Using the Delaunay to Compute Lake Volume, Part 2

In Part 1 of this turtorial, I introduced a technique for computing lake volume using the Constrained Delaunay Triangulation. In that article, I tried to focus on the ideas underlying the technique, and to avoid giving too much attention to Tinfour itself. That being said, the reason that I investigated the technique in the first place is that it provided an excellent opportunity for illustrating some of the less obvious aspects of the Tinfour implementation.

In Part 2 of the tutorial, I will walk through the details of the LakeVolumeExample class and explain the most important design elements of the code. In particular, I will show how the Tinfour implementation supports adding and accessing constraints in the Delaunay Triangulation.

**Note:** In the text below, I will sometimes use the acronymn TIN for Triangulated
Irregular Network. The Delaunay Triangulation is a specific kind of TIN
that has important geometrical properties.

The first step to performing the lake-volume calculation was to acquire data that could be read in and processed by the existing Tinfour code. Fortunately, I found a source of Shapefile data posted by Greg Silsbe (Silsbe, 2015) at the website Lake Victoria Shapefiles.

Tinfour is not a Geographical Information System (GIS). The fact that so many of the examples that show up in the Tinfour documentation are geophysical in nature is mainly a matter of convenience and the availability of data. To access some of that data, Tinfour does partially implement a Shapefile reader. Shapefiles are a GIS industry standard for storing information about geographic features such as lakes, islands, roads, geopolitical borders, etc. A Shapefile product is packaged in a set of closely coupled files that all share the same root name, but use different extensions. Geometry information for the entities in the data set are given in a file with the extension "SHP". Metadata information for each entity in the geometry file is given in a file with the extension "DBF".

In the case of the Silsbe data, the soundings were given in a Shapefile named "finbath". The geometry file indicated the position of the soundings and the metadata file indicated the depth. This information was extracted directly from the Shapefiles using the Tinfour ShapefileReader class. Both position and depth data were given in meters. The lake outline data and island polygons were also read directly from the Shapefiles. Again, the position data was specified in meters using the same coordinate system as the soundings.

The logic for these operations is implemented in the class BathymetryData in the Tinfour example code.

As described in Part 1 of this tutorial, the calculation used the outline of the lake and the island polygons as constraints on the Delaunay Triangulation. These elements were needed so that when the triangular mesh was processed, the application code would have a way of knowing which triangles were water and which were land.

To establish this information, the first thing the LakeVolumeExample application does is to loop through the constraints and populate them with a boolean value indicating which constraints are what. The lake-outline Shapefile gives a single polygon that encloses the water area. Any triangles outside that polygon would be land. With the exception of islands, the area (and triangles) inside that polygon would be water. The constraint classes in the Tinfour package allow an application to add "application data" in the form of a Java Object using the setApplicationData() method as shown below.

List<IConstraint> lakeConstraints = data.getLakeConstraints(); List<IConstraint> islandConstraints = data.getIslandConstraints(); for (IConstraint con : lakeConstraints) { con.setApplicationData(true); } for (IConstraint con : islandConstraints) { con.setApplicationData(false); }

In its present implementation, Tinfour has a limitation. Constraints can only be added to the TIN structure once. And, since we have two separate lists of constraints, we need to combine them into one.

List<IConstraint> allConstraints = new ArrayList<>(); allConstraints.addAll(lakeConstraints); allConstraints.addAll(islandConstraints);

This shortcoming will be addressed in future releases of the Tinfour code. But for now, it is a necessary step.

With the input data in place, the example code initializes the TIN. There are two different incremental-TIN implementations in Tinfour. The IncrementalTin class and the SemiVirtualIncrementalTin class. The standard IncrementalTin class is the faster of the two, but it uses more memory for each edge. The semi-virtual implementation is slower than its counterpart, but it uses about half the memory. So for sample sets exceeding a certain size, the example code uses the semi-virtual version. The cut-off value, 500 thousand, is arbitary.

IIncrementalTin tin; if (soundings.size() < 500000) { tin = new IncrementalTin(1.0); } else { tin = new SemiVirtualIncrementalTin(1.0); } tin.add(soundings, null); tin.addConstraints(allConstraints, true);

The LakeData class is a class for storing the results of the analysis. It also implements a Java Consumer that is designed to process triangles. The Tinfour TriangleCollector class was originally contributed to the Tinfour project by a user, Martin Janda, and it is one of my favorite tools for processing data in Tinfour. The visitSimpleTriangles() method loops through the triangles in the TIN structure, passing them to the Consumer (LakeData) on at a time.

LakeData results = new LakeData(tin); TriangleCollector.visitSimpleTriangles(tin, results);

Now we come to the heart of the calculation. As you can see in the
code below, LakeData implements a Java Consumer that accepts objects
of type SimpleTriangle. The SimpleTriangle consists of three
Tinfour edges. The first thing that LakeData does when it receives
a triangle from Tinfour is to extract the edges and see if they
are enclosed in a constraint that defines "water". If at least one of the three edges
is unambiguously in the interior of the constraint, then the code knows that the
whole triangle lies within the polygon defining the lake outline. If that condition
is met, the triangle is used to make a contribution to the volume calculation.
If not, it is ignored. The word *unambiguous* is important here. If an edge
lies on the boundary of a constraint polygon, it does not provide enough information
for the process to know if if the triangle is inside or outside the constraint.
So the process looks for edges that are inside the polygon and not touching
the border except, perhaps, at an endpoint.

Here's most of the code for LakeData. If you'd like, you can give it a quick scan. I will break it out into parts in the discussion below.

class LakeData implements Consumer<SimpleTriangle> {

boolean []water; final GeometricOperations geoOp; int nTriangles;

KahanSummation volumeSum = new KahanSummation(); KahanSummation areaSum = new KahanSummation();

LakeData(IIncrementalTin tin) { List<IConstraint> constraintsFromTin = tin.getConstraints(); water = new boolean[constraintsFromTin.size()]; for(IConstraint con: constraintsFromTin){ water[con.getConstraintIndex()] = (Boolean)con.getApplicationData(); } Thresholds thresholds = tin.getThresholds(); geoOp = new GeometricOperations(thresholds); }

@Override public void accept(SimpleTriangle t) { IQuadEdge a = t.getEdgeA(); IQuadEdge b = t.getEdgeB(); IQuadEdge c = t.getEdgeC(); if (isWater(a) || isWater(b) || isWater(c)) { Vertex vA = a.getA(); Vertex vB = b.getA(); Vertex vC = c.getA(); double zA = -vA.getZ(); double zB = -vB.getZ(); double zC = -vC.getZ(); double zMean = (zA + zB + zC) / 3; double area = geoOp.area(vA, vB, vC);

nTriangles++; volumeSum.add(zMean*area); areaSum.add(area); } }

Here's a fragment of the LakeData constructor. When it is called, it gets the list of constraints from the TIN structure and loops through them checking their application data to see whether they are water or land. Each constraint has a unique integer index in the range 0 to N-1 where N is the number of constraints. Later on, these indices will be used to access the water array.

boolean []water;

LakeData(IIncrementalTin tin) { List<IConstraint> constraintsFromTin = tin.getConstraints(); water = new boolean[constraintsFromTin.size()]; for(IConstraint con: constraintsFromTin){ water[con.getConstraintIndex()] = (Boolean)con.getApplicationData(); } }

The other constructor elements that are worth review are the two Kahan summations and the GeometricOperations object. The KahanSummation class provides an extended precision tool for adding up the surface area and volume contributions of the triangles. The total volume of the lake is so large that using the summations seemed like a good idea. In practice, they turned out to not be strictly necessary for the small number of soundings in the Silsbe data set. But I kept them in the application in case future data sets required extra precision. The GeometricOperations object is used to compute triangle area. The thresholds obtained from the TIN are used by the class to decide whether the slower extended precision math is needed for calculations or whether standard floating-point operations will suffice.

KahanSummation volumeSum = new KahanSummation(); KahanSummation areaSum = new KahanSummation(); Thresholds thresholds = tin.getThresholds(); GeometricOperations geoOp = new GeometricOperations(thresholds);

Inside the accept method, the code extracts the edges from the triangle and checks to see if they indicate that they are contained within a water polygon. If they are, it uses them to compute the area and volume contribution of the triangle. As mentioned in Part 1 of this turtorial, the volume is just the average of the three water depths times the area of the triangle (area based on its x/y coordinates, ignoring its z coordinates). The accept method was given above and I won't repeat it, except to show the structure of the conditional block

if (isWater(a) || isWater(b) || isWater(c)) { // compute area and volume and add it to the sums }

One of the challenges in the design of Tinfour is that it had to be able to process a very large number of sample points (10 million samples is not uncommon). A Delaunay Triangulation containing N vertices has, on average, 3 times N edges. So the memory footprint of the Tinfour edge implementation had to be as small as possible. Ideally, when an edge is associated with a constraint, it would carry a reference to that constraint. And in cases where an edge served as the border between two constrained regions (such as land and water), it would need to carry two references. Unfortunately, in the current Tinfour architecture, adding such reference to the edge class definition would increase the size of a single edge by 16 bytes for the standard IncrementalTin and 8 bytes for the semi-virtual implementation. However, when the IQuadEdges were created, there was room to add an integer without increasing the in-memory size of the objects. So instead of carrying a reference to the constraints, the Tinfour IQuadEdge classes carry their integer index.

When a polygon constraint is added to the Tin, all the edges that lie in its interior are populated with the index of the constraint. The edges also know whether they are the border of the polygon, or unambiguously in its interior. If an edge lies on the border of the polygon, it doesn't give us sufficient information to know whether the triangle it belongs to is a water triangle or a land triangle. But if it is in the interior of a water polygon, the isWater() method returns true.

A triangle will be a water triangle if at least one of its edges returns true for the isWater() method. The other two edges might be borders, but if at least one is in the interior, then the entire area of the triangle is in the interior.

boolean isWater(IQuadEdge edge) { if (edge.isConstrainedRegionBorder()) { return false; } if (edge.isConstrainedRegionInterior()) { int index = edge.getConstraintIndex(); return water[index]; } return false; }

The isWater() method also comes in handy when estimating the distance between soundings. The Delaunay triangulation structure organizes the input data so that each sample vertex is connected to its nearby neighbors. So the triangulation provides a meaningful way of knowing which pairs of vertices to consider when tabulating the average distance between soundings. We simply look at the lengths of the connecting edges. But, we only want to consider those edges that lie in water areas and connect pairs of soundings. The isWater() method will help us exclude edges that lie outside the constrained areas or on their borders. It turns out that for the Lake Victoria sample data, the sample spacing measured using the Delaunay triangulation and restricting the tally to "water edges" is about two thirds of the value estimated from the raw data using simple arithmetic.

That's about it. Here's the output from the LakeVolumeExample application. Since the Shapefiles for the lake and island areas define polygons, it is possible to compute their area using simple methods. Those calculations are included in the output as a reference. At first I was puzzled by the fact that the Net Area from the Shapefiles does not exactly match the Surface Area computed using the triangulation. It turns out that some of the polygon features in the source data are simple triangles that do not contain soundings. Because triangles don't contain internal edges (all edges are constraint borders), the isWater() method does not detect that they are water features and they are not added to the tally.

Input Data Soundings Count: 12679 Min (x,y,z): 345486.7, -332106.7, -79.50 (feature 2197) Max (x,y,z): 705660.9, 54640.2, -0.50 (feature 5276) width,height: 360174.2, 386746.9 Nominal Spacing: 3561.7

Data from Shapefiles Lake area 6.93840107e+10 69,384,010,673 m2 69384.0 km2 Island area 2.61629720e+09 2,616,297,199 m2 2616.3 km2 Net area (water) 6.67677135e+10 66,767,713,474 m2 66767.7 km2 Lake shoreline 3.84310964e+06 3,843,110 m 3843.1 km Island shoreline 3.23382902e+06 3,233,829 m 3233.8 km Total shoreline 7.07693866e+06 7,076,939 m 7076.9 km N Islands 741

Computations from Constrained Delaunay Triangulation Volume 2.66773715e+12 2,667,737,147,208 m3 2667.7 km3 Surface Area 6.67677170e+10 66,767,717,048 m2 66767.7 km2 Avg depth 39.96 m N Triangles 69717 Est. Sample Spacing 2361.11 m

Time to load data 148.3 ms Time to build TIN 259.2 ms Time to compute lake volume 54.2 ms Time for all analysis 313.4 ms Time for all operations 461.7 ms

Silsbe, G. (2015). *Lake Victoria Shapefiles.* Data downloaded October, 2018 from
https://figshare.com/articles/Lake_Victoria_Shapefiles/1494839/1