Introducing HTDP 3.1 to transform coordinates across time and spatial reference frames

Abstract

The National Geodetic Survey, an office within the National Oceanic and Atmospheric Administration, recently released version 3.1 of the Horizontal Time-Dependent Positioning (HTDP) utility for transforming coordinates across time and between spatial reference frames. HTDP 3.1 introduces improved crustal velocity models for both the contiguous United States and Alaska. The new HTDP version also introduces a model for estimating displacements associated with the magnitude 7.2 El Mayor–Cucapah earthquake of April 4, 2010. In addition, HTDP 3.1 enables its users to transform coordinates between the newly adopted International Terrestrial Reference Frame of 2008 (ITRF2008) and IGS08 reference frames and other popular reference frames, including current realizations of NAD 83 and WGS84. A more convenient format to enter a list of coordinates to be transformed has been added. Users can now also enter dates in the decimal year format as well as the month-day-year format. The new HTDP utility, explanatory material and instructions are available at http://www.ngs.noaa.gov/TOOLS/Htdp/Htdp.shtml.

Appendices

Appendix: Transforming coordinates between ITRF2008 and NAD 83

This appendix presents equations for transforming positional coordinates between ITRF2008 and NAD 83 (CORS96). We also present equations for transforming positional coordinates between ITRF2008 and NAD 83 (PACP00), as well as equations for transforming positional coordinates between ITRF2008 and NAD 83 (MARP00). These equations were incorporated into HTDP 3.1. In the following text, the names—NAD 83 (CORS96), NAD 83 (PACP00), and NAD 83 (MARP00)—will be truncated to NAD 83 when referring to these three NAD 83 realizations collectively.

Let x(t)_{NAD 83}, y(t)_{NAD 83} and z(t)_{NAD 83} denote the NAD 83 positional coordinates for a point at time t as expressed in a 3-D Cartesian earth-centered, earth-fixed coordinate system. These coordinates are expressed as a function of time to reflect the reality of the crustal motion associated with plate tectonics, land subsidence, volcanic activity, postglacial rebound and so on. Similarly, let x(t)_ITRF, y(t)_ITRF and z(t)_ITRF denote the ITRF2008 positional coordinates for this same point at time t. The given ITRF2008 coordinates are related to their corresponding NAD 83 coordinates by a similarity transformation that is approximated by the following equations

$$ \begin{aligned} x( t )_{\text{NAD 83}} & = T_{x} ( t ) + [ {1 + s( t )} ] \cdot x( t )_{\text{ITRF}}\,+ \omega_{z} (t) \cdot y ( t )_{\text{ITRF}} -\omega_{y} ( t ) \cdot z( t )_{\text{ITRF}} \\ y( t )_{\text{NAD 83}} & = T_{y} ( t )-\omega_{z} ( t ) \cdot x( t )_{\text{ITRF}} + [ {1 + s( t )} ] \cdot y( t )_{\text{ITRF}} + \omega_{x} ( t ) \cdot z( t )_{\text{ITRF}} \\ z( t )_{\text{NAD 83}} & = T_{z} ( t ) + \omega_{y} ( t ) \cdot x( t )_{\text{ITRF}} -\omega_{x} ( t ) \cdot y( t )_{\text{ITRF}} + [ {1 + s( t )} ] \cdot z( t )_{\text{ITRF}} \\ \end{aligned} $$

(1)

Here, the symbols T _x (t), T _y (t), and T _z (t) are translations along the x-, y-, and z-axes, respectively; ω _x (t), ω _y (t), and ω _z (t) are counterclockwise rotations about these same three axes; and s(t) is the differential scale change between ITRF2008 and NAD 83. These approximate equations suffice because the three rotations have small magnitudes. Note that each of the seven quantities is represented as a function of time because space-based geodetic techniques have enabled scientists to detect their time-related variations with some degree of accuracy. These time-related variations are assumed to be mostly linear, so that the quantities may be expressed by the following equations,

$$ \begin{aligned} T_{x} \left( t \right) & = T_{x} \left( {t_{0} } \right) + \dot{T}_{x} \cdot \left( {t-t_{0} } \right) \hfill \\ T_{y} \left( t \right) & = T_{y} \left( {t_{0} } \right) + \dot{T}_{y} \cdot \left( {t-t_{0} } \right) \hfill \\ T_{z} \left( t \right) & = T_{z} \left( {t_{0} } \right) + \dot{T}_{z} \cdot \left( {t-t_{0} } \right) \hfill \\ \omega_{x} \left( t \right) & = [\varepsilon_{x} \left( {t_{0} } \right) + \dot{\varepsilon }_{x} \cdot \left( {t-t_{0} } \right)] \cdot m_{r} \hfill \\ \omega_{y} \left( t \right) & = [\varepsilon_{y} \left( {t_{0} } \right) + \dot{\varepsilon }_{y} \cdot \left( {t-t_{0} } \right)] \cdot m_{r} \hfill \\ \omega_{z} \left( t \right) & = [\varepsilon_{z} \left( {t_{0} } \right) + \dot{\varepsilon }_{z} \cdot \left( {t-t_{0} } \right)] \cdot m_{r} \hfill \\ s\left( t \right) & = s\left( {t_{0} } \right) + \dot{s} \cdot \left( {t-t_{0} } \right) \hfill \\ \end{aligned} $$

(2)

where m _r  = 4.84813681 · 10⁻⁹ is the conversion factor from milliarc seconds (mas) to radians.

Here, the symbol t ₀ denotes a fixed, prespecified time of reference. Hence the seven quantities T _x (t ₀), T _y (t ₀),…, s(t ₀) are all constants. The seven other quantities $\dot{T}_{x}$, $\dot{T}_{y}$,…, $\dot{s}$, which represents rates of change with respect to time, are also assumed to be constants.

The transformation from ITRF2008 to NAD 83 (CORS96), denoted (ITRF2008→NAD 83 (CORS96)), is defined in terms of the composition of five distinct transformations, applied sequentially. First, positional coordinates are transformed from ITRF2008 to ITRF2005, then from ITRF2005 to ITRF2000, then from ITRF2000 to ITRF97, then from ITRF97 to ITRF96, and finally from ITRF96 to NAD 83 (CORS96). This composition may be symbolically expressed via the following equation:

$$ \begin{aligned} \left( {{\text{ITRF2}}00 8\to {\text{NAD 83 }}\left( {\text{CORS96}} \right)} \right) & = \left( {{\text{ITRF2}}00 8\to {\text{ITRF2}}00 5} \right) \\ & + \left( {{\text{ITRF2}}00 5\to {\text{ITRF2}}000} \right) + \left( {{\text{ITRF2}}000 \to {\text{ITRF97}}} \right) \\ & \quad + \left( {{\text{ITRF97}} \to {\text{ITRF96}}} \right) + \left( {{\text{ITRF}}96 \to {\text{NAD 83 }}\left( {\text{CORS96}} \right)} \right) \\ \end{aligned} $$

(3)

where (ITRF2008→ITRF2005) denotes the transformation from ITRF2008 to ITRF2005, (ITRF2005→ITRF2000) denotes the transformation from ITRF2005 to ITRF2000, and so forth.

For (ITRF2008→ITRF2005), HTDP uses the parameter values adopted by the International Earth Rotation and Reference System Service (IERS) for t ₀ = 2005.00 (~1 January 2005) (Altamimi et al. 2011). We have converted the IERS-adopted values for t ₀ = 2005.00 to their corresponding values for t ₀ = 1997.00. Table 7 displays both sets of values.

Table 7 Transformation from ITRF2008 to NAD 83 (CORS96)

For (ITRF2005→ITRF2000), HTDP uses the parameters adopted by the IERS for t ₀ = 2000.00 (Altamimi et al. 2007). The corresponding values for t ₀ = 1997.00 are given by Pearson et al. (2010).

For (ITRF2000→ITRF97), (ITRF97→ITRF96), and (ITRF96→NAD 83 (CORS96)), HTDP uses the parameter values adopted by NGS for t ₀ = 1997.00 (Soler and Snay 2004). Table 7 summarizes the values adopted for all transformations at t ₀ = 1997.00.

Because the values for the parameters associated with each of the five transformations, appearing on the right side of equation (A3), are rather small in magnitude, the values for the parameters of (ITRF2008→NAD 83 (CORS96)) at t ₀ = 1997.00 may be computed with sufficient accuracy by adding the corresponding values for these five transformations at t ₀ = 1997.00. The right-most column of Table 7 displays the resulting values used by HTDP. It should be noted that many of the values, given in Table 7, are expressed to more significant digits than the accuracy to which they can be resolved using current geodetic observations. Nevertheless, HTDP uses these values. The inverse transformation (NAD 83 (CORS96)→ITRF2008) at t ₀ = 1997.00 is obtained by changing the sign for each of the 14 values appearing in the right-most column of Table 7.

Transforming coordinates between ITRF2008 and NAD 83 (PACP00)

Snay (2003) introduced the NAD 83 (PACP00) realization of a Pacific plate-fixed version of the NAD 83 reference frame so that points located in the stable interior of the Pacific tectonic plate would experience little or no horizontal motion relative to this frame. NAD 83 (PACP00) is defined in terms of a transformation from ITRF2000 of the form of Equation A2. Adopted values for the parameters of these equations are listed in Table 8 for t ₀ = 1993.62. Table 8 also presents equivalent values for t ₀ = 1997.00.

Table 8 Transformation from ITRF2000 to NAD 83 (PACP00)

The transformation from ITRF2008 to NAD 83 (PACP00), denoted (ITRF2008→NAD 83 (PACP00)), is defined in terms of the composition of three distinct transformations, applied sequentially. First, positional coordinates are transformed from ITRF2008 to ITRF2005, then from ITRF2005 to ITRF2000, and then from ITRF2000 to NAD 83 (PACP00). This composition may be symbolically expressed via the equation

$$ \begin{aligned} \left( {{\text{ITRF2}}00 8\to {\text{NAD 83}}\left( {{\text{PACP}}00} \right)} \right) & = \left( {{\text{ITRF2}}00 8\to {\text{ITRF2}}00 5} \right) \, \\ & \quad + \left( {{\text{ITRF2}}00 5\to {\text{ITRF2}}000} \right) + \left( {{\text{ITRF2}}00 0\to {\text{NAD 83}}\left( {{\text{PACP}}00} \right)} \right) \\ \end{aligned} $$

(4)

Table 9 displays values for the various parameters at t ₀ = 1997.00, where values for the transformation from ITRF2005 to ITRF2000 for t ₀ = 1997.00 were computed by Pearson et al. (2010) from those published by Altamimi et al. (2007) for t ₀ = 2000.00. The right-most column in this table is obtained by adding the previous three columns across each row.

Table 9 Transformation from ITRF2008 to NAD 83 (PACP00)

Transforming coordinates between ITRF2008 and NAD 83 (MARP00)

Snay (2003) introduced the NAD 83 (MARP00) realization of a Mariana plate-fixed version of the NAD 83 reference frame so that points located on the Marianna tectonic plate would experience little or no horizontal motion relative to this frame. NAD 83 (MARP00) is defined in terms of a transformation from ITRF2000 of the form of Equation A2. Adopted values for the parameters of these equations are listed in Table 10 for t ₀ = 1993.62. Table 10 also presents equivalent values for t ₀ = 1997.00.

Table 10 Transformation from ITRF2000 to NAD 83 (MARP00)

The transformation from ITRF2008 to NAD 83 (MARP00)) is defined in terms of the composition of three distinct transformations, applied sequentially. First, positional coordinates are transformed from ITRF2008 to ITRF2005, then from ITRF2005 to ITRF2000, and then from ITRF2000 to NAD 83 (MARP00). This composition may be symbolically expressed via the equation

$$ \begin{aligned} \left( {{\text{ITRF2}}00 8\to {\text{NAD 83}}\left( {{\text{MARP}}00} \right)} \right) & = \left( {{\text{ITRF2}}00 8\to {\text{ITRF2}}00 5} \right) \, \\ & \quad + \left( {{\text{ITRF2}}00 5\to {\text{ITRF2}}000} \right) + \left( {{\text{ITRF2}}00 0\to {\text{NAD 83}}\left( {{\text{MARP}}00} \right)} \right) \\ \end{aligned} $$

(5)

Table 11 displays values for the various parameters at t ₀ = 1997.00. The right-most column in this table is obtained by adding the previous three columns across each row.

Table 11 Transformation from ITRF2008 to NAD 83 (MARP00)

Related transformations

The International GNSS Service(IGS) recently adopted a reference frame called IGS08 that is based on ITRF2008. IGS08 coordinates for some reference stations differ from their corresponding ITRF2008 coordinates because the IGS08 coordinates were computed using more current calibrations of GPS antennas than were used in computing ITRF2008 coordinates. Nevertheless, IGS considers the 14-parameter transformation between ITRF2008 and IGS08 to be the identity function, that is, each of the 14 transformation parameters is zero in value. As a result, the value of the 14 parameters in the transformation between IGS08 and a given NAD 83 realization is the same as the values of the 14 parameters in the corresponding transformation between ITRF2008 and this NAD 83 realization.

Also, NGS recently adopted three new reference frames called NAD 83 (2011), NAD 83 (PA11), and NAD 83 (MA11). To create these reference frames, NGS reprocessed its collection of GPS data from the CORS network to compute IGS08 coordinates for reference stations in this network. The corresponding NAD 83 (2011) coordinates were then obtained by applying the (ITRF2008 → NAD 83 (CORS96)) transformation given in this paper, which as mentioned above is the same as the (IGS08 → NAD 83 (CORS96)) transformation. As a result, the 14-parameter transformation between NAD 83 (CORS96) and NAD 83 (2011) is the identity function. In most cases, NAD 83 (CORS96) coordinates for a given reference station will differ from the corresponding NAD 83 (2011) coordinates for this same reference frame because the NAD 83 (2011) coordinates were computed using more GPS data, more recent models for systematic errors (like ocean loading), and improved mathematical algorithms than were involved in the NAD 83 (CORS96) computations many of which were performed in 2002. Thus, the fact that the 14-parameter transformation between two reference frames is the identity function does not mean that corresponding coordinates in the two frames agree exactly in value. It only means that these corresponding coordinates agree on average when considering all the reference stations in the network, which have coordinates in both reference frames.

In the same manner, the transformation between ITRF2008 (or IGS08) and NAD 83 (PA11) is the same as the transformation between ITRF2008 and NAD 83 (PACP00), and the transformation between ITRF2008 and NAD 83 (MA11) is the same as the transformation between ITRF2008 and NAD 83 (MARP00). Thus, the transformation between NAD 83 (PA11) and NAD 83 (PACP00) is the identity function, and the transformation between NAD 83 (MA11) and NAD 83 (MARP00) is also the identity function.