+
C D+
+
=
Figure 5.6: The end-member abundance layers sum to one. Graphs A to D represent end-member abundance layers. The horizontal axes give location, and the vertical axis represents abundance. The layers sum to one, as shown in the lower graph.
CHAPTER 5. MAPPING
=
+
+
+
Figure 5.7: Validation of Gaussian Distributions on a single observation repeated in 250 locations. The horizontal axes are offsets in metres, and the vertical axis represents proportional contribution at the offset location. Observation was a: 0.71, b: 0.05, c: 0.05, d: 0.19.
5.5. VALIDATING THE ALGORITHM
5.5 Validating the Algorithm
The Gaussian distribution of observations into the classification abundance layers must have the property that the sum of abundances of the layers at any location should equal 100%, as described in Equation 3.3.3. The generated maps should be an accurate representation of the seafloor, even in the presence of inaccuracy in vehicle position.
Confirmation that the software satisfies these requirements is performed below through a number of tests. These were performed by software written in Python by the author.
5.5.1
Procedure
A sample grid 1000 m2 with cell-size 10 m2 was generated, with one abundance
layer for each end-member. The validation was tested with a varying number of end-members ranging from 4 to 16, with similar results for all. Two hundred and fifty random observations were generated, consisting of a mix of end-member abundance fractions. These were located at random positions within the grid. Each observation consisted of a normalised random mix of the end-members, such that the sum of the proportions was unity (Equation 3.3.3). Each observation was added to the grid using a random standard deviation ranging from 20 m to 50 m, representing uncertainty in position. A random confidence value between 0 and 1 was also used, representing confidence in quality of the observation. At the end of this process, each abundance layer consisted of a grid of cells containing a sum and a count. Each abundance layer was normalised by dividing the sum by the count (see Figure 5.5), for each cell. These normalised layers were plotted, along with a summing layer showing the sum of all the normalised layers. The summing layer is observed to be uniform value of 1 (representing a mix of 100%), within the limits of the mathematical accuracy of the software and accounting for randomness of the observations, indicating that the assumptions behind the technique, and operation of the software are correct.
For each observation, the distribution curve is first generated, based on the standard deviation of the uncertainty in position (σ), and the confidence of the measurement. The curve is of size 3σ (99.7% of values lie within±3σ of the mean in a normal distribution), and sub-sampled into a square grid with cell size twice the resolution of the cell size of the main mapping grid (per Nyquist). The sum of the sub-sampled points is calculated (effectively the estimate of area under the curve), and this total is used as a divisor to normalise the sub-sampled array such
CHAPTER 5. MAPPING
that the sum of the sub-sampled points is unity. This sub-sampled curve now represents a single sample as a series of sub-sampled probability points which can be distributed over the abundance layers. A final modification to the curve is to apply the confidence, which is a simple multiplication of the height of all sub- sampled points, attenuating the contribution of an observation to the abundance layers based on the confidence level of the observation.
In the case of image reconstruction, the three abundance layers are the red, green, and blue planes of the image pixels. After all observation have been distributed over the individual layers, they are converted to a colour-image by representing the proportions of red, green, blue pixels as the abundance of each layer at the pixel position. These are combined to form the colour pixel for display in the final map.
5.5.2
Sum-to-one Constraint
This section validates the sum-to-one constraint through the use of randomly generated observations and examination of the reconstructed layers. Figure 5.6 show the results of a test using random observations consisting of four end-member types. The graphs display a 1000 m2 area on the (x, y) plane, with the z axis showing the proportion of each end-member.
The random observations contained four end-members. Each observation had the sum-to-one constraint enforced. The requirement is that the Gaussian distribution of observations onto the map produces variable abundances of end-members for any map-cell, but the sum of abundance of end-members in any cell is 1.
The lowest graph in the figure shows a combined map where the proportions of each end-member at each (x, y) position is summed, representing the total allocation of end-members. The expected result is that a 100% allocation applies at all (x, y) locations, and this is confirmed by the plane atz = 1. The colour variation on the graph has been emphasised to highlight the extremely small variation in combined values. These variations are caused by accuracy constraints in the representation of small values in software.
5.5.3
Proportion Preservation
A test was performed using a single randomised observation mix for all samples (i.e., the same observation was distributed over the entire area). In this situation, the expected result is that each normalised layer should demonstrate the original