371

1 INTRODUCTION

With the development of advanced Global Navigation

Satellite System (GNSS) infrastructures, hardware, and

data processing strategies, real-time high precision

positioning with centimetre level accuracy has been

widely available to support diversified applications,

such as surveying and mapping [1], precise agriculture

[2], seismology [3], autonomous vehicle [4],

atmospheric sensing [5], and scientific research [6], [7].

Conventional Real-Time Kinematic (RTK) technologies

require a geodetic infrastructure with a regional

reference station network and low-cost communication

link (i.e. mobile communication network) in the

coverage region [8], [9]. This will not available for

offshore applications. Currently, for offshore high

precision positioning applications, there are mainly

two technologies, including Wide Area Differential

GNSS (WADGNSS, i.e. US WAAS, European EGNOS

systems) and Precise Point Positioning (PPP) Systems

(i.e., BeiDou, Galileo, Trimble RTX) from different

governmental or industrial organizations [10]–[14].

WADGNSS systems is originally designed to satisfy

high reliability navigation services for civil aviation

A Novel Network RTK Technique for Mobile Platforms:

Extending High-Precision Positioning to Offshore

Environments

W. Chen, J. Ding, Y. Wang, X. Mi & T. Liu

The Hong Kong Polytechnic University, Hong Kong

ABSTRACT: Network Real-Time Kinematic (NRTK) positioning, as the most mature real-time high-precision

positioning technology, is widely recognized for its centimetre-level accuracy, operational efficiency, and

extensive application potential. However, conventional NRTK systems rely on reference stations anchored to

bedrock-based infrastructure, limiting their coverage to terrestrial areas within Continuous Operating Reference

Station (CORS) networks. This architectural limitation renders conventional NRTK inapplicable for offshore and

marine environments. To overcome this geographical constraint, we propose an innovative NRTK framework for

mobile platforms featuring (1) simultaneous estimation of atmospheric delays and baseline dynamics to get

precise relative coordinate movements, (2) the regularization method is applied to de-correlate the positional and

atmospheric parameters and the regularization coefficients are optimized by mean square error minimization,

and (3) integration of Precise Point Positioning (PPP) at a main base station to maintain an absolute position

reference for the network. Experimental validation using Hong Kong's terrestrial CORS network demonstrates

that the proposed marine-adapted system achieves positioning accuracy comparable to conventional bedrock-

based NRTK, with three-dimensional (ENU) errors measuring (2.90, 3.22, 4.32) cm and (2.90, 2.88, 6.70) cm in two

operational scenarios. This methodological advancement enables the deployment of buoy-based NRTK systems

in marine environments, with significant implications for maritime applications including port traffic

management, fishing fleet navigation, and offshore resource exploration. By extending NRTK's operational

domain beyond terrestrial boundaries, our technique not only enhances positioning reliability for marine

operations but also creates new paradigms for oceanic resource management.

http://www.transnav.eu

the International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 19

Number 2

June 2025

DOI: 10.12716/1001.19.02.04

372

which can achieve meter level positioning accuracy

with high integrity [15]. Kinematic PPP technology

utilizes the high-quality orbit and satellite clock error

products from a global network (i.e. International

GNSS Services (IGS)) to provide decimetre level

positioning worldwide with a single user receiver [16],

[17]. However, with the PPP approach the vertical

positioning accuracy is relatively lower (decimetre

level) and the convergence time is relatively long

(typical 30 min), compared with conventional land

based RTK technology with the ambiguity fixing time

of a few seconds [18]–[20]. These restrict some

applications requiring higher accuracy (centimetre

level) and quick setup in offshore applications [21]–

[25]. Another possible way is the moving-base

positioning mode, while it is not designed for ocean

positioning, and existing studies on mobile base

stations focus on moving base positioning models,

which mainly focus on ambiguity fixation of a single

ultra-short moving baseline, with application scenarios

of two moving cars [26]–[28], or unmanned aerial

vehicles (UAVs) [29], [30]. These scenarios usually do

not estimate atmospheric parameters and do not

provide external positioning services like NRTK.

Therefore, they cannot provide solutions and effective

decorrelation methods need to be introduced to

improve the accuracy of the solutions.

A promising way to extend GNSS technology to the

offshore area is to equip floating objects at sea with

GNSS receivers, such as shipboard GNSS [31]–[33] and

buoy-based GNSS [34], [35]. The former usually only

stays in a certain area for a short time and has only a

single receiver, or multiple receivers on a single ship,

and cannot provide stable positioning services to the

area. The latter is more costly as the buoy needs to be

cemented to the anchor chain to prevent loss of the

buoy due to the action of ocean currents. Existing

marine GNSS studies have focused on reconstructing

seafloor horizontal displacement [36], acoustic seafloor

positioning precision [37], [38], testing the effectiveness

of short message communication [39], significant wave

height retrieval [40], [41], sea surface salinity retrieval

[42], meteorological tsunami monitoring [43], and

water level measurements [44]. These studies typically

give little consideration to communication between

multiple offshore GNSS receivers and have never

considered the construction of an offshore GNSS

network to provide continuous positioning service to

the region. In addition, the high coupling qualities of

vertical position movement estimates with

tropospheric delays in a dynamic setting have not been

extensively studied.

In this study, we aim to develop a new GNSS

positioning method which extends high-precision

NRTK positioning for mobile platforms for offshore

industry, where conventional network RTK is not

covered. This method effectively combines the

advantages of PPP and RTK. By utilizing the services

provided through this method, users can achieve

precise absolute positioning comparable to PPP, while

retaining the rapid convergence advantage

characteristic of NRTK. All GNSS users around the

kinematic reference stations in offshore environment

will be able to achieve real-time, and centimetre level

positioning accuracy, similar to the performance of

conventional land-based NRTK operation. Table 1

shows the comparison of several existing methods of

offshore positioning, including PPP, traditional NRTK,

moving-base positioning solution and the method this

research proposed.

Table 1. Comparison of several methods of offshore

positioning

Solutions

High

accuracy

Fast

convergence

With

regularization

Offshore

PPP







NRTK





Moving-base

 (relative)







This research



We introduce the traditional NRTK method for

fixed base stations in Section 2, followed by its

adaptation for mobile base stations. Then the

regularization technique to decorrelate unknown

parameters and detail the computation of the

regularization parameter is described. Section 3

compares parameter estimation of baseline solutions

between fixed and mobile base station NRTK methods

using the Hong Kong CORS network data, evaluates

the impact of regularization, and presents positioning

validation results. Finally, we give conclusions and

outlooks in Section 4.

2 METHODS

In this section, we first introduce the traditional NRTK

for fixed-base station platforms, then describe how it

can be extended to NRTK for mobile platforms, next

describe the introduction of regularization methods to

de-correlate between the parameters to be estimated,

and finally introduce the method for determining the

regularization parameters.

2.1 RTK for fixed base stations

Observation equations of GNSS code pseudorange (P)

and carrier phase () measurements from a satellite s

to a receiver r with frequency f can be expressed as:

, , , ,

, , , , ,

s s s s s s s

r f r r r r f P r f r rel

s s s s s s s s

r f r r r r f f r f r f r rel

P t t T I b b M

t t T I b b N M

  

   



= + − + + + − + + +

 = + − + − + − + + + +

(1)

in which



is the geometric distance between the

phase centres of satellite and receiver antennas. tr and

are the receiver clock and satellite clock respectively.

and

are tropospheric delay and ionospheric

delay respectively. b is the device delay and M is the

multipath delay.



and



are relativistic effects and

observational noise. N are the phase ambiguities,



the wavelength of signal with frequency f.

By combining the observation equations of

frequency f1 and f2, we can get the double difference

(DD) observation equation:

1 1 1 1

2 2 2 2

f f P

f f f f

P I T

I T N



  



 =  +  +  + 

 =  −  +  +  + 

(2)

373

where



is the double difference sign and the rest

are the same as before. By using the double-difference

method, the majority of common errors in

conventional RTK positioning are eliminated.

Then, the Melbourne–Wubbena (MW) combination

is used to determine wide-lane (WL) integer

ambiguities:

MW WL NL WL WL P



 =  − =   + 

(3)

f P f P

− 

 =

−

(4)

f P f P

 + 

 =

(5)

WL f f

N N N =  −

(6)



−

(7)

After that, the ionosphere-free combination (IF)

approach is used to estimate the L1 ambiguity [45]. The

only remaining unknown parameters to calculate are

the geometric distance and the tropospheric delay.

( )

IF WL NL WL

f NL

R T N







   



 − 

 = − + 

−

=  +  +  +  + 

(8)

()



(9)

For the traditional NRTK, the DD geometric

distance is solved by using the known fixed base

station coordinates. Then the relative tropospheric

delay remaining as an unknown parameter matrix X:

[]T=x

(10)

Then, the tropospheric delay is estimated by the

following least squares (LS) method:

x x x

−

N A PA

W A Pl

X N W

(11)

where A is the coefficient matrix, P is the weight matrix

and I is their noise vectors.

2.2 RTK for mobile base stations

As for the mobile platform, all base stations are located

at sea and are in constant motion, making their exact

coordinates unknown. Consequently, the relative

movement of the base stations' coordinates is also

incorporated into the matrix of unknown parameters.

Figure 1 shows a minimal triangulation NRTK diagram

from a fixed platform to a mobile platform.

[ , , , ]x y z T=    x

(12)

At this point, the number of observations is still

greater than the number of unknown parameters, so

the equation is solved. But the change in the number of

unknown parameters obviously weakens the strength

of the equation. In addition, due to the poor geometric

configuration of the satellite in the vertical direction,

the unknown parameters in the U-direction are usually

strongly coupled with the tropospheric parameters,

and this coupling will lead to a decrease in the accuracy

of the solution when the base stations are all in a

constant state of movement.

Figure 1. Schematic diagram of fixed base station NRTK to

mobile base station NRTK

2.3 RTK for mobile base stations with regularized solution

We introduce the following regularization methods to

de-correlate the parameters and thus improve the

accuracy of the solution.



=+N A PA S

(13)



−

=X N W

(14)

where  is the regularization parameter and S is the

regularization matrix. They are defined as [46]:









(15)

where I3 is 3x3 identity matrix ， the LS method

calculates the unknown estimate under the

regularization criterion as:

( ) ( ) min



−

− − + =Ax y Q Ax y x x

(16)

The solution for the estimate is:

( )

1 1 1

2 1 1









−

− − −

−−

x A Q A I

A Q y N A Q y

Σ N N N

(17)

The accuracy of the regularization solution is

usually evaluated by the mean square error taking into

account the influence of the deviation.

(

)

1 2 2 1





−−

= + = +

M Σ g g N N x N

(18)

374

2.4 Determination of regularization parameter

Choosing the regularization parameter demands a fine

balance to ensure that the solution is both efficient and

accurate. Setting it too high can overshadow the

importance of observations, increasing deviation,

while setting it too low might not effectively reduce

noise. Techniques such as the discrepancy principle

[47], the general discrepancy principle (GDP) method

[47], the generalized cross-validation (GCV) [48] and

the L-curve method [49], [50] are all used to determine

this parameter. The L-curve method is popular but can

lead to over-regularization if not carefully applied [51].

It is important to recognize that each method has its

unique strengths and weaknesses, and they are best

suited to specific scenarios. There is no one-size-fits-all

criterion that is universally superior across all

applications. In this study, we adopted the method of

[52], which is valid for this study and the description is

given in the following.

To ensure the validity of the parameter estimates,

the mean square error is minimized to compute the

regularization parameter:

( )

argmin tr





(19)

For





( ) /





holds, i.e. there exists a

unique extreme point of the function. Therefore, the

simplest method is to give a reasonable search interval

and step size for





. According to [53], the spectral

decomposition equation of the mean-variance matrix

trace can be obtained:

( )





   



(20)

where vi is the i-th column vector of V. Solve the

following equations:

( )

(

)

( )

tr / / 0

i i i

     



  = − + =







M v x

(21)

The optimal valuation of



is obtained. The

equations can be solved using the theorem based on the

median of continuous functions on a closed interval.

However, the determination of the regularization

parameter requires the true value of the parameter, but

the true value of the parameter is unknown, which

constitutes the difficulty of the regularization method.

Commonly, the LS estimation is employed instead of

the true value, that is:

1 1 1 1

− − − −

xx x x N A Q yy Q AN

(22)

However, if the un-iterated LS valuation is

employed, the regularization parameter



becomes too

small because its value is prone to be large, which

makes it challenging to achieve a regularization effect.

If the iterated LS estimation is utilized, then:

( ) ( )

T T T



 = =vv vv ee Q

(23)

2 1 1 1 1 2 1 2

0 0 0

  

− − − − −

 = =

x x x

xx x x N A Q QQ AN N Q

(24)

In the ill-posed model, the covariance matrix

of the parameter LS valuation remains unstable.

Therefore, other methods or observations can be used

to calculate the initial value of the parameter

and its

covariance matrix

, i.e.:



xx ΣQ

(25)

Then the regularization parameter can be gotten:

1 2 1

argmin ( ( ) )



−−

N N Q N

(26)

Given that the geometry of the satellites is roughly

uniformly distributed horizontally, the higher

coupling is primarily attributed to coordinate error in

the vertical direction and tropospheric delay, we

propose the following changes to the regularization

parameter and regularization matrix:

 

( )

diag ,0

h h v m



(27)

where Rm is the rotation matrix from local East, North,

Up (ENU) coordinates to the Earth Centred Earth Fixed

(ECEF) coordinates, and



h and



v are the

regularization parameters in the horizontal and

vertical directions, respectively.

3 EXPERIMENT RESULTS AND DISCUSSIONS

In this section, we will first present the experimental

design, which encompasses experimental data and

data processing methods. Subsequently, we compare

the results obtained under two settings: one where the

relative motion of the baseline is estimated and the

other where it is not estimated (i.e. traditional method).

Then, we compare the results of baseline solution

achieved by using regularization method with those

obtained without using regularization method. Finally,

comparisons of user positioning results in different

scenarios are given.

3.1 Experimental setup

The experimental data comes from a total of 8 hours

from 16:00 to 24:00 on 25 March 2024 at 15 Hong Kong

SatRef stations. The results of the HKST-HKPC

baseline are used to show. The baseline length of

HKST-HKPC is 19.4 km (see Fig. 2). The acquisition

rate of the selected data is 1 second. Figure 3 shows the

distribution of the 15 Hong Kong CORS GNSS stations

used in the experiment, and the baseline network

formed by these GNSS stations. The length of these

baselines ranges from a few kilometres to more than

two dozen kilometres. The correction models and

estimation strategies for NRTK of server side and user

side are show in Table 2. It should be noted that Fig. 3

shows the baseline distribution at the epoch when all

the stations are able to receive data normally. There are

some epochs where data from some stations are

interrupted, at these epochs the triangulation network

is reconfigured, and the full baselines can be seen in

Table 3.

375

Figure 2. The test stations (red dots) selected for the

experiment and corresponding baselines (yellow lines). The

blue numbers represent the baseline length in kilometres. The

blue square symbol corresponds to the user station HKSC

used for positioning verification and does not participate in

NRTK network construction.

Table 2. Correction models and estimation strategies for

NRTK positioning

Items

Server side

User side

GNSS systems

GPS, Galileo, BeiDou

Data sampling

1 s

30 s

A priori noise

Pseudorange: 0.3 m; carrier-phase: 0.003 m

GNSS orbit and

clock

Broadcast ephemeris

Combination mode

Ionosphere-free

combination

Double difference

L1&L2

Elevation mask

20°

Estimator

Sequential least squares

Extended Kalman

filter

Weight of

observation

Elevation-dependent weight

Phase ambiguties

WL-L1 cascade fixing

with Constants for each

continuous observation

arc

L1&L2 partial fixing

with Constants for

each continuous

observation arc

Receiver coordinate

Fixed, Mobile

Kinematic

Ionospheric delays

First order eliminates,

higher order ignore

Corrected by linear

interpolation model

(LIM)

Tropospheric

delays

Hydrostatic component:

modeled by

Sasstamoinen with Neil

mapping function

(NMF);

Wet component: hourly

constants with process

noise of 1 cm h and

NMF

Corrected by linear

interpolation model

(LIM)

3.2 Comparison of estimating and not estimating the

relative coordinates

Figure 3 shows the results of the baseline HKST-HKPC.

The upper panels show the time series of tropospheric

delay (∆ZWD) and double-difference atmosphere (DDI

and DDT) without estimating the relative movement of

the base stations. The lower panels of the figures show

the time series of the above three parameters as well as

the three additional coordinates (∆e, ∆n, ∆u) when

estimating the relative movement of the base stations.

Where e, n, and u are the north, east, and up directions,

respectively.

From the figures, we can see that, after estimating

three additional parameters, DDT and DDI become

slightly larger, but not significantly. And the values of

∆ZWD became significantly larger. This also indicates

that the increase in the number of unknown

parameters leads to a weaker structure of the observed

equations, which in turn leads to a worse precision of

the estimated unknown parameters. In addition, in the

three newly added parameters, the errors in the north

and east directions are smaller than in the up direction.

3.3 Comparison of using and not using regularization

methods

As demonstrated in the preceding section, the

introduction of three additional unknown parameters

has diminished the robustness of the observational

equations, consequently leading to a decrease in the

precision of the solutions. To counteract this, we have

implemented regularization techniques to improve

accuracy. To evaluate the impact of the regularization

method on enhancing the precision of experimental

outcomes, we have used the same data as in the

previous section, choosing the results of the baselines

established by the HKPC and HKST stations as a case

study. The time series of regularization parameter α of

baseline HKST-HKPC over time is showed in Fig. 4,

and the time series of the solutions for the three

baselines are depicted in Fig. 6. A too small α will not

fully utilize the contribution of the observations, while

a α that is too large will lead to over-regularization,

making the solution more biased. The regularization

parameter computed by the mean square error

minimization criterion that we adopt maximizes the

utilization of the contribution of the solution. From the

results in Fig. 5, the regularization parameter itself

varies significantly between epochs, which is related to

the variation of observation conditions. It is important

to note that the regularization parameter is very

sensitive to observational condition and quality of the

data. The results shown in Figure 4 are not universally

applicable, we recommend that users calculate

customized regularization parameters using their own

data based on the methodology outlined in this study.

Figure 3. Time series of tropospheric delays (ΔZWD), relative

coordinates (Δe, Δn and Δu), and double-difference

atmospheres (DDI and DDT) with and without estimation of

relative coordinates for baseline HKST-HKPC (smaller is

better).

󵾈󵾉󵽹󵽒󵽅󵽃󵽅󵽉󵽅󵽍󵾂

󵾈󵾉󵽹󵽒󵽅󵽃󵽅󵽇󵽊󵽌󵾂 󵾈󵾉󵽹󵽒󵽅󵽃󵽅󵽆󵽎󵽇󵾂 󵾈󵾉󵽹󵽒󵽅󵽃󵽅󵽌󵽈󵽆󵾂

󵽛󵽾󵾍󵽺󵽹󵼵󵾅󵾁󵽶󵾉󵽻󵾄󵾇󵾂

󵽢󵾄󵽷󵽾󵾁󵽺󵼵󵾅󵾁󵽶󵾉󵽻󵾄󵾇󵾂

󵾈󵾉󵽹󵽒󵽅󵽃󵽅󵽆󵽅󵽅󵾂

376

Figure 4. Regularization parameter α of baseline HKST-

HKPC over time.

We show the results without regularization in the

up panels of Fig. 5 (labelled “Without regularization”),

and with regularization in the bottom panels (labelled

“With regularization”). The standard deviation of the

time series is also given in the figures. From the results

shown in the figures, it is evident that the DDIono and

DDTrop residues exhibit no significant changes after

applying the regularization method. Further analysis

confirms that only a slight reduction in the number of

valid observations occurs. In addition, the accuracies of

the tropospheric delay and the relative movement of

the base station are significantly improved, especially

when the tropospheric delay ∆ZWD, and the

movement in the up direction ∆u, are significantly

reduced. In addition, ∆e and ∆n also become

significantly smaller, but the magnitude of the change

is smaller than that in the up direction. This suggests

that the regularization method is mainly used for the

decorrelation of coordinates in the tropospheric delay

and up directions, but is also effective in improving the

accuracy of coordinates in the north and east

directions.

Figure 5. Time series of tropospheric delays (ΔZWD), relative

coordinates (Δe, Δn and Δu), and double-difference

atmospheres (DDI and DDT) with general method and

regularization method for baseline HKST-HKPC (STD

smaller is better).

Table 3 presents a comparison of the baseline results

with and without the regularization method. The

number of baselines in the table is greater than that

shown in Fig. 2. This is because the Delaunay

triangulation network is constructed in real-time. Some

epochs have missing data, which leads to the update of

the baselines. The table is arranged in descending order

of baseline length from top to bottom. We have marked

in red in the table the results with values larger than 5

cm. It can be observed that the vast majority of the

STDs of ∆ZWD and ∆u are larger than 5 cm without

using the regularization method, and nearly half of the

∆e and ∆n are larger than 5 cm. However, after using

the regularization method, almost all the results are

smaller than 5 cm, and most of them are smaller than 5

cm. This indicates that the errors in the upward

direction and tropospheric delays are most correlated,

and the regularization method is mainly used to

decorrelate them. Of course, the regularization method

also contributes to improving the accuracy of the north

and east-oriented coordinates, but the contribution in

this regard is weaker than the former.

Table 3. Comparison of using and not using regularization

methods

Baseline

Length

STD of general method

(cm)

STD of regularization

method (cm)

(km)

ΔZWD

Δe

Δn

Δu

ΔZWD

Δe

Δn

Δu

HKTK-HKLT

27.3

6.02

7.98

4.29

6.01

3.13

2.35

2.09

3.11

HKNP-HKLM

23.5

2.98

1.30

0.94

3.23

2.29

1.23

0.81

2.65

HKOH-HKWS

23.5

3.77

2.07

1.40

5.08

2.34

1.70

1.14

3.63

HKST-HKLM

20.7

4.51

3.23

1.85

4.98

2.70

1.33

1.25

2.37

HKST-HKPC

19.4

5.17

11.06

3.04

5.76

6.23

2.67

2.73

2.79

HKKT-HKPC

18.0

5.48

2.23

1.41

5.56

2.60

1.58

1.35

3.29

HKWS-HKTK

17.0

6.44

4.62

4.15

5.94

3.14

2.43

1.68

3.47

HKLT-T430

16.9

3.05

1.76

1.16

4.72

2.30

1.35

1.03

3.47

HKPC-HKLT

15.4

4.61

1.93

1.13

5.27

2.18

1.33

1.15

2.88

T430-HKSS

15.2

5.54

4.91

2.47

5.50

2.34

2.29

1.21

4.45

HKSL-HKNP

14.1

3.44

2.10

1.21

4.87

1.93

1.35

0.80

2.96

HKPC-HKCL

13.4

4.57

6.09

3.50

4.56

6.51

2.24

1.34

3.27

HKLM-HKMW

12.7

3.62

1.93

1.39

4.49

3.07

1.38

1.01

3.08

HKLM-HKOH

11.6

4.82

2.04

2.59

3.89

2.17

1.42

1.01

2.57

HKNP-HKMW

11.3

3.01

2.03

1.14

4.34

2.17

1.32

0.98

3.19

HKMW-HKCL

10.8

4.34

6.32

3.96

5.03

2.52

1.45

1.00

2.90

HKSS-HKST

9.6

5.01

3.53

2.07

5.29

2.93

2.15

1.38

4.10

HKKT-T430

9.2

4.84

4.70

3.02

5.37

1.90

1.39

0.94

3.32

HKCL-HKSL

8.7

5.65

4.74

4.31

6.00

3.01

1.40

1.09

2.73

HKLT-HKSL

8.7

3.36

6.29

3.69

4.43

5.57

1.19

0.97

2.55

HKLT-HKKT

7.8

5.65

2.24

4.06

5.47

1.89

1.36

0.96

2.90

HKWS-HKSS

6.8

4.84

2.41

1.50

5.19

2.93

2.15

1.18

3.99

HKNP-HKCL

5.4

2.94

3.41

1.61

3.77

2.96

1.54

1.17

2.74

HKMW-HKPC

4.8

3.68

2.59

1.90

5.01

2.54

1.40

1.04

3.44

*The red are values greater than 5.00 cm.

Figure 6 presents the solution results for 24

baselines, illustrating the standard deviations (STDs)

of ∆e, ∆n, ∆u, and ∆ZWD both with and without the

regularization method. The baselines are ordered from

left to right in reverse order of their lengths, with the

longest baseline, HKTK-HKLT, at 27.3 km on the far

left, and the shortest baseline, HKMW-HKPC, at 4.8 km

on the far right. The figure demonstrates that the

regularization method significantly enhances the

accuracy of the unknown parameters, particularly

∆ZWD and ∆u. There is no significant correlation

between this accuracy improvement and the baseline

length. However, it is observed that the regularization

method may reduce the accuracy of ∆e and ∆n for

baselines longer than 15 km (left side of baseline HKSL-

HKPC).

377

Figure 6. Comparison of STDs in the ΔE, ΔN, and ΔU

directions of coordinates and ΔZWD using the regularization

method and without applying the regularization method.

3.4 Verification of positioning results

The comparison of the baseline parameter estimation

results has demonstrated the effectiveness of the

algorithm. To further validate the proposed method,

we perform positioning validation from the user side.

We convert fixed station observations into mobile

station observations by adding periodic noise. The

generation of periodic noise is as follows:

( ) ( )

3 3 3 4 4 4

5 5 5 6 6 6

1 1 1 2 2 2

sin sin

sin cos

cos cos

e A B t C A B t C

n A B t C A B t C

u A B t C A B t C

 = + + +

(28)

where A, B and C are parameters of the periodic

function, which have different values at different

stations to simulate the different movements of

different sites. After generating the periodic noise, it is

projected to each satellite direction and then added to

the pseudo-range and phase observations of the fixed

station (as the followed equations), after which the

station is converted to a mobile station.

[ , , ] 2 ([ , , ])

([ , , ])

s s s

r mobile r fixed r

s s s

r mobile r fixed r

x y z enu xyz e n u

P P mf x y z

mf x y z

   =   

= +   

 =  +   

(29)

where enu2xyz is the coordinate conversion function,

r mobile

and

r mobile



are pseudorange and carrier

phase from satellite r to receiver s, the

is the

mapping function from the coordinates to line of sight

of the signals.

To verify the effectiveness of this simulated mobile

station, we used precise point positioning (PPP) and

real-time kinematic (RTK) to conduct positioning

experiments on the user station. The RTK experiment

is divided into two sub-solutions: fixed base and

mobile base according to the different base stations

used as references. In addition, we also conducted

NRTK positioning experiments, which are also divided

into fixed and mobile solutions according to different

base stations. In the mobile solution, there are two sub-

solutions according to whether the baseline of server

side uses regularization method. These experimental

settings are summarized in Table 4, and are named

solutions S1–S6, respectively.

Table 4. Experiment setting of six solutions

Solution

Base

Rover

Method

Regularization

---

mobile

PPP

---

fixed

mobile

RTK

---

mobile

RTK

---

fixed

mobile

NRTK

---

mobile

NRTK

Without regularization

mobile

NRTK

With regularization

Figure 7 shows the PPP result (Solution S1). The

points in the left panels of the figure are the error time

series with the fixed coordinates as the reference, the

line is the periodic noise generated according to Eq.

(28), and the right panels of the figure are the time

series of the difference between the two, that is, the

positioning error with the position of the mobile

coordinates as the reference. From the left panel, it can

be noticed that the horizontal positioning result of PPP

is very consistent with the noise added to the

observation value, and the U direction is slightly

worse. The positioning errors in the three directions of

E, N, U, are 0.74 cm, 0.43 cm, 1.74 cm.

Figure 7. Precise point Positioning (PPP) results (Solution S1)

of the mobile user station (with periodic movements). The left

panels are the error time series with reference to the fixed

station coordinates (i.e., points, the lines are periodic

motions), and the right panels are the residual series with

reference to the coordinates after periodic motion. The upper,

middle, and lower panels are the E, N, and U directions,

respectively.

Figure 8 shows the RTK result (Solution S2), the

points and lines in this figure have the same meaning

as Figure 8. From the results, the RTK positioning

results are worse than those of PPP, because only float

solutions can be obtained in some epochs (marked with

gray background). The errors in the three directions of

E, N, U are 1.92 cm, 2.79 cm, 4.21 cm, the fixing rate is

97.74%.

Figure 8. Real-time kinematic (RTK) results (Solution 2) of the

mobile user station with periodic movements, with fixed base

station. The rest is the same as Figure 7.

378

Figure 9 is the same as Figure 8, except that the base

station is replaced by a mobile station (Solution S3). In

other words, it is the RTK positioning result of all

stations that are mobile. The waveform in the figure is

different from the previous one because the movement

of the base station is also transmitted to the positioning

result along with the observation value. It can be found

that the RTK positioning results in all three directions

when all the stations are mobile, and the fixing rate also

drops to 88.06%.

Figure 9. Real-time kinematic (RTK) results (Solution S3) of

the mobile user station with periodic movements, with

mobile base station. The rest is the same as Figure 7.

Figure 10 shows the positioning result of NRTK

with the fixed base station, which is the general NRTK

(Solution S4). It can be found that the result is less

accurate than Solution S3, but the fixing rate is higher,

but still lower than Solution S2.

Figure 10. Network real-time kinematic (NRTK) results

(Solution S4) of the mobile user station with periodic

movements, with fixed network base station. The rest is the

same as Figure 7.

Figure 11 and Figure 12 are the NRTK positioning

results of the user side when the server does not use the

regularization method and when it uses the

regularization method (Solution S5 and S6). Both

figures are the results of the base station and the user

station being the mobile station. There is no significant

difference between the two results, which is

predictable, because the regularization method is

mainly used by the server-side baseline solution to

decorrelate the baseline movement and the

atmosphere, and the atmosphere itself is not sensitive

to small movements during the interpolation process.

However, the positioning result is still within the

normal accuracy that RTK can achieve, that is, 2-3 cm

horizontally and about 5 cm in vertical direction, which

can prove the effectiveness of this new method.

Figure 11. Network real-time kinematic (NRTK) results

(Solution S5) of the mobile user station with periodic

movements, with mobile network base station and without

regularization. The rest is the same as Figure 7.

Figure 12. Network real-time kinematic (NRTK) results

(Solution S6) of the mobile user station with periodic

movements, with mobile network base station and with

regularization. The rest is the same as Figure 7.

We show the result statistics of the 6 solutions in

Table 5, i.e., the accuracy and fixing rate in the three

directions of E, N, and U. Experimental analysis reveals

that moving the fixed reference stations induces

measurable degradation in positioning accuracy and

ambiguity resolution rates. However, the resultant

precision (remaining within practical operational

tolerances) validates the methodological framework

proposed for oceanic NRTK implementation. This

finding confirms the technical feasibility of extending

NRTK capabilities to marine environments through

adaptive station deployment strategies.

Table 5. Positioning results and fixing rate of six solutions

Solution

E (cm)

N (cm)

U (cm)

Fixing rate

(%)

Convergence time

0.74

0.43

1.74

---

~ 30 min

1.92

2.79

4.21

97.74

Seconds

3.33

3.40

4.82

88.06

Seconds

2.90

3.22

4.32

92.78

Seconds

2.98

3.05

7.03

87.78

Initialization: ~30 min

Follow-up: seconds

2.90

2.88

6.70

88.02

Initialization: ~30 min

Follow-up: seconds

4 CONCLUSIONS

In this contribution, we propose a novel network RTK

solution for mobile platforms to extend high-precision

379

network RTK positioning service to marine areas. In

this solution, all base stations are assumed to be

anchored to ocean buoys, which are constrained by an

anchor chain to move with the waves over a limited

range of the sea surface. The proposed method

incorporates baseline motions, specifically ∆x, ∆y, and

∆z, into the unknown parameters of the baseline

solution, then the regularization method is

subsequently employed to mitigate the degradation of

the equation structure caused by the introduction of

additional unknowns, while also serving to decorrelate

these parameters and enhance the accuracy of the

solutions for the unknown parameters. Finally, PPP is

performed on one or more of the stations to provide an

absolute position reference for the network.

Observations from 15 CORS stations in the Hong

Kong were utilized to experimentally validate the

feasibility of the proposed method. The experimental

results indicate that while the introduction of unknown

parameters does result in a degradation of

tropospheric delay accuracy, it has no significant

impact on DD troposphere delay and DD ionospheric

delay. Furthermore, the regularization technique

proves effective in enhancing the accuracy of these

unknown parameters, with the average standard

deviations of e, n, u, and ZWD for 24 baselines

decreasing from 4.44/3.50/2.38/4.96 cm to

2.83/1.62/1.16/3.18 cm. From the positioning results,

although the positioning accuracy of the mobile

platform is reduced compared to the fixed platform, it

can still reach the centimetre level, which is still at the

level of normal NRTK accuracy. In addition, the new

NRTK service initially requires a longer convergence

time similar to PPP during startup, but subsequently

achieves rapid convergence comparable to fixed base

NRTK, delivering consistent and stable service

thereafter.

This technology can be combined with existing

fixed base station NRTK (such as HK CORS) to expand

the existing high-precision positioning services on land

to offshore areas. Our next step will be to gather data

from the offshore marine GNSS buoys to further

develop this maritime NRTK system.

ACKNOWLEDGMENTS

The authors would like to thank the Geodetic Survey Section

of Survey and Mapping Office of Hong Kong for providing

GNSS observation data.

FUNDING

This research is funded by the General Research Fund of

Hong Kong (Grant No. 15229622) and the Innovation and

Technology Fund of Hong Kong (Grant No. ITP/019/22LP).

DATA AVAILABILITY

The software platform of this paper is Network RTK software

“Venus”. The GNSS observations data are obtained from the

Hong Kong Satellite Positioning Reference Station Network

(SatRef): https://www.geodetic.gov.hk/en/satref/satref.htm.

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