1 Introduction

In recent years, the positional accuracy attainable from GNSS technology is at a millimetre level [10]. With this accuracy, coordinates change in a high pace over time due to plate tectonic motion and other geophysical phenomenon. Current and previous International Terrestrial Reference Frame (ITRF) realizations take into account tectonic plate motion and other deformation such as earthquakes. Therefore, coordinates of points with every new realization of ITRF change even at a later epoch of same realization.

The coordinates of geodetic datum are the fundamentals for positioning. However, GIS and surveying software as well as spatial data do not consider continuous changes in coordinates, meaning that national datum and coordinates are assumed fixed with time. Similarly, geodetic networks are known to form the basis of investigating the shape, dimension and in many cases the gravity field of the earth [10, 26]. Therefore, positioning is a major stakeholder in modern day society in that it is of interest in navigation and guidance, datum realization, crustal deformation, plate tectonic studies, amongst others.

Geodetic datums are curved reference surfaces used to express position with adopted ellipsoid of revolution, size and shape [10, 29]. Generally, local geodetic datums whose ellipsoid does not coincide with the earth’s centre of mass and geocentric datum whose ellipsoid coincides with the earth’s centre of mass are the two fundamental categories of geodetic datum [10]. Geodetic datum can be static, dynamic and semi-dynamic [9, 25, 27]. Traditional geodetic datums are assumed static in nature. This is because they consider the constantly changing earth to be static which is very untrue because they are affected by geological and tectonic activities.

Dynamic datum, on the other hand, is a function of time. This means that coordinates vary with time. Example of a dynamic datum is the ITRF [2, 3]. Therefore, to take into account the changing earth, the ITRF is updated after every 5 years to accommodate advances in processing and data improvement. Implementation of a dynamic datum is a very difficult task at national a level. This has to be done monthly or weekly, thereby making the choice of the correct epoch for referencing observation extremely complex [13].

Semi-dynamic datum considers the constantly changing earth’s motion, but coordinates are referred to a single reference epoch [9, 25, 27]. Coordinates in a semi-dynamic datum can be propagated from the pre-defined reference epoch to an epoch of interest. What needs update is the deformation model.

Rapid advances in space-geodetic technique such as GNSS, very-long-baseline interferometry (VLBI), Global Navigation Satellite System (GNSS), Doppler Orbitography and Radiopositioning Integrated by Satellite (DORIS) and their found application in geodesy and geomatics have led to a significant improvement in modern positioning and allied applications. The tectonic motion of the NUBIA plate which might result in volcanism, earthquakes and earth tremors and other deformation sources such as subsidence and soil creep [10] and un-modelled measurement biases [6] affect geodetic infrastructure (e.g. NigNet), and therefore, decrease the accuracy of reference station coordinates, thereby leading to inconsistencies in legal traceability of coordinates of NigNet coordinates over time [11].

Since the MINNA datum for Nigeria was realized based on classical methods of surveying, it is known to be subjected to biases as a result of poor observations and techniques [8]. Also, Nigerian MINNA datum is faced with biases such as inaccuracies of the scale factor by compression of the Clarke 1880, poor definition of origin, geoidal height model absence and difficulty in datum parameters determination [30]. Therefore, with these inherent errors, there is the need to move away from the still-in-use MINNA datum and adopt a new datum that is geocentric in nature as done in countries like Australia (GDA2020), China (CTRF2000), Indonesia (DGN1995), Malaysia (GDM2000), New Zealand (NZGD2000), USA (NAD83), amongst others [7, 20]. Studies of [5, 8, 12, 18, 25] further justify the need for this study, in that coordinates of points anywhere in the world will change with time. Therefore, by integrating velocity, epoch and episodic deformation information, locations of points can be kept on track. Therefore, this study presents and proposes a new time-dependent geocentric datum transformation parameters for Nigeria by processing 5 years (2011–2015) of the 14 NigNet tracking stations (Fig. 1) [19] and 9 IGS stations (Fig. 2) [22] using GAMIT for producing loose constrain estimate of position and covariance matrix and GLOBK for reference frame definition.

Fig. 1
figure 1

Spatial distribution of NiGNet tracking stations with the station at KANO excluded

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Fig. 2
figure 2

Spatial distribution of IGS stations adopted for the study

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2 Need for transformation parameters and coordinate update in Nigeria

With series of earth tremor occurrences in Nigeria, epoch-by-epoch realization of ITRF is essential so that spatial data of global, local, national and regional origin can be integrated with ease. Over the years, the development of transformation parameters for Nigeria particularly, the classical transformation which involves 7-parameter similarity (Helmert) transformation, has been met with several technical challenges such as inaccuracies of the scale factor by compression of the Clarke 1880, poor definition of origin, geoidal height model absence and difficulty in datum parameters determination as highlighted earlier. However, with the evolution of ITRF, the 7-parameter transformation has extended to 14-parameter transformation. The additional 7 parameters describe the transition of the initial 7 parameters with time [20]. Furthermore, depending on the need, surveys in Nigeria are reported in various coordinates that include Nigeria Transverse Mercator Projection (NTM), MINNA, World Geodetic System 1984 (WGS84) and ITRF. Ensuring the compatibility and uniformity of coordinates, [7] reported that the use of different coordinate system poses danger and a difficult task to achieve.

Therefore, the optimum way to achieve centimetre-level accuracy is to relate GNSS measurements to ITRF. Therefore, ITRF-based transformation parameters will go a long way towards realization of centimetre accuracy since deformation of the earth is taken into cognisance.

Most importantly, with modern space-geodetic techniques, such as the GNSS CORS network in Nigeria, the biases due to classical observational techniques and the phenomenon of plate tectonic motion can be mitigated and a more accurate geocentric datum transformation parameters between MINNA DATUM and the known ITRF can be developed and adopted. This study therefore aims at proposing time-dependent geocentric datum transformation parameters for Nigeria using 5-year (2011–2015) NigNet GNSS CORS data.

3 Materials and method

3.1 Test site description

In Nigeria, an initiative to establish a Continuously Operating Reference Stations (CORS) called NIGerian Reference GNSS NETwork (NIGNET) (see Fig. 1) which is a network of Continuous GNSS stations kicked off in 2008 by Office of the Surveyor General of the Federation (OSGoF). The initiative was aimed at contributing to the African Reference Frame (AFREF) and serve as a primary fiducial network that will define and materialize a new reference frame based on space-geodetic technique [19]. The study area is Nigeria located on the western part of Africa Plate between latitude 4° and 14°N and longitude 2° and 15°E. The details of the dataset used for this study are presented in Table 1. The spatial distribution of the NigNet and IGS stations used for the study is presented in Figs. 1 and 2, respectively.

Table 1 Summary of dataset and sources adopted for the study
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3.2 GNSS data processing

GAMIT/GLOBK software release 10.6 was used for processing [14,15,16]. GAMIT/GLOBK is a comprehensive GNSS analysis package developed at Massachusetts Institute of Technology (MIT), the Harvard Smithsonian Center for Astrophysics (CFA) and the Scripps Institute of Oceanography (SIO) for estimating station coordinate and velocities, stochastic or functional representation of post-seismic deformations, atmospheric delays, satellite orbits and Earth orientation parameters [15, 32].

Depending on the task at hand, processing in GAMIT can be in single session or automatic batch processing (invoked when there is considerable large amount of data or multiple session of data and time is needed to be saved) using scripts for example sh_gamit in GAMIT or sh_glred in GLOBK. In the automatic batch processing, which was adopted in this study, the only preparation is assembling and preparation of control files like sestbl, sittbl, station.info, session.info, etc. Models applied to account for dynamic factors include Vienna Mapping Function (VMF1) for tropospheric mapping of dry and wet mapping function, FES2004 ocean tide loading model and IERS03 solid earth tide model.

The results obtained in automatic batch processing of GAMIT are generally loose constrain estimate of position and covariance matrix associated with a survey station. Table 2 depicts the processing parameters adopted for this study. The ASCII h-files containing loose-constrained weighted least squares estimate of sites coordinates and variance–covariance produced by GAMIT are converted to binary H-files readable by GLOBK to produce time series, velocity and reference frame definition.

Table 2 Basic processing parameters [4]
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3.3 Reference frame definition

Frame realization was carried out in GLOBK by adopting the International Terrestrial Reference Frame (ITRF) as global constraints. In this study, solutions from GAMIT were constrained to ITRF2008 [2] and ITRF2014 [3] while estimating seven Helmert parameters (3 translation, 3 rotation and 1 scale) and their respective rates for each solution from GAMIT analysis. The study constrained NigNet tracking stations to global stations consisting of 9 selected IGS (see, for example, Fig. 2) sites in ITRF2008 and ITRF2014 using GLOBK. The solutions from GLOBK include velocity solution, reference frame and Euler plate motion parameters, amongst others.

3.4 Coordinates transformation method

Transformation of GNSS coordinate time series from one reference frame to another can be realized with Helmert 14-parameter or 7-parameter transformation. The Helmert 14-parameter was adopted for this study. These include 3 translations (which affects position of coordinate origin), 3 rotations (which affects orientation of coordinate axes) and 1 scale (affects length of coordinate axes), and their rates (achievable with a velocity model). The Euclidian similarity 14-parameter datum transformation model (Eqs. 1a–1c) which includes three rotations, \(T_{x} (t)\), \(T_{y} (t)\), \(T_{z} (t)\), three translation (\(R_{x} (t)R_{y} (t)R_{z} (t)\)) and a scale factor \(S_{c} (t)\) [6, 10, 31], was adopted and tested to transform station coordinates and respective velocities at epoch 2015.9685 (2015/12/20) from ITRF08 and ITRF14 to a proposed Geocentric Datum of Nigeria (GDN). Though, Euler plate motion parameters can also be used [26].

$$\begin{aligned} X(t)_{\text{GDN}} & = T_{x} (t) + [1 + s(t)]X(t)_{{{\text{ITRF}}_{yy} }} \\ & \quad + R_{z} (t)Y(t)_{{{\text{ITRF}}_{yy} }} - R_{y} (t)Z_{{{\text{ITRF}}_{yy} }} \\ \end{aligned}$$
(1a)
$$\begin{aligned} Y(t)_{\text{GDN}} & = T_{y} (t) - R_{z} (t)X(t)_{{{\text{ITRF}}_{yy} }} \\ & \quad + [1 + s(t)]Y(t)_{{{\text{ITRF}}_{yy} }} + R_{x} (t)Z(t)_{{{\text{ITRF}}_{yy} }} \\ \end{aligned}$$
(1b)
$$\begin{aligned} Z(t)_{\text{GDN}} & = T_{z} (t) + R_{y} (t)X(t)_{{{\text{ITRF}}_{yy} }} \\ & \quad + [1 + s(t)]Z(t)_{{{\text{ITRF}}_{yy} }} - R_{x} (t)Y(t)_{{{\text{ITRF}}_{yy} }} \\ \end{aligned}$$
(1c)

The time-related variations are assumed to be linear; therefore, the quantities can be expressed by Eq. (2) [23, 24]. Equation (2) is inputted into Eq. (1) to finally get a 7-parameter transformation model.

$$\begin{aligned} & T_{x} = t_{x} (t_{0} ) + \dot{t}_{x} (t - t_{0} ) \\ & T_{y} = t_{y} (t_{0} ) + \dot{t}_{y} (t - t_{0} ) \\ & T_{z} = t_{z} (t_{0} ) + \dot{t}_{z} (t - t_{0} ) \\ & S_{c} = s_{c} (t_{0} ) + \dot{s}_{c} (t - t_{0} ) \\ & R_{x} = r_{x} (t_{0} ) + \dot{r}_{x} (t - t_{0} ) \\ & R_{y} = r_{y} (t_{0} ) + \dot{r}_{y} (t - t_{0} ) \\ & R_{z} = r_{z} (t_{0} ) + \dot{r}_{z} (t - t_{0} ) \\ \end{aligned}$$
(2)

where the 7 Helmert parameters \(r_{x} (t_{0} )\), \(r_{y} (t_{0} )\), \(r_{z} (t_{0} )\) (\(t_{x} (t_{0} )\), \(t_{y} (t_{0} )\), \(t_{z} (t_{0} )\) and \(s_{c} (t_{0} )\)) are rotations, translation and scale parameters in the position domain, respectively, at a reference epoch, which are constant. Their respective first time derivations in the rate domain are \(\dot{t}_{x}\), \(\dot{t}_{y}\), \(\dot{t}_{z}\), \(\dot{r}_{x}\), \(\dot{r}_{y}\),\(\dot{r}_{z}\) and \(\dot{s}_{c}\). The Helmert parameters and their first time derivatives are obtained by invoking a generalized constraint in glorg module of GLOBK.

3.5 Test statistics for validating transformation parameters

The accuracy, precision and reliability of the newly obtained transformation parameters were evaluated using standard model performance indicators. The normalized mean absolute error (NMEA) [28], root mean square error (RMSE), reliability index (RI) [21] and correlation coefficient (r) are given in Eqs. (3)–(6) [17].

$${\text{NMAE}} = \frac{{\sum\nolimits_{i = 1}^{n} {\left( {\left| {{\text{residuals}}_{i} |} \right.} \right)} }}{{n\bar{o}}}$$
(3)
$${\text{RMSE}} = \sqrt {\frac{{\sum\nolimits_{i = 1}^{n} {\left( {{\text{residuals}}_{i} } \right)^{2} } }}{n}}$$
(4)
$${\text{RI}} = \sqrt[{\exp }]{{\frac{{\sum\nolimits_{i = n}^{n} {\left( {\log \frac{{o_{i} }}{{p_{i} }}} \right)^{2} } }}{n}}}$$
(5)
$$r = \frac{{\sum\nolimits_{i = 1}^{n} {\left( {p_{i} - \bar{p}} \right)\left( {o_{i} - \bar{o}} \right)} }}{{\left[ {\sum\nolimits_{i = n}^{n} {\left( {p_{i} - \bar{p}} \right)^{2} } \sum\nolimits_{i = 1}^{n} {\left( {o_{i} - \bar{o}} \right)^{2} } } \right]^{1/2} }}$$
(6)

where \(n\) is the number of NigNet tracking stations used, \({\text{residuals}} = p_{i} - o_{i}\) and \(o_{i}\) and \(p_{i}\) are the ith computed and model estimated. Similarly, \(\bar{o}\) and \(\bar{p}\) are the mean computed and model estimates.

4 Results and analysis

4.1 Results

While estimating translation, rotation and scale (Helmert parameters) in GLOBK, a rigorous approach of frame definition was realized through a generalized constraints, where the study minimized the adjustments of coordinates of the frame-defining sites [15] as discussed earlier in Sect. 3.4. By minimizing the adjustment of coordinates of the frame-defining sites, all of the reference sites are free to adjust, thereby revealing bad data or coordinates solutions. Table 3 shows the summary of the time-dependent geocentric transformation parameters for Nigeria.

Table 3 Summary of ITRF08 and ITRF14 to geocentric datum of Nigeria 2015 (GDN15) transformation parameters and their uncertainties at reference epoch 2015.9685
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Similarly, Tables 4 and 5 are the coordinate solutions of the adopted NigNet stations in earth-centred earth-fixed (ECEF) and geodetic coordinate system, respectively. The residuals plots of the two coordinates systems in both reference frames are presented in Fig. 3a, b, with station RUST having the highest residual. The reason for this high residual is unknown in the present study. Probably, bad stabilization might be the cause, but the post-root mean square error (post-RMS) in ITRF14 for velocity and position system stabilization yielded 0.029 mm/year and 0.087 mm, respectively. In ITRF08, the post-root mean square error (post-RMS) for velocity and position system stabilization yielded 0.049 mm/year and 0.016 mm, respectively. [15] recommended value of 1–5 mm for position system stabilization. Therefore, bad stabilization is written off as the cause.

Table 4 Geocentric (ECEF) coordinates of NigNet in ITRF08 and ITRF14 at epoch 2015.9685
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Table 5 Geodetic coordinates of NigNet in ITRF08 and ITRF14 at epoch 2015.9685
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Fig. 3
figure 3

Residual plot of coordinate of a (in ECEF, Table 4) solutions in ITRF08 and ITRF14; b (in geodetic, Table 5) solutions in ITRF08 and ITRF14

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4.2 Analysis

To this end, Tables 4 and 5 present the geocentric and geodetic coordinates of the NigNet GNSS stations at epoch 2015.9685, respectively. Furthermore, the time-dependent 14-parameter datum transformation relation as presented in Table 3 has been developed at epoch 2015.9685 (2015/12/20). First, to assess the accuracy and validity of the transformation parameters, coordinates of some of the stations at epoch 2015.9685 as obtained from GLOBK in this study were transformed to an epoch (2011.00) in which the Office of the Surveyor General of the Federation (OSGoF) last published their coordinates. Tables 6 and 7 present the ECEF and geodetic coordinates and from OSGoF and those obtained using the transformation parameters in Tables 3, 4 and 5, respectively.

Table 6 Transformed geocentric (ECEF) coordinates of NigNet in ITRF08 to epoch 2011.00 using that obtained from OSGoF
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Table 7 Transformed geodetic coordinates of NigNet in ITRF08 to epoch 2011.00 using that obtained from OSGoF
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4.3 Statistics for validating transformation parameters

The accuracy, precision and reliability of the transformation parameters in Table 3 were evaluated by comparing solution obtained from the transformation parameters in ITRF08 and from OSGF using the performance indicators discussed in Sect. 3.5. Figure 4 is the scatter plot in this regard.

Fig. 4
figure 4

a–c Scatter plot of the correlation between coordinates obtained from OSGF and that computed using the developed transformation parameters in Table 3

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The correlation value R, root mean square error (RMSE), normalized mean absolute error (NMAE) and reliability index (RI) are at the bottom right: (a) the plot in the X, (b) the plot in Y and (c) the plot in Z directions, respectively, of ECEF.

RMSE measures average square error. Zero values or near indicate close match. Therefore, the RMSE (0.00411 m) of the coordinates from the two data sources is an indication of close match. NMAE assesses the absolute deviation of computed coordinates from GLOBK and that of OSGF. Values of zero or near indicate close match and vice versa. The NMAE (1.267E−10 m) of the coordinates obtained from both sources showed a close match. The RI is an indication of how the two coordinate sources differ from each other; a near one value also indicates a close match. Therefore, from the two coordinate sources, the two data sources showed close match. The correlation coefficient of the data sources also yielded 1.0, which is a perfect correlation.

Similarly, [1] stated that the origin and scale between ITRF2014 and ITRF2008 are less than 5 mm. Therefore, the transformation parameters for ITRF2014 as developed in this study are also valid since the difference is within 5 mm.

5 Conclusions and recommendations

In this study, time-dependent geocentric datum transformation parameters for Nigeria using the NigNet tracking stations have been developed. The accuracy, precision and reliability of the transformation parameters using the R2, RMSE, NMAE and RI statistics validate the transformation parameters developed in this study. Also the origin and scale differences between ITRF2008 and ITRF2014 found to be within 5 mm also portray the reliability of the transformation parameters.

The significance of these results is fundamental in providing a variety of applications in areas such as geophysical hazard monitoring and assessment, sea-level monitoring, mining engineering, location based services, land boundary definition (international and local boundaries), environmental mapping, navigation, civil engineering and cadastral applications.

Based on the results of the study, the followings are hereby recommended:

  1. 1.

    The current trend by many countries is the realization of geocentric datum based on ITRF realization. Since the old MINNA datum is filled with many deficits, Nigeria should adopt the ITRF-based geocentric datum.

  2. 2.

    There should be awareness within the geospatial community on the need to move away from the old MINNA datum. This is a task vested on geospatial stakeholders such OSGoF.