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Lab 5



How is the high topography of the Himalaya and Tibetan Plateau supported at depth?

 

mountain range cartoon image mountain range cartoon image
mountain range cartoon image mountain range cartoon image

This lab/assessment provides an opportunity to use the knowledge, techniques and forward modelling approach we have covered in the previous lab sessions, reading and during lectures on a real world example. In this case the aim of the lab session is to use the semi-infinite slab approximation model to help interpret measured gravity data extracted from three parallel transects across the Himlaya-Tibet mountain front.

The key question we need to address here is; How is the topography of the Himalaya-Tibet mountain range supported at depth?

The key to tackling the modelling part of this lab/assessment is to recognise that it is essentially an extension/adaption of what we did in Lab 4, but in this case we have real data, not synthetic data.

 

Plotly plot test
Tibet gravity data [test]

 

Reading

Lillie, Whole Earth Geophysics, Chapter 8 Gravity & Isostasy, Read/review these sections:

Isostasy p. 237-243;

Semi-infinite slab model p. 248-250;

Example of mountain problem p. 257-259.

Task 1

Examine and compare the gravity data for the three transects across the Himalya mountain front. For all tasks in the assessment you must use the personalised version of the data that were provided to you. These data were extracted from the WGM2012 data set which a high resolution global dataset of Earth's gravity anomalies (Bouguer, isostatic and surface free-air), computed at global scale in spherical geometry. See the BGI website for further details; http://bgi.omp.obs-mip.fr/data-products/Grids-and-models/wgm2012

The raw gravity data have been processed following the standard procedures you used in previous labs to calculate the Free Air and Bouguer anomalies. For each transect the mean gravity anomalies (and the calculated standard deviation) were extracted from 20km wide (measured at right angles to the transect line) and 5km long (measured along line of transect) windows along the line of the transect. This yields estimates of the gravity anomalies and the standard deviation at 5km intervals along each profile.

Inspect the gravity anomaly profiles for each line by plotting graphs of the anomaly values and the topography (adjust the scale of the topography, or use a secondary y-axis option in Excel for the elevation values, as required so you can plot the curves on one graph) on the y-axis versus centre distance on the x-axis. Compare the shape and magnitude of the anomalies with respect to the topography (i.e. elevation) for each transect.

Compare the observed anomalies for the Himalaya-Tibet mountain range to those analysed in the previous lab session for a mountain in Airy isostatic equilibrium.

You will see on inspection of the data files that the raw gravity data has already been processed, and the Free Air and Bouguer gravity anomaly values are provided along with an estimate of the standard deviation for these estimates.

The first part of any analysis is clearly to inspect and understand the data you have available. So in this case it would be a good idea to draw graphs of the Free Air and Bouguer anomalies along each of the three transects (line 1, line 2 and line 3) as well as the topography along these transects (see topography profiles example below).

That is, plot the gravity anomalies on the y-axis versus central distance along the x-axis (as shown below for the elevations)...I find it easiest to compare/evaluate these data by plotting the gravity and elevation values on the SAME graph...so it is easy to see how each varies relative to the other. Makes seeing patterns much easier.

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mountain range cartoon image

Note 1.1

You should first assess/examine each transect separately by plotting the FAA and the BA profiles along with the topography (three sets of data) on the same x-y graph.

Then you might consider calculating and viewing the average FAA and BA for the three transects. Using the average of the three transects would "smooth" out some of the details, but this may be useful given that our model is very simple.

Note 1.2

The maximum topographic values on the plateau are circa 5000m while the maximum gravity anomaly values are much lower (circa 500 mGal), so it may be useful (essential?) to scale the topographic values so that you can see the variation/pattern of both clearly. Excel has an option to add a secondary y-axis with a different scale to the primamry y-axis, so this is an elegant way to solve this issue.

Task 2

Use an Airy isostatic model and a simplified crustal structure such as illustrated in the sketch below to calculate the thickness of the crustal root (hm) and the total crustal thickness (hM ) under the Tibetan Plateau assuming the surface topography is completely supported by the crustal root, i.e. it is in Airy isostatic equilibrium (ρc = 2.67, ρm =3.1, ρa = 0 all in g.cm-3).

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You will need to estimate an appropriate height for the topography (h1=ha in the diagram here) from the actual data (look at your graphs of topography from Task 1...BUT, remember to use the ACTUAL values, not the scaled values.

You will also need to calculate the thickness of the crustal root, hm (h2=hm in the diagram here) assuming Airy-type local isostatic compensation. So setup and solve the simultaneous equations for equal height and equal pressure to obtain an expression for hm. Follow the same procedure as used in Lab 4 to do this.

Note 2.1

Make sure you include a sketch/diagram to illustrate your model structure (like the one shown here). If you need to review how the parameters of the semi-infinite slab model are defined re-read p. 248-250 of Chapter 8 of Lillie.

Task 3

Use the semi-infinite slab approximation (see pages 250, 257 and 258, Chapter 8, Lillie) to model the Free Air gravity anomaly and the Bouguer gravity anomaly across the Himalaya-Tibet mountain front assuming a simple Airy isostatic model. It will be useful and convenient to set up your model so that the x-distance intervals match those in the observed datasets (i.e. from c. -300 to 400 km). If you copy the column of x-distances from the observed data you could use these directly to set yu model up, or simply replace the x-distances in an existing model spreadsheet (e.g. from Lab 4) with the new values for your observed data.

Test your model predictions of the BA and FAA by comparing the model anomalies with the observed anomalies along each of the three transects, lines 1, 2 and 3. You could do this by testing your model against each transect separately.

Given that our model design/geometry is very simple, you might consider whether an "average model", i.e. a model that attempts to fit the mean of the observations calculated across all three lines is appropriate. That is calculate the mean Free Air anomaly and the mean Bouguer anomaly (based on the three transect values available) and fit a single model to these mean values. Averaging the observations made along the three transects will "smooth" some of the variation arising from spatial differences in density, topography etc.

Ensure that you calculate the Root Mean Square Deviation (RMSD), i.e. the average misfit between your model predictions and the observations, for each model you test and compare this with the mean standard deviation on the observations for the transect.

Also, make sure you plot a set of suitable graphs to illustrate the observed anomaly data, topography and your model results for each transect/line, ensuring that a plot of the residuals is also included to aid the evaluation and assessment of the model.

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This task/section essentially requires the same approach we took in Lab 4, so review your spreadsheet/model/notes for Lab 4 to refresh your memory on how to set-up the semi-infinite slab model if you need to.

Note 3.1

Model details are in Lillie, Chapter 8, p. 248-250.

Task 4

 

Box, George E.P. (1976) Science and Statistics, Journal of the American Statistical Association, Vol. 71, No. 356, pp. 791-799.

Please note that the deadline for submission of the assessment report is 1000 Monday 26th Nov 2018.

Note 4.1

Please read the comment/advice notes concerning the assessment as it contains information about WHAT to include in your report, and how to go about it.

“All models are wrong, but some are useful.”

George Box