pDEM - Paleo-Digital Elevation Model

Welcome to the pDEM - Paleo-Digital Elevation Model Marathon Match! The problem we approach today is the reconstruction of prehistoric Earth terrain (also including ocean floors) from a sparse set of geological observations that indicate the elevation and slope of terrain at certain points in geological time. These observations originate, for example, from drilling a hole in the ground, and analysing the composition of extracted material at different depths. Today creation and updates of such elevation models is a labor-intensive largely manual process performed by highly-skilled researchers, our goal is to come up with algorithms to do the same in an accurate, fast, and fully automated manner.

Prizes

1st: $10 000
2nd:  $7 500
3rd:  $5 000
4th:  $3 800
5th:  $2 500
6th:  $2 000
7th:  $1 500
8th:  $1 000
9th:    $700
10th:   $500
 

Dataset Details

The training dataset is provided in the match forum, accessible after the registration. It is a CSV file with following columns:

• X, and Y are the longitude and latitude coordinates at Earth surface in degrees according to WGS84. The points are distributed evenly on a regular rectangular grid covering the Earth's surface in a WGS84 geographic projection. You will find however, that our data are sparse, and the values in other columns are present only for some points where they are known. Your ultimate goal in this challenge is to output for each X, Y point in the input a single number: the estimated paleo-terrain level at that coordinate, expressed in meters.
• DEM contains the ground truth terrain elevation values (measured in meters) for western hemisphere (X = -180 to 0; Y = -90 to 90). These are the values you aim to reconstruct for the entire globe using information from other columns. During provisional and final scoring this column will be empty in the input. Further below you will find more details on differences between training, provisional, and final datasets.
• Elev_value, Elev_error, Elev_min, Elev_max, and Slope_azimuth are accurate estimations of terrain level you aim to predict, obtained from geologic observations. These are hard restrictions your solution must satisfy at the points where they are given. Specifically:
  • Elev_value and Elev_error are the estimated paleo-terrain level and its estimation error, both in meters. At X, Y coordinates where they are given your output terrain level estimation Z_xy must satisfy |Z_xy - ElevValue_xy| <= 2 ElevError.
  • Elev_min and Elev_max are the estimated lower and maximum bounds on the paleo-terrain level, both in meters. At X, Y coordinates where either of them is given your output terrain level estimation Z_xy must satisfy Z_xy > ElevMin_xy and Z_xy < ElevMax_xy.
  • Slope_azimuth is the estimated direction of paleo-terrain slope, given as the angle between the direction to north, and the direction of positive terrain level gradient, measured in degrees in the clockwise direction. At X, Y coordinates where the slope azimuth is provided, the azimuth calculated from your output terrain level must match with it within 5 degrees.
    
    Fig. 1. Geometry of the problem. X and Y axes are directed to the East and North; the points are sampled at the regular grid with equal steps dx = dy = 0.5°. The Z axis (terrain elevation) is directed upward (to the viewer). At a selected point (X, Y) the slope gradient g is the vector with coordinates  (i.e. the standard definition of gradient for a function of two variables). To estimate derivatives at (x, y) point of the grid we use two points difference: , . The slope azimuth a is defined as the angle between North direction and the slope gradient, measured clockwise.
  • It is a good place to mention that a correctly reconstructed terrane level should be continuous and smooth (i.e. there must be no sudden jumps of height, nor of its first derivatives, between the neighbouring grid points). See further details in the scoring section.
• L_Class specifies the type of Earth crust at each point: 0 = oceanic; 1 = continental; 2 = oceanic island arcs.
  
• CA_Values is the known age of crust in millions of years, which may correlate with topography.
• PB_Class provides line features that delineate different types of plate-tectonic boundaries.
  
• GC_Class specifies rock forming environments, where known: 0 - basaltic volcanism; 1 - intermediate-acidic volcanism; 2 - other.
• Trend Depth and Trend Error are empirically estimated paleo-terrain levels and its error, both in meters. It is known for most of the Earth surface, and in theory it is the same terrain level you aim to reconstruct; the catch is the accuracy of this empirical estimation is very poor, compared to the values originating from geologic observations (see Elev_value above). Thus, all known Trend Depth values are provided as input, and you are allowed to use them as you see fit, but your goal is to reconstruct the terrain level with a higher accuracy matching Elev_value and related values.

As usual in Topcoder Marathon Matches, we rely on different datasets for training (provided to competitors), and for provisional (during the competition) and final scoring at Topcoder server. However in this case we split data in a special way, explained in the following table. As you will see, the X,Y grid, and some data are provided for the entire Earth surface in every dataset, some only for selected geographic regions. Your solution in every case should output the reconstructed terrain level for the entire Earth surface, but it will be scored based on different areas.

Submission Format

You will submit a Dockerized code following this generic template. A sample submission customized for this very match is provided in the match forum, and you are welcome to ask there for further help with the dockerization and submission format.

It should be possible to build and run your submission executing these Docker commands in the code folder of your solution:

docker build -t submission .
docker run --rm -v /path/to/data:/data:ro -v /path/to/workdir:/workdir submission ./test.sh /data/input.csv /workdir/output.csv

The entry point script of your solution, test.sh, will get two arguments: the path to input and output CSV files. The output file must have three columns: X, Y, DEM. The first two columns must match exactly the same columns of the input file; to the DEM column you should output the reconstructed terrain elevation at each point.

Scoring Details

The outputs of your submissions will be scored in three steps. A failure on the first or second step means the zero overall score.

1. Fail or Pass.
   1. A sanity check that your output contains exactly the same points as the input; i.e. the X, and Y columns in your output must match exactly the X, and Y columns from input.
   2. A sanity check that in each row (for each X, and Y) you have output a valid DEM number - the estimated terrain level elevation in meters.
   3. A sanity check that DEM values and its derivatives are continuous over the entire Earth surface. For that end we estimate first and second derivatives of DEM from your output and check that their absolute values at any point are smaller than 12000 m/deg for the first-order derivatives, and 12000 m/deg2 for the second-order derivatives. We also check that derivatives of each kind are bound by a smaller 4000 m/deg absolute value threshold at least at 99% of points (4000 m/deg2, in case of second-order derivatives).
2. Fail or Pass. We check that your solution satisfies the hard conditions on Elev_value, Elev_error, Elev_min, Elev_max, Slope_azimuth (explained in the Dataset Details section above), at all points where these values were provided in the input.
3. Numeric Score. Assuming your solution has passed the checks (1) and (2), we calculate its score based on RMS deviation of your output DEM from the ground-truth DEM within the geographic area used for scoring.
   
   Mathematically, we want to calculate RMS as (in the formula that is a spherical integral over some area of the Earth’s surface):
   
   as in our case DEM values are sampled at a regular rectangular grid in the geographic projection, we approximate the integral in this formula by the following sum over X,Y points in the area of interest, thus (N is the total number of points in our grid):
   
   We map the resulting RMS value to the leaderboard score as: , where C = 500 is just a normalization constant, not affecting the relative ranking of solutions at the leaderboard.

The dockerized version of scorer is provided in the challenge forum. It can be built and executed with the following commands (assuming you keep you solution output, and the ground truth in the same workdir folder on your host machine):

docker build -t scorer .
docker run --rm -v /path/to/workdir:/workdir scorer ./run-scorer.sh /workdir/solution.csv /workdir/ground-truth.csv /workdir

As the ground-truth.csv you should provide the training input CSV file. The score at step #3 will be calculated over the points containing non-empty DEM values in that training input. The result score will be written out into the result.txt file in the workdir.

• This match is rated (TCO-eligible), and individual (no teaming allowed).
• The provisional and final scoring is performed at t2.medium AWS EC2 machine. The solution runtime limit is 1 hour. Solutions not completed within the runtime limit will be failed with zero score.
• Only positive scores (anything passing the hard "fail or pass" checks) on final dataset are eligible for prizes.
• Any 3rd party components of your solution must be available under permissive open source licenses, similar to MIT License. In case of doubts, please ask in the challenge forum, or contact the match copilot directly, to approve the use of libraries licensed under more restrictive terms.
• To claim a prize, in addition to getting a prize-eligible score in the final testing, within a week after the announcement of the final results you must submit a detailed write-up explaining your solution, its implementation, and any other details relevant to its usage and further development.