods, resulting in a better baseline solution in certain scenarios.
In addition to these most common methods, there are the maximum-ambiguity-fixed-
rate method [27] and the STAR method [8]. However, the former uses the outcome of the
solution as a basis for selection and cannot provide a pre-defined option for the independent
baseline solution as other methods. The STAR method is commonly used for local networks
rather than global ones. Therefore, only OBS-MAX and SHORTEST from the traditional
methods are involved in the comparison. In future work, the performance of different
constellations including positioning accuracy, number of observations, and signal quality
could also be used as another baseline searching criteria.
The baseline solution precision is closely related to the station location and density,
the shape of the network, and the local atmospheric environment. Different baseline search
strategies can be adapted to specific situations. For example, baseline solutions at low
latitudes, equatorial and polar regions are usually affected more heavily by ionospheric
effects [28,29], especially during a solar maximum period. Thus, more consideration should
be given to making the baselines shorter during such periods.
It should be noted that the stations selected for this experiment are globally distributed.
The results of these methods may be less different in a local area network where all stations
have comparable observations. For example, for a local area network [6] or network RTK
(Real-Time Kinetic) [30–32], the different baseline selection methods are theoretically close
to being equivalent, especially with a large number of observations of multiple systems [33].
While all stations are close to each other, the number of co-viewing satellites between them
is also similar. The baselines selected by different methods may differ from each other, but
the total length of the baseline and the total number of satellite observations will not vary
significantly.

5. Conclusions
In light of the limitations of current independent baseline selection methods, such as
OBS-MAX and SHORTEST, an alternative optimized scheme named OBS-DEN is proposed
for GNSS network solutions. It is characterized by maximum co-viewing satellites per unit
distance. Since the SHORTEST pursues only short baselines, there is a risk of introducing
low-precision baselines with small co-observations numbers; OBS-MAX aims only for
more observations and will potentially introduce baselines with large tropospheric and
ionospheric differences. OBS-DEN considers both shorter paths and more DD observations
in an independent baseline network. It compensates for the shortcomings of SHORTEST
and OBS-MAX and does not require empirical weighting. It can be a new independent
baseline search strategy for baseline selection in GNSS software, e.g., Bernese.
In both the single-day and annual solutions, OBS-DEN demonstrates its ability to
obtain comparable or even higher 3D accuracies. In the single-day solution, the distribution
of OBS-DEN is more concentrated. The RMS is smaller than OBS-MAX and SHORTEST.
In the statistical results of annual solutions, the 3D RMS of OBS-DEN has the highest
probability to be less than 9.67 mm, i.e., the average 3D RMS of all the four methods,
compared to other methods.
Remote Sens. 2022, 14, 4717 11 of 12

Due to the uncertainty of the error distribution, OBS-DEN would not be better than
other methods in all cases. Different network types and application scenarios correspond
to different optimal baseline schemes. In scenarios where the traditional methods are both
limited, OBS-DEN can be considered as the preferred scheme.

Author Contributions: Conceptualization, Y.D.; methodology, T.L.; software, T.L. and W.N.; valida-
tion, T.L., Y.M., and J.L.; formal analysis, T.L., Y.M., and J.L.; writing—original draft preparation, T.L.;
writing—review and editing, Y.D.; visualization, T.L.; funding acquisition, G.X. and W.N. All authors
have read and agreed to the published version of the manuscript.
Funding: This research was funded by the Open Fund of the Key Laboratory of Urban Land Re-
sources Monitoring and Simulation, Ministry of Natural Resources, grant number (KF-2021-06-104);
Guangdong Basic and Applied Basic Research Foundation, grant number (2021A1515012600); The
Opening Project of Guangxi Wireless Broadband Communication and Signal Processing Key Lab-
oratory (No. GXKL06200217); The National Nature Science Foundation of China (No. 42004012);
The Natural Science Foundation of Shandong Province (No. ZR2020QD048); Wenhai Program of the
S&T Fund of Shandong Province for Pilot National Laboratory for Marine Science and Technology
(Qingdao) (NO. 2021WHZZB1004, 2021WHZZB1004_01).
Data Availability Statement: The data and products are downloaded from (www.igs.org, Weihai,
China, 1 June 2018).
Acknowledgments: Thanks for the support from the Navigation and Remote Sensing Group of
Shandong University. Mowen Li and Zhenlong Fang provided part of the code for data batch
downloading; Tianhe Xu, Chunhua Jiang, and Yan Xu provided valuable discussions. This experiment
was conducted at the Supercomputing Center of Shandong University in Weihai, China. We also
thank Ta-Kang Yeh of Taipei University and Baoqi Sun of the National Time Service Center-Chinese
Academy of Sciences, who provided assistance in using and compiling the Bernese software.
Conflicts of Interest: The authors declare no conflict of interest.

References
1. Rodriguez-Solano, C.; Hugentobler, U.; Steigenberger, P.; Bloßfeld, M.; Fritsche, M. Reducing the draconitic errors in GNSS
geodetic products. J. Geod. 2014, 88, 559–574. [CrossRef]
2. St˛epniak, K.; Bock, O.; Bosser, P.; Wielgosz, P. Outliers and uncertainties in GNSS ZTD estimates from double-difference processing
and precise point positioning. GPS Solut. 2022, 26, 74. [CrossRef]
3. Sun, M.; Liu, L.; Yan, W.; Liu, J.; Liu, T.; Xu, G. Performance analysis of BDS B1C/B2a PPP using different models and MGEX
products. Surv. Rev. 2022, 1–12. [CrossRef]
4. Yang, Y.; Mao, Y.; Sun, B. Basic performance and future developments of BeiDou global navigation satellite system. Satell. Navig.
2020, 1, 1. [CrossRef]
5. Zajdel, R.; Sośnica, K.; Dach, R.; Bury, G.; Prange, L.; Jäggi, A. Network effects and handling of the geocenter motion in
multi-GNSS processing. J. Geophys. Res. Solid Earth 2019, 124, 5970–5989. [CrossRef]
6. Specht, C.; Specht, M.; Dabrowski,
˛ P. Comparative analysis of active geodetic networks in Poland. Int. Multidiscip. Sci. GeoConf.
SGEM 2017, 17, 163–176.
7. Xu, G.; Xu, Y. GPS: Theory, Algorithms and Applications; Springer: Berlin, Germany, 2016.
8. Dach, R.; Walser, P. Bernese GNSS Software, Version 5.2; University of Bern: Bern, Switzerland, 2015.
9. Herring, T.; King, R.; McClusky, S. Introduction to Gamit/Globk; Massachusetts Institute of Technology: Cambridge, MA, USA, 2010.
10. Gilad, E.-T. Graph theory applications to GPS networks. GPS Solut. 2001, 5, 31–38.
11. Graham, R.L.; Hell, P. On the history of the minimum spanning tree problem. Ann. Hist. Comput. 1985, 7, 43–57. [CrossRef]
12. Cui, Y.; Chen, Z.; Li, L.; Zhang, Q.; Lu, Z. An efficient parallel computing strategy for the processing of large GNSS network
datasets. GPS Solut. 2021, 25, 36. [CrossRef]
13. Stepniak, K.; Bock, O.; Wielgosz, P. Reduction of ZTD outliers through improved GNSS data processing and screening strategies.
Atmos. Meas. Tech. 2018, 11, 1347–1361. [CrossRef]
14. Liu, T.; Xu, T.; Nie, W.; Li, M.; Fang, Z.; Du, Y.; Jiang, Y.; Xu, G. Optimal Independent Baseline Searching for Global GNSS
Networks. J. Surv. Eng. 2021, 147, 5020010. [CrossRef]
15. Krypiak-Gregorczyk, A. Ionosphere response to three extreme events occurring near spring equinox in 2012, 2013 and 2015,
observed by regional GNSS-TEC model. J. Geod. 2019, 93, 931–951. [CrossRef]
16. Chen, Z.; Zhiping, L.; Yang, C.; Hao, L. Parallel computing of GNSS data based on Bernese processing engine. J. Geod. Geodyn.
2013, 33, 79–82.
Remote Sens. 2022, 14, 4717 12 of 12

17. Li, L.; Lu, Z.; Chen, Z.; Cui, Y.; Sun, D.; Wang, Y.; Kuang, Y.; Wang, F. GNSSer: Objected-oriented and design pattern-based
software for GNSS data parallel processing. J. Spat. Sci. 2019, 66, 27–47. [CrossRef]
18. Jiang, C.; Xu, T.; Du, Y.; Sun, Z.; Xu, G. A parallel equivalence algorithm based on MPI for GNSS data processing. J. Spat. Sci.
2019, 66, 513–532. [CrossRef]
19. Naidoo, K. MiSTree: A Python package for constructing and analysing Minimum Spanning Trees. arXiv 2019, arXiv:1910.08562.
[CrossRef]
20. Kruskal, J.B. On the shortest spanning subtree of a graph and the traveling salesman problem. Proc. Am. Math. Soc. 1956, 7, 48–50.
[CrossRef]
21. Spira, P.M.; Pan, A. On finding and updating spanning trees and shortest paths. SIAM J. Comput. 1975, 4, 375–380. [CrossRef]
22. Prim, R.C. Shortest connection networks and some generalizations. Bell Syst. Tech. J. 1957, 36, 1389–1401. [CrossRef]
23. Cui, Y.; Lv, Z.; Li, L.; Chen, Z.; Sun, D.; Kwong, Y. A Fast Parallel Processing Strategy of Double Difference Model for GNSS Huge
Networks. Acta Geod. Cartogr. Sin. 2017, 46, 48–56.
24. Paziewski, J.; Kurpinski, G.; Wielgosz, P.; Stolecki, L.; Sieradzki, R.; Seta, M.; Oszczak, S.; Castillo, M.; Martin-Porqueras, F.
Towards Galileo+ GPS seismology: Validation of high-rate GNSS-based system for seismic events characterisation. Measurement
2020, 166, 108236. [CrossRef]
25. Fotiou, A.; Pikridas, C.; Rossikopoulos, D.; Chatzinikos, M. The effect of independent and trivial GPS baselines on the adjustment
of networks in everyday engineering practice. In Proceedings of the International Symposium on Modern Technologies, Education
and Professional Practice in Geodesy and Related Fields, Sofia, Bulgaria, 5–6 November 2009; pp. 201–212.
26. Geng, J.; Mao, S. Massive GNSS network analysis without baselines: Undifferenced ambiguity resolution. J. Geophys. Res. Solid
Earth 2021, 126, e2020JB021558. [CrossRef]
27. Hua, C. Application Research of Method of Large GNSS Network Realtime Data Rapid Solution. Ph.D. Thesis, Wuhan University,
Wuhan, China, 2010.
28. Liu, T.; Yu, Z.; Ding, Z.; Nie, W.; Xu, G. Observation of Ionospheric Gravity Waves Introduced by Thunderstorms in Low Latitudes
China by GNSS. Remote Sens. 2021, 13, 4131. [CrossRef]
29. Dao, T.; Harima, K.; Carter, B.; Currie, J.; McClusky, S.; Brown, R.; Rubinov, E.; Choy, S. Regional Ionospheric Corrections for
High Accuracy GNSS Positioning. Remote Sens. 2022, 14, 2463. [CrossRef]
30. Bakuła, M.; Przestrzelski, P.; Kaźmierczak, R. Reliable technology of centimeter GPS/GLONASS surveying in forest environments.
IEEE Trans. Geosci. Remote Sens. 2014, 53, 1029–1038. [CrossRef]
31. Wang, P.; Liu, H.; Yang, Z.; Shu, B.; Xu, X.; Nie, G. Evaluation of Network RTK Positioning Performance Based on BDS-3 New
Signal System. Remote Sens. 2021, 14, 2. [CrossRef]
32. Liu, J.; Zhang, B.; Liu, T.; Xu, G.; Ji, Y.; Sun, M.; Nie, W.; He, Y. An Efficient UD Factorization Implementation of Kalman Filter for
RTK Based on Equivalent Principle. Remote Sens. 2022, 14, 967. [CrossRef]
33. Siejka, Z. Validation of the accuracy and convergence time of real time kinematic results using a single galileo navigation system.
Sensors 2018, 18, 2412. [CrossRef]