# Equivalent Finite Rotations and Cross-overs

Author: Sabin Zahirovic

EarthByte Research Group, School of Geosciences, University of Sydney, Australia

Equivalent Finite Rotations and Cross-overs

Aim

Background

Exercise 1 – Plate circuits

Appendix

Euler Rotations

Stage vs Finite rotations

References

## Aim

This tutorial is designed to teach the user more advanced techniques of plate rotation, including how to: (1) draw and visualise plate motion hierarchies, (2) re-calculate equivalent finite rotations, and (3) to insert and change cross-over rotations.

## Background

Plate reconstructions in GPlates require a set of instructions, stored in the rotation file (*.rot), that allow the software to rotate and move geometries through time. Each motion is defined by a finite rotation; each line in the rotation file containing a Plate ID, an assigned time, Euler rotation and relative Plate ID. Such a framework allows plate motion models to be hierarchically structured, with relative plate motions embedded in an absolute reference frame – where Africa tends to be trunk of the hierarchy. The resulting absolute plate motions are a cumulative motion of all plates higher up in the rotation hierarchy. The complicating factor is a scenario where one tectonic element changes from moving relative to one plate to moving with another. This is particularly the case when continental fragments collide and accrete, and their motion “crosses-over” to move relative to another plate. In such situations, the rotation file must contain two entries for a single time. Both entries are equivalent finite rotations with different Relative Plate IDs. We will cover a number of hypothetical scenarios of Borneo’s (Plate ID 614) motion to cover the most common examples of calculating equivalent finite rotations and cross-overs. You need to download GPlates 1.3 and use the Sample Data that is provided.

### Basics of a rotation file

The entries below (in bold) are an example of a rotation file for Borneo (Plate ID is the first column) at three times (present-day, 10 and 25 Ma). In this case, Borneo is moving relative to Plate 673 (6th column), which is Sumatra in the EarthByte rotations. The ages (2nd column) are in million years, and are followed by the finite Euler rotation (3rd to 5th column). The columns are space or tab-delimited. GPlates treats anything following the exclamation mark (“!”) as a comment, and ignores any lines that have a Plate ID of 999. This is a legacy code from the PLATES rotation scheme where lines beginning with 999 are ignored. This will become useful later, as it will be necessary to “comment out” particular lines in the rotation file. Note that GPlates can deal with Plate IDs exceeding three digits unlike the PLATES format, as long as they are integers.

The extracted portion of the rotation file from the GPlates 1.3 Sample Data indicates that there is no relative motion between Borneo and Sumatra during the 0 to 10 Ma time interval, while there is some motion between 10 and 25 Ma. If consecutive lines in a rotation file have the same Euler rotation (i.e. columns 3 to 5), it means that there has been no relative motion between the plate pairs during that time interval. Remember, in GPlates we deal with finite rotations, and not stage rotations. However, stage rotations can be exported from GPlates.

614 0.0 0.0 0.0 0.0 673 !KLM-NSM Kalimantan(Borneo)-North Sumatra

614 10.0 0.0 0.0 0.0 673 !KLM-NSM

614 25.0 3.03 101.54 -10.0 673 !KLM-NSM small ccw rot according to pmag, Muller et. al. 2008

 Column 1 Column 2 Column 3 Column 4 Column 5 Column 6 Column 7+ Plate ID (Integer) Age (Float/Integer in Ma) Lat Lon Angle of rotation Relative (conjugate) Plate ID (Integer) Comment

Table 1. Summary table showing the basic layout of entries required in a rotation file. In practice, each entry is separated by a space or tab in order to differentiate between each type of entry.

### Included Files

The tutorial dataset includes the following files:

Coastlines file: Seton_etal_ESR2012_Coastlines_2012.1

Rotation file: Seton_etal_ESR2012_2012.1

## Exercise 1 – Plate circuits

In this exercise we are going to quickly review how plate circuits and plate hierarchies work. This has been covered in previous tutorials so may be skipped for intermediate to advanced GPlates users.

At any particular reconstruction time, one can draw a plate motion hierarchy (AKA plate circuit) by following the trail of plate pairs (i.e. Plate ID to Relative Plate ID) to get back to the “trunk” of the hierarchy, which is usually Africa (701) or the Pacific (901). Africa moves relative to the spin-axis (000), and this motion is most commonly referred to as the Absolute Reference Frame. During times when the Pacific cannot be linked to the Indo-Atlantic plate circuit, the Pacific moves relative to the spin axis as defined by motions derived from hotspot tracks. Remember that the plate circuit changes through time, mainly due to necessary cross-overs in the plate hierarchy.

Once a rotation model is loaded in GPlates, and a reconstruction time is set, one can look up the plate circuits by going to: Reconstruction > View Total Reconstruction Poles > Plate Circuits to Anchored Plate

Switch to the ‘Equivalent Rotations rel. Anchored Plate’ tab to see the finite rotation that would be required to move any plate relative to the Anchored Plate (which by default in GPlates is the spin axis, i.e. 000) at that particular reconstruction time (Figure 1). In this case, we are presented with the absolute rotation since it is relative to the spin axis, rather than the relative plate pair. If you wish to isolate motions relative to a specific Plate ID, go to GPlates and click Reconstruction > Specify Anchored Plate. If we select the anchored plate to be 673 (Sumatra), we see that the equivalent rotation for the Borneo plate (614) is “indeterminate” (Figure 2). This means the equivalent rotation is zero and no motion exists between Borneo and Sumatra between 6 and 0 Ma.

Figure 1. Viewing plate circuits on the Total Reconstruction Poles window. This shows the absolute rotation required to move a plate relative to the Anchored Plate (chosen as the spin axis 000 by default) between 6 and 0 Ma.

Figure 2. The Total Reconstruction plates window with a user-chosen Anchored Plate ID (673) on which relative plate movement is based upon. Indeterminate rotation means that no motion exists between that plate (614) and the Anchored Plate (673).

To visualise the plate hierarchy that leads to the Anchored Plate, switch to the ‘Plate Circuits to Anchored Plate’ tab in the Total Reconstruction Poles window (Figure 3). Scroll down to Borneo (614) and expand the line. Since we anchored Sumatra (673), we only see one entry.

Figure 3. The ‘Plate Circuits to Anchored Plate’ tab allows you to visualise the plate hierarchy that leads to the Anchored Plate. Anchoring a plate other than the spin axis (614) will results in only one entry being shown.

If we change the anchored plate back to the default spin axis (‘0’ or ‘000’), we will see the complete plate circuit (Figure 4).

Figure 4. Anchoring the spin axis (‘0’ or ‘000’) allows us to visualise the complete plate circuit in the ‘Plate Circuits to Anchored Plate’ tab.

## Exercise 2 – Re-calculating equivalent rotations and inserting a cross-over

If we follow the Borneo scenario, we know that in the Seton et al. (2012) model that is included in the GPlates 1.3 Sample Data that Borneo (614) moves relative to Sumatra (673) from 0 to 100 Ma. Say that we want to change the rotation model in a way that Borneo moves relative to Sumatra only between 0 and 10 Ma, and then crosses-over to Indochina, we need to calculate the equivalent rotation at 10 Ma between Borneo and Indochina using GPlates.

1. Make sure the rotation file is loaded in GPlates.

2. Reconstruction > Specify Anchored Plate ID > Enter 604 and click OK. Here we are setting a new relative plate. The new relative Plate ID for this example will be 604 (Indochina). If you reconstruct through time, you will notice that Indochina remains fixed.

Note: Plate IDs can be queried interactively in GPlates by clicking the ‘Feature Inspection’ tool and then clicking a geometry feature, such as those contained in the coastline file.

3. Reconstruct to the time of the cross-over, here being 10 Ma.

4. To view the equivalent rotation at 10 Ma between 614 and 604, go to Reconstruction > View Total Reconstruction Poles (Figure 5). Go to the ‘Equivalent Rotations rel. Anchored Plate’ and scroll down to 614. The entry for Borneo (614) displays the Euler rotation that will need to be entered into the rotation file: -16.8497 -76.8497 -0.593467. Copy these numbers to a text file, or write them down somewhere.

Figure 5. The ‘Equivalent Rotations rel. Anchored Plate’ tab in the Total Reconstruction Poles window, showing the equivalent rotation between plates 614 and 604 at 10 Ma.

5. We can now insert this rotation into the rotation file, and create a new model for Borneo’s rotation – such as assuming that Borneo did not move relative to Indochina before 10 Ma. Alternatively, we can insert new published rotations for Borneo, or build them interactively in GPlates. For the sake of this part of the exercise, we assume Borneo has had no motion relative to Indochina before 10 Ma. To create the cross-over, open the rotation file in a text editor. You may want to work on a copy of your file. Keep the rotations at 10 Ma and younger, since we only want to alter the rotation for earlier times. Create a new entry at 10 Ma, remembering that cross-overs have two Euler rotations at that reconstruction time – one with the existing relative motion, and one for the new plate pair. Below is the original rotation file entry for Borneo:

614 0.0 0.0 0.0 0.0000 673 !KLM-NSM Kalimantan(Borneo)-North Sumatra

614 10.0 0.0 0.0 0.0 673 !KLM-NSM

614 25.0 3.030 101.54 -10.0 673 !KLM-NSM small ccw rot according to pmag, Muller et. al. 2008

614 40.0 3.030 101.54 -10.4 673 !KLM-NSM Muller et. al. 2008

614 50.0 3.030 101.54 -15.4 673 !KLM-NSM Muller et. al. 2008

614 70.0 3.030 101.54 -15.4 673 !KLM-NSM Muller et. al. 2008

614 100.0 27.39 141.79 -20.28 673 !KLM-NSM Muller et. al. 2008

614 100.0 26.74 133.02 -21.75 604 !KLM-ICH Kalimantan-Indochina cross-over Muller et. al. 2008

614 200.0 26.74 133.02 -21.75 604 !KLM-ICH Muller et. al. 2008

The new entries will look something like this:

614 0.0 0.0 0.0 0.0000 673 !KLM-NSM Kalimantan(Borneo)-North Sumatra

614 10.0 0.0 0.0 0.0 673 !KLM-NSM

614 10.0 -16.8497 -76.8497 -0.593467 604 !KLM-ICH Kalimantan-Indochina cross-over, Tutorial 2013

614 200.0 -16.8497 -76.8497 -0.593467 604 !KLM-ICH Assuming no motion between KLM and ICH

You will notice that there are now two entries for 10 Ma, and that the second entry (belonging to the older set of rotations from 10 to 200 Ma) contains the Euler rotation calculated in the previous step and that the relative Plate ID is 604 (Indochina). Note that the Euler rotation at 10 and 200 Ma are the same, which implies that there has been no motion Borneo-Indochina motion during this time interval.

6. Say we wanted to incorporate the position of Borneo at 40 Ma from the original file, but with the new plate pair, then we simply reconstruct to 40 Ma and repeat Step 4 using our original rotation file. The equivalent finite rotation relative to Indochina (604) is 2.97531 (lat), 103.633 (lon) with an angle of -23.7881. You can now create an entry in the new rotation file at 40 Ma with these numbers, and copy them to the 200 Ma time to ensure no relative motion occurs between 40 and 200 Ma. Make use of the comment fields to help you remember what you did. Remember, anything following the “!” is treated as a comment. Make sure the columns are space or tab-delimited, including a space or tab between the Relative Plate ID and the comment field symbol (“!”). With the new entries, the rotation file should now contain something like this:

614 0.0 0.0 0.0 0.0000 673 !KLM-NSM Kalimantan(Borneo)-North Sumatra

614 10.0 0.0 0.0 0.0 673 !KLM-NSM

614 10.0 -16.8497 -76.8497 -0.593467 604 !KLM-ICH Kalimantan-Indochina cross-over, Tutorial 2013

614 40.0 2.97531 103.633 -23.7881 604 !KLM-ICH Equivalent rotation from original 614-604 plate pair

614 200.0 2.97531 103.633 -23.7881 604 !KLM-ICH Assuming no motion between KLM and ICH

At this stage, make sure these entries are pasted into your new rotation file. Make sure to overwrite the existing rotations. If you keep the old rotations and insert new ones, GPlates will not know which ones to use and you will end up with velocity artefacts and bizarre motions of your plates. Once you have entered your new rotations, save the rotation file (make sure it has a .rot file extension) and (re-) load it in GPlates (File > Manage Feature Collections).

WARNING 1: You can corrupt your rotation file if you make changes using GPlates (i.e. move a plate and save changes) and simultaneously edit the rotation file in the text editor. Therefore, if you make changes using GPlates, make sure you save the file and then re-open it in your text editor. Some text editors will automatically re-load changed files, but most will use the original stored in memory.

Once you have successfully made the cross-over and changed to the rotation file, you can enter new rotations for Borneo with the 614-604 plate pair, or create a new rotation interactively in GPlates. There will be instances where you need to introduce new rotations at cross-over times, such as 10 Ma in the Borneo example. In most cases like this, you will need to comment out the rotations belonging to the older rotations (i.e. below the 10 Ma lines) because GPlates may not know which of the two plate pairs you want to apply the new rotation. GPlates will choose one of the plate pairs, most likely the entry belonging to the younger rotations.

## Exercise 3 – Change and re-calculate rotation at cross-over time

1. For this scenario, we are modifying the rotations of Borneo at 10 Ma, belonging to the young plate pair (614-604). Open the rotation file in your text editor and comment out all the lines that belong to the 614-604 plate pair. Remember that lines that have a Plate ID of 999 will be ignored and will be commented out by GPlates. Save your changes and re-load the rotation file in GPlates (this is important, see Warning 1). Your entries for Borneo will now look like this:

614 0.0 0.0 0.0 0.0000 673 !KLM-NSM Kalimantan(Borneo)-North Sumatra

614 10.0 0.0 0.0 0.0 673 !KLM-NSM

999 10.0 -16.8497 -76.8497 -0.593467 604 !KLM-ICH Kalimantan-Indochina cross-over, Tutorial 2013

999 40.0 2.97531 103.633 -23.7881 604 !KLM-ICH Equivalent rotation from original 614-604 plate pair

999 200.0 2.97531 103.633 -23.7881 604 !KLM-ICH Assuming no motion between KLM and ICH

2. Modify rotation of Borneo interactively at 10 Ma by selecting Borneo’s geometry and then altering the rotation interactively using the Modify Reconstruction Pole tool (Figure 6). See ‘Tutorial 1.3 Interacting with Features’ for detailed instructions on interactively rotating plates. For this example we have created a hypothetical case that implies Borneo has moved counter-clockwise between 10 and 0 Ma, thus we need to rotate Borneo’s geometry in a clockwise fashion at 10 Ma. Click Apply and OK to change the rotation (Figure 7). Remember that we are modifying the younger rotations belonging to the 614-673 plate pair. Save your changes by going to File > Manage Feature Collections and clicking on the Save button.

Figure 6. To modify the rotations of a plate, select the plate’s geometry and then use the Modify Reconstruction Pole tool to interactively rotate it. In this example, the rotation of Borneo (614) has been interactively modified at 10 Ma.

Figure 7. This summary window will appear after you interactivity modify the rotation of a plate and click Apply. To confirm your changes to the plate rotation, click OK.

3. Now we want to re-calculate the cross-over at 10 Ma. Change the Anchored Plate to be Indochina (604) by following Step 2 to 5 from Scenario 1. The Euler rotation should be -0.348348 (lat),-69.3538 (lon) and with an angle of 22.4763. Overwrite the old cross-over rotation for 614-604 at 10 Ma, and uncomment the rotations for Borneo. Save your changes in the text editor and re-load the rotation file in GPlates. You will notice that Borneo will no longer “jump” at 10 Ma because the new cross-over ensures a smooth transition to the new plate pair. Your entries should now look like this:

614  0.0   0.0        0.0        0.0  673 !KLM-NSM Kalimantan(Borneo)-North Sumatra

614 10.0  -0.73  -69.62   23.04  673 !Calculated interactively by GPlates

614 10.0 -0.348348 -69.3538 22.4763  604 !KLM-ICH new Kalimantan-Indochina cross-over, Tutorial 2013

614 40.0 2.97531 103.633 -23.7881 604 !KLM-ICH Equivalent rotation from original 614-604 plate pair

614 200.0 2.97531 103.633 -23.7881 604 !KLM-ICH Assuming no motion between KLM and ICH

## Exercise 4 – Change cross-over rotation belonging to an older plate pair

There will be instances where you will need to change rotations of older plate pairs (614-604), meaning that the cross-over belonging to the younger plate pair (614-673) would need to be re-calculated. Although the general steps are similar, there is a critical step that is different than in Scenario 1 or 2. In this example, we will make such changes at 10 Ma for Borneo by interactively changing the reconstruction in GPlates.

WARNING 2: Remember that you will no longer preserve the stage rotation of the younger pair (i.e. between 0 and 10 Ma for Borneo), as we are going to recalculate the equivalent finite rotation based on the older plate pair. This is especially important when dealing with seafloor magnetic anomalies where we have very well-defined stage rotations. In such cases it is best to work from present to past to ensure that stage rotations are preserved. A number of Generic Mapping Tools (GMT) utilities allow the calculation of stage rotations from finite rotations, and these utilities should be used to make sure that well-constrained stage rotations are preserved when re-calculating cross-overs.

1. Open your rotation file in your text editor. Delete the 0 to 10 Ma lines belonging to the existing 614-673 (Borneo-Sumatra) plate pair. Insert a new entry at 0 Ma, with zero rotations (remember, that at present-day we should not have a non-zero finite rotation) but with the Relative Plate ID being 604 (Indochina). Save your changes and re-load your rotation file in GPlates. Your rotation file should look like this:

614  0.0   0.0        0.0        0.0  604 !KLM-ICH Kalimantan(Borneo)-Indochina

614 10.0 -0.348348 -69.3538 22.4763  604 !KLM-ICH new Kalimantan-Indochina cross-over, Tutorial 2013

614 40.0 2.97531 103.633 -23.7881 604 !KLM-ICH Equivalent rotation from original 614-604 plate pair

614 200.0 2.97531 103.633 -23.7881 604 !KLM-ICH Assuming no motion between KLM and ICH

2. In GPlates, set the Anchored Plate ID to be 673 (the target for the new cross-over rotation) and reconstruct to 10 Ma.

3. Change the rotation of Borneo interactively following Step 2 in Scenario 3 (Figure 8). Alternatively, you can introduce a new rotation (such as a published rotation) in the text editor – but make sure you save the file in the text editor and re-load it in GPlates. Here we have changed the rotation interactively at 10 Ma. Click Apply and OK (Figure 9), and then save your changes using the Manage Feature Collections dialog.

Figure 8. Again to modify the rotations of a plate, select the plate’s geometry and then use the Modify Reconstruction Pole tool to interactively rotate it. In this example, the rotation of Borneo (614) has been interactively modified at 10 Ma.

Figure 9. Again this summary window will appear after you interactivity modify the rotation of a plate and click Apply. To confirm your changes to the plate rotation, click OK.

4. Look up the equivalent finite reconstruction poles for Borneo (614) relative to Sumatra (673) at 10 Ma by going to Reconstruction > View Total Reconstruction Poles. The equivalent rotation should be -0.727708 (lat), -69.6143 (lon) with an angle of 23.0449.

5. Open your rotation file in a text editor. Change the Relative Plate ID at 0 Ma to be 673, and insert a line at 10 Ma with the results from the previous step. Save your file and re-load it in GPlates. You should have smooth motion of Borneo that incorporates the changes you made to the older rotation sequence. Your rotation entries for Borneo should look something like this:

614  0.0   0.0        0.0        0.0  673 !KLM-NSM Kalimantan(Borneo)-North Sumatra

614 10.0 -0.727708 -69.6143 23.0449 673 !KLM-NSM recalculated rotation from 10 Ma 614-604 plate pair

614 10.0  -0.35  -69.35   22.48  604 !KLM-ICH new Kalimantan-Indochina cross-over, Tutorial 2013

614 40.0   2.98  103.63  -23.79  604 !KLM-ICH Equivalent rotation from original 614-604 plate pair

614 200.0   2.98  103.63  -23.79  604 !KLM-ICH Assuming no motion between KLM and ICH

WARNING 3: If you have multiple cross-overs to re-calculate, start with the youngest cross-over highest up in your plate hierarchy (i.e. closer to the absolute reference frame) and work your way to older rotations and down the hierarchy, away from the absolute reference frame.

## Appendix

### Euler Rotations

The motion of any rigid body on the surface of a sphere can be described by an Euler rotation. Such a ‘rotation’ is defined by the axis of rotation, and an angle. The axis of rotation is defined by a single co-ordinate. The rotation angle then defines the rotation magnitude of this body around this axis. Rotation direction (clockwise or counterclockwise) is determined by the sign preceding the angle of rotation.

In the example below, Plate 501 moves relative to Plate 511 from 0 to 20.2 Ma. All the rotations in the rotation file are finite rotations (see next section). Between 0 and 9.9 Ma, Plate 501 rotates 2.75° clockwise about a pole of rotation that has a co-ordinate of 8.7°S, 76.9°E. That is, the axis of rotation is at 8.7°S, 76.9°E.

501        0.0   0.0   0.0  0.0   511 !IND-CIB India Craton-Central Indian Basin

501        9.9  -8.7  76.9  2.75  511 !IND-CIB AN5 Muller et.al. 1997

501   20.2  -5.2  74.3  5.93  511 !IND-CIB Royer & Chang 1991

### Stage vs Finite rotations

A stage rotation is a rotation from one timestep to another. A finite rotation is the sum of all the stage rotations from present-day (0 Ma) to the time of interest. So for example, a stage rotation can be one from 50 to 300 Ma. To obtain the finite rotation at 300 Ma, then the stage rotation from 50 to 300 Ma needs to be added to the finite rotation from 0 to 50 Ma. The finite rotation for Plate 101 (relative to 714) from 0 to 9.7 Ma is the sum of the finite rotation between 0 and 2.7 Ma and the stage rotation from 2.7 to 9.7 Ma. Thus the finite rotation is always a cumulative sum of all stage rotations of all the rotation time intervals.

e.g.