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Rev. Téc. Ing. Univ. Zulia, 2025, Vol. 48, e254808

Two-Dimensional Affine Transformation (AT2D) for

Geodetic Velocity Interpolation utilizing SIRGAS-CON

Superior. Escuela de Ingeniería Geodésica. Universidad del Zulia. Av. 16 (Guajira) con calle 67

Recepción: 02 abril 2025 | Aceptación: 23 octubre 2025 | Publicación: 31 octubre 2025.

Abstract

Geodetic velocity models are critical for transforming coordinates from a reference epoch to an observation

epoch. However, developing these models is often time-consuming, requiring frequent updates to maintain validity.

To address the issue of its validity over time, this study examines the use of the Two-Dimensional Affine

Transformation (AT2D) method for interpolating velocities at GNSS stations in South America. The AT2D relies on

triangulation from surrounding SIRGAS-CON stations, which have well-known velocities, over a specific time

interval. The resulting velocities from the AT2D method and VEMOS models were compared against their well-

contexts relevant to surveying and geodetic engineering.

Keywords: Affine Transformation; Geodetic Velocities; GNSS; Interpolation; SIRGAS-CON.

Transformación Afín Bidimensional (AT2D) para

Interpolación de Velocidades Geodésicas utilizando

Soluciones Semanales SIRGAS-CON

Resumen

Los modelos de velocidades geodésicas son cruciales para la transformación de coordenadas de una época de

referencia a una época de observación. Su desarrollo suele requerir mucho tiempo y actualizaciones frecuentes para

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triangulación desde estaciones SIRGAS-CON circundantes, con velocidades conocidas, durante un intervalo de

tiempo específico. Las velocidades resultantes del método AT2D y de los modelos VEMOS, sobre cincuenta

estaciones, se compararon contra sus velocidades conocidas, lo que permitió el cálculo de residuos para la evaluación

Palabras clave: Interpolación; GNSS; SIRGAS-CON; Transformación Afín; Velocidades Geodésicas.

Os modelos de velocidade geodésica são cruciais para a transformação de coordenadas de uma época de

referência para uma época de observação. Seu desenvolvimento costuma ser demorado e requer atualizações

frequentes para manter sua validade. Para abordar a questão da validade, este estudo examina o uso da Transformação

Afim Bidimensional (AT2D) para interpolar velocidades em estações GNSS na América do Sul. A AT2D baseia-se

na triangulação a partir de estações SIRGAS-CON vizinhas com velocidades conhecidas em um intervalo de tempo

específico. As velocidades resultantes do método AT2D e dos modelos VEMOS, em cinquenta estações, foram

comparadas com suas velocidades conhecidas, permitindo cálculos residuais para avaliação da qualidade. Em 82%

Geodetic velocities refer to the rate of change of a point's position on the Earth's surface over time, expressed in

a terrestrial reference frame and derived from continuous or episodic geodetic observations. Researchers typically

derive geodetic velocities through various geodetic techniques, including Very Long Baseline Interferometry (VLBI),

Global Navigation Satellite Systems (GNSS), Interferometric Synthetic Aperture Radar (InSAR), Light Detection and

Ranging (LiDAR), Photogrammetry, leveling, tiltmeters, and total station measurements (Yan et al., 2022).

Geodetic velocities find broad applications in various geophysical and engineering contexts, including

(Hoyer et al., 1998), the weekly coordinates from SIRGAS continuously operating network (SIRGAS-CON) provide

a robust dataset with precisely known positions (referred to a specific reference epoch) and their changes over time

(station velocities) (Sánchez, 2023; SIRGAS Analysis Centre at DGFI-TUM, 2024a). Based on SIRGAS reference

frame solutions, a regularly updated velocity model for SIRGAS called VEMOS (Drewes et al., 2024) has been

developed since 2003 (Drewes and Heidbach, 2005). However, generating velocity models takes considerable time,

and ongoing geodynamic processes inherently limit their validity.

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geodetic engineering projects. The proposed method utilizes SIRGAS-CON station coordinates to calculate velocities

over a defined period, based on the principle that velocities are determined as the rate of coordinate change between

two epochs within a consistent ITRF. This linear, uniform displacement model implicitly assumes the absence of

significant trajectory anomalies such as abrupt jumps, antenna changes, receiver changes, or data discontinuities

for geodetic velocity estimation in a local context (range < 1,000 km), with a dense GNSS network, such as SIRGAS-

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SIRGAS weekly position differences (*.NEU) files, which include the reference frame, epoch in decimal years

and position differences (∆E, ∆N), were also extracted, for the period 2014.011 to 2024.011 (Sánchez et al., 2022).

All SIRGAS weekly products presented and discussed in this paper are publicly available at the following website

(SIRGAS Analysis Centre at DGFI-TUM, 2024b): https://www.sirgas.org/en/stations/station-list/.

*.TXT files of the VEMOS2017 (Drewes and Sánchez, 2020) and VEMOS2022 (Drewes et al., 2024) velocity

models. VEMOS2017 was derived from pointwise station velocities inferred at 515 geodetic sites from January 1,

2014, to January 28, 2017, using a geodetic least-squares collocation approach with empirically determined covariance

functions. The average uncertainty of VEMOS2017 is ±1.0 mm/yr in the north-south direction and ±1.7 mm/yr in the

east-west direction. VEMOS2022 was computed using the same methods as the previous VEMOS2017, covers the

period from February 1, 2017, to April 30, 2022, and its average uncertainty is assessed to be ±0.8 mm/yr in the north-

using the Two-Dimensional Affine Transformation SGC_SIRGAS-AT2D_Interp.exe (Suárez, 2023) based on

Python's NumPy library (Harris et al., 2020).

This research examines the Two-Dimensional Affine Transformation (AT2D) method (Ghilani, 2018), to

interpolate the horizontal velocity components of any point to be interpolated (Pi) within the area of a triangle formed

by three surrounding SIRGAS-CON reference stations (Sr) (Figure 2). The geodetic coordinates (Xr, Yr), and

velocities (VEr, VNr) are known for a defined time interval (tf - ti), within the same ITRF. Figure 2 presents an

example of triangulation involving three surrounding SIRGAS-CON reference stations—Sr, CUIB, ROJI, and

SCRZ—with the interpolation target station (Pi), MTLA, positioned at the center. The distances between these stations

range from 350 to 560 km.

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In this study, the authors selected SIRGAS-CON stations based on three main criteria: geodynamics, geometrics,

and data availability. The authors located all stations (Pi and Sr) on the same tectonic plate to ensure consistent velocity

interpolation. Furthermore, for maximum accuracy, the configuration of the selected SIRGAS-CON reference stations

With the reference stations selected, the authors proceed with their velocity’s calculation. It is important to

highlight that the authors did not derive velocities through a regression model of the coordinate time series. Instead,

the authors calculated the SIRGAS-CON stations' velocities over a defined period corresponding to the rate of change

of their coordinates (position differences) between two observation epochs within the same ITRF (Drewes, 2020). The

calculation of horizontal velocities (VEr, VNr) of the SIRGAS-CON reference stations (Sr) was based on weekly

position difference files (*.NEU), using the coordinate changes (∆Er, ∆Nr) between two observation epochs (tf and

ti), representing discrete positional shifts over a defined time interval. The governing equations are shown below (Eq.

1, 2).

󰇛󰇜

The authors interpolated the velocities at the point of interest (Pi) by using the Two-Dimensional Affine

Transformation (AT2D), which applies two translations of the origin, and a rotation about the origin, plus a small

nonorthogonality correction between the X and Y axes (Ghilani, 2018). AT2D utilizes six parameters (a, b, c, d, e, f),

including two translations (c, f), two independent scales, one rotation, and one distortion, allowing for modeling not

only rotations and translations but also stretching and distortions in two different directions. X and Y are the well-

known coordinates (longitude and latitude) extracted from *.PLH files. Equations (3) and (4) show the observation

   

(5)

Once the authors explicitly solve the observation equation system AX = L by calculating X=A−1L using matrix

inversion, and all its unknown parameters (a, b, c, d, e, f) are determined. The authors interpolate the velocities at the

point of interest (Pi) with known coordinates (Xi, Yi) using equations (6) and (7).

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  (6)

10, 11).

󰇛󰇜 󰇛󰇜 (8)

 󰇛󰇜 󰇛󰇜 (9)

 󰇛󰇜

 󰇛󰇜

CON velocity values. The authors performed the first comparative analysis of velocities for the period 2014.011–

2017.011, simultaneous with VEMOS2017, using fifteen triangular configurations formed by SIRGAS-CON stations

[15(Pi) + 45(Sr)]. In this analysis, the distance between the Pi and its corresponding Sr varies from 120 to 870 km

(Figure 3).

From Table 2, the AT2D method exhibited the lowest RMSE (VN: ±1.1 mm/yr, VE: ±1.4 mm/yr) among three

velocity interpolation methods (AT2D, VEMOS2017, VEMOS2022) compared against reference SIRGAS-CON

velocities. In addition, AT2D's direction (sign) of the interpolated velocities matches the SIRGAS-CON velocities for

all fifteen stations. VEMOS17 failed at station BLPZ, and VEMOS2022 failed at stations BLPZ, RIO2, and RNMO.

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Figure 3. SIRGAS-CON stations [15(Pi) + 45(Sr)], period 2014.011–2017.011.

Table 2. Interpolated velocities and residuals [mm/yr] (period 2014.011-2017.011).

SIRGAS-CON

AT2D

Res.

VEMOS2022

δVNi

14.02

0.91