Practical Notes on Coordinate Transformation

This article serves as a practical guide to the background, requirements, and processes for aligning geodetic reference frames both temporally and spatially. It emphasizes clear descriptions of fundamental technical issues and highlights key practical considerations. For a more detailed overview, with worked examples and mathematical background on coordinate conversion and transformation read the full white paper.

Expressing Location

There are three common ways of expressing location on the earth: Cartesian, geodetic and map grid coordinate systems. 

Cartesian Coordinates

The simplest way to express the location of an object on Earth is to use the 3D Cartesian axes (x,y,z). The advantage of the Cartesian system is that the three components of the position vector are expressed in linear units (meters), allowing all position-related calculations to be performed using vector geometry. However, its disadvantage lies in the non-intuitive nature of Cartesian coordinates, as it is difficult to conceptualize location in this form due to the lack of clear 'horizontal' and 'vertical' components.

Geodetic Coordinates

Alternatively, we can use the geodetic (or geographic) coordinate system, which consists of latitude, longitude, and height, represented as (𝜙, 𝜆, ℎ). Geodetic coordinates are defined by the normal to the ellipsoid at a given point. Latitude is the angle between this normal and the equatorial plane, while longitude is the angle between the zero meridian (Greenwich) and the meridian containing the normal. The ellipsoid height is the distance along the normal from the surface of the ellipsoid to a given point.

The advantage of the geodetic system is that locations are easier to conceptualize and visualize. However, its main drawbacks include the use of curvilinear coordinates (degrees) and the complexity of spatial calculations on the reference ellipsoid’s surface. For instance, calculating the ellipsoid distance between two points in terms of latitude and longitude is not straightforward, even though it is more practically meaningful than the vector distance derived from Cartesian coordinates.

Figure: The Cartesian and geodetic coordinate systems

Map Grid Coordinates

To avoid the complexity of working on the curved surface of the ellipsoid and the non-intuitive nature of Cartesian coordinates, planar or map grid coordinates are often used. These coordinates are obtained by applying the appropriate map projection formulas to the corresponding geodetic coordinates. While various map projections exist, the Transverse Mercator (TM) and Universal Transverse Mercator (UTM) projections are most commonly used. Map grid coordinates are typically represented as (𝐸, 𝑁, ℎ).

Technically, ellipsoidal height is not part of the map grid system, as it is derived from the geodetic coordinates. However, for practical purposes, the ellipsoidal height is often converted into height above the geoid—an equipotential reference surface such as mean sea level—by subtracting the geoid undulation (the difference between the ellipsoid and the geoid). Map grid coordinates are expressed in linear units relative to the origin of a specific map grid zone covering the area of interest.

The advantage of the map grid system lies in its planar nature, which simplifies calculations. However, every map projection introduces some form of distortion (e.g., in scale, area, or orientation), which must be accounted for in spatial calculations, adding complexity and potential for error.

Reference Systems and Reference Frames

What is a Reference System?

In geodetic terms, a reference system provides the theoretical foundation for realizing station coordinates within an accessible reference frame. Its design encompasses four key elements: axes, location, orientation, and scale.

What is a Reference Frame?

A geodetic reference frame provides a practical implementation of a reference system by computing and publishing station coordinates relative to the system's definition. A well-known contemporary example is the International Terrestrial Reference Frame 2014 (ITRF2014), which includes 3D station coordinates for 975 distinct global sites. It incorporates data from four complementary geodetic observation techniques:

• Very Long Baseline Interferometry (VLBI)
• Satellite Laser Ranging (SLR)
• Doppler Orbitography and Radio-positioning Integrated by Satellite (DORIS)
• Global Navigation Satellite Systems (GNSS)

The International Earth Rotation and Reference Systems Service (IERS) is responsible for establishing the International Terrestrial Reference System (ITRS) and for the ongoing realization and maintenance of the ITRF.

Why are there Different Reference Frames?

While a geodetic reference system can be defined once and remains valid thereafter, this is not the case for a geodetic reference frame. The dynamic nature of the Earth's crust means that points on its surface are constantly shifting due to tectonic plate movement and other sources of crustal deformation, such as earthquakes. In practical terms, within an Earth-Centered, Earth-Fixed (ECEF) reference frame like ITRF2014, the coordinates of surface points are continually changing.

It is for this reason that IERS publishes station velocities along with station coordinates at a specified epoch for each realization of the frame. These velocities allow station coordinates to be moved through time within the reference frame, so long as the assumption of linear motion remains valid. However, the longer the time between the reference epoch and the computation date, the less accurate the propagated coordinates will be. 

Additionally, as time progresses, more stations can be incorporated into the reference frame solution, leading to the inclusion of improved observations, longer time series, and more sophisticated error modeling and computational methods. Consequently, each realization of the ITRF is more robust and accurate than its predecessor.

Coordinate Conversion & Transformation

Coordinate Conversion

Coordinate conversion is the process of changing from one form of coordinates to another while remaining within the same reference frame and at the same epoch. Standard formulas exist to convert between map grid coordinates and geodetic coordinates, although these formulas vary depending on the type of map projection used (e.g., UTM). These conversion formulas can be easily found in standard geodetic textbooks or online. Similarly, Cartesian coordinates can be derived from geodetic coordinates and vice versa. However, it is not possible to convert directly between map grid coordinates and Cartesian coordinates; these conversions must always go through the geodetic system.

Linear Temporal Transformation

Temporal transformation is the process of moving coordinates from one epoch to another, whileremaining in the same reference frame. Typically, cartesian coordinates are used in temporal transformation calculations. Most commonly, such movement is due to linear tectonic plate motion. Non-linear movement through time can also arise and will be discussed in more detail in the next section. 

The underpinning assumption is that tectonic plate motion is essentially linear and able to be modeled either by point velocities or rotation rates. This assumption is generally valid (at least for relatively short periods of time). In Australia, for example, where tectonic motion is highly stable, empirical data suggests the  assumption of linearity can be adopted for approximately 20 years. However, there are areas where the assumption fails much more quickly, most typically in regions of earthquake activity, such as Japan, New Zealand and along the San Andreas fault of the US. In such cases, point velocities must be augmented with a model to account for the non-linear deformation. 

Non-Linear Temporal Transformation

While linear motion from tectonic plate movement is well understood and routinely accounted for, modeling nonlinear crustal motion (deformation) is more complex and challenging to detect. Deformation can arise from natural causes, such as post-seismic relaxation, or from human activities like oil and gas extraction, and underground mining. To effectively address time-dependent coordinates, regional deformation models are necessary to capture this non-linear motion alongside linear tectonic movement. Efforts are underway in Australia to tackle this issue, but operational solutions remain a work in progress. 

Measuring and modeling non-linear crustal motion requires higher spatial resolution and temporal frequency due to the localized nature of the movement; typical CORS station densities of tens to hundreds of kilometers are insufficient. Consequently, remote sensing techniques such as InSAR and LiDAR are being evaluated as primary methods for assessing regional crustal deformation. Once measured, this motion must be modeled to complement linear plate movement, often using a grid-based approach. New grids can be created with each measurement, allowing for the summation of individual grids to determine total deformation continuously.

Total Temporal Transformation

Total temporal transformation is a process that encompasses a full temporal transformation, accounting for both linear and non-linear components. The linear component reflects tectonic motion from the reference epoch to the current epoch. Periodically, local deformation is measured, and these measurements are interpolated onto a grid to provide a continuous representation of the deformation field over the area of interest between measurement epochs. By adding these deformation grids for each epoch, the cumulative non-linear deformation from the reference epoch to any subsequent epoch can be calculated. Thus, the total crustal motion at any given epoch is the sum of the linear motion and the non-linear deformation.

Spatial Conformal Transformation

Spatial transformation is the process of converting coordinates from one frame of reference to another at a common epoch. For instance, a user of the Swift solution may need to transform coordinates from ITRF2014 to ITRF2008 at the common epoch of 2010.0. A spatial-conformal transformation, which does not involve a time component, preserves the shape of the transformed object. The parameters for this transformation are known and can be applied independently of the reference epoch. Typically, a 7-parameter transformation, also known as 'Similarity,' 'Conformal,' or 'Helmert,' is used for this purpose.

Spatial Non-Conformal Transformation

Legacy reference frames can exhibit significant spatial distortions, typically arising from measurement and computational errors during their realization. For example, AGD66 in Australia was established based on sparse, sinuous geodetic traversing and disparate measurements taken with less accurate terrestrial surveying equipment, along with non-rigorous computational models and procedures. As a result, AGD66 produced a solution that was inferior to the newer GDA94 frame.

To transform coordinates from AGD66 to GDA94 requires not just a conformal transformation, but a complex, spatially variable correction model that allows for the removal of distortions from the historic AGD66 coordinates. Generally, a distortion model is represented by a grid of 2D distortion components (Δ𝜙, Δ𝜆), derived from spatial analysis and modeling of coordinates in both the old and new reference frames. These grids are typically provided in the NTv2 format and usually include both conformal and distortion components at each grid node, enabling a comprehensive transformation (conformal + distortion) in a single computational step.

Total Spatial Transformation

As mentioned in the previous section, when performing a spatial transformation that includes both a conformal and a non-conformal (distortion) component, it is common practice to combine these components and represent them in gridded form for practical application. There are instances when spatial transformation from one frame of reference to another must occur via an intermediate frame. For example, in Australia, transforming from AGD66 to GDA2020 requires using GDA94 due to the availability of transformation parameters and grids. Thus, a two-stage transformation is necessary. Unlike total temporal transformation, it is incorrect to simply add distortion grids; they must be applied sequentially, as they are specific to each reference frame.

Over time, the need for distortion models will diminish as historical reference frames are phased out. For instance, distortions between AGD66 and GDA94 exceeded 1 meter, while those between GDA94 and GDA2020 are typically less than 0.1 meters. As modern reference frames replace legacy systems worldwide and eliminate the distortions characteristic of older frames, the demand for distortion models will decrease. The necessity for Swift to accommodate distortion models will need to be evaluated on a case-by-case basis, depending on the magnitude of the distortions relative to the accuracy of the Swift solution, the accuracy requirements of the user, and whether the digital map product has been affected by the distortions in the survey control network from which the distortion model was derived.

Combined Temporal and Spatial Transformation

With the emergence of time-dependent reference frames, it is increasingly common to change from one reference frame to another while simultaneously propagating between epochs. For instance, transforming coordinates from ITRF08@2005.0 to ITRF2014@2010.0 is referred to as a combined transformation, typically utilizing a 14-parameter model. This model enhances the standard 7-parameter transformation by incorporating rates of change for each parameter, allowing for adjustments in translation, rotation, and scale to reflect the time difference between reference frame realizations.

It is important to note that the 14-parameter model assumes linear crustal motion and does not account for non-linear deformation. Non-linear motion requires a separate gridded approach, as it cannot be included in a single-step transformation.

The Complete Transformation Picture

The complete transformation process is complex, necessitating careful attention to ensure the reliability, repeatability, and traceability of results. Due to the variety of formulas required for the total transformation and the potential for multiple transformation layers, there can be computational advantages in gridding the individual components and combining these grids as needed to achieve the final transformation solution. Conceptually, this approach is illustrated in the figure below.

Figure: The concept of a fully gridded transformation solution, with each component represented in gridded form and the total solution computed by sequential addition of the individual grids

Read the full white paper

Glossary of Terms:

Geodesy is the science that underpins the determination of the size and shape of the earth, the precise location of points and objects on and near the earth, and the 4D modeling of the earth’s gravity field.

Reference systems are the theoretical and conceptual basis for the unique and repeatable description of location. The International Terrestrial Reference System (ITRS) is a well known example.

A reference frame is the practical realization of a reference system, achieved by computing and publishing station coordinates relative to the frame definition. The International Terrestrial Reference Frame 2014 (ITRF2014) is a contemporary example

Geodetic datum is a term used interchangeably with the term reference frame. 

Coordinate conversion is the mathematical process whereby one form of coordinates is changed into another form within the same reference frame. For example, changing a Cartesian coordinate to a geodetic one.

Coordinate transformation is the generic term for the mathematical process whereby the reference frame and/or the reference epoch of a given location is changed to a different reference frame and/or epoch.

Spatial transformation refers to the component of the coordinate transformation process that deals with a change of reference frame at a common epoch.

Temporal transformation refers to the component of the coordinate transformation process that deals with a change of reference epoch, while remaining in the same reference frame.

Combined transformation is the explicit combination of both the spatial and temporal elements of the coordinate transformation process.

Crustal motion encapsulates how the surface of the earth is in constant motion due to the movement of tectonic plates and, in some locations, local and regional crustal deformation. The latter may be caused by natural processes (e.g., post-seismic deformation) or induced by human activity (e.g., groundwater extraction or underground mining).

Distortion occurs when errors in underlying measurements and limitations in data processing affect the accuracy of station coordinates in a realized reference frame, a problem especially common in older frames.

Accuracy is the closeness to the truth, quantified by the mean or average difference between the true and measured locations.

Precision is the statistical repeatability of the solution, quantified by the standard deviation of the data sample.

Map transformation is a process for shifting the user’s digital map data to spatially align it with the Swift navigation solution.

Navigation transformation refers to a process for shifting the Swift solution to spatially align it with the user’s map data.