Volumetrics and Photogrammetry: Sources of Error
This brief is to, at a high level, consider a few sources of error when calculating volumetrics from 3D models derived from photogrammetry. Photogrammetry is based on images and therefore can only derive data from what is ‘seen’ by the camera. Consequently, if images produced by field personnel are of low quality or low overlap, then any point cloud data (thus 3D models) derived from these images during post processing will potentially reflect additional error.
While there are quite a few potential sources of error, this brief will be of limited focus to those affecting accuracy of point cloud generation.
This high level overview of the topic is provided as guidance mainly for our pilots, mission planners and photogrammetrist, but should be considered by all team members.
Prior to continue reading this brief it may prove beneficial to review several items listed below.
A subject area containing multiple stockpiles is used for our discussion.
A DJI Phantom 3 Pro was used to capture the images. The mission was flown at approximately 100ft AGL elevation, about 70% LAT / 60% LON overlap, perpendicular legs and the camera position was NADIR. Each stockpile of interest was separately flown using a POI mission with approximately 30-meter radius and an oblique camera position (~40 degree). Time of day for the flight was about 1200 hours (GMT -5), which provided high solar irradiance of our surface target.
The additional step of the POI oblique is important as the sides of the stockpile can be prone to error when resolving in Pix4D using nadir only images. Shadows cast onto a surface by the stockpile itself and from other objects are also a potential source of introduced error.
Initial processing for our data set was conducted using Pix4D. Settings that deviate from the standard 3D Map template are: Image Scale was set to 1 Original Size, no default classification of data points (as this will be handled separately as needed). The point cloud used in this discussion is georeferenced WGS 84 and measurement references are in meters.
Pictured below (Image 1) is the Ortho derived from the mission. This image is provided to orient our readers to the target area and provide visual reference for our discussion. In the image below there are visible several stockpile targets. The areas of interest are marked.
Area A: Main Stockpile, conical shape.
Area B: Aggregate Stockpile (gravel), irregular form.
Area C: Dirt Stockpile, irregular form.
Image 1 - Ortho of Stockpile Areas
Analyzing the quality of the point cloud is the first step in assessing any potential error introduced from our image collection. With an exposed surface such a stockpile, we will expect highly coherent point cloud. Low coherence of the point cloud has the potential to create elevated deltas when attempting to calculate actual volumes.
Using the resulting point cloud generated from the captured images, a quick analysis of the cohesion along the z axis of available points is shown below in Image 2. Blue areas represent the areas of highest cohesion (<0.005m deviation). Red areas represent areas that are the least coherent (> 0.09m deviation).
Note that these are very small differentials overall and reflect a well derived model.
Target Area A (circled in Red on right) shows decreasing cohesion near the stockpile apex. However, target Area B (also circled in Red on left) shows uniformly high cohesion. Area C is a mostly well derived point cloud, but has several areas that are weak (likely shadows).
Image 2 – Point Cloud Deviation in Z axis.
Area A Cross Section Analysis
To see what the above image is telling us, let's zoom in on a cross section of Area A. In Image 3 a mensuration line of width 0.05m is visible. The points contained within this very narrow width is sliced from our model for analysis. This line is also overlaid onto the Deviation Z (Image 4) at scale to make it easier for the reader to reference how it intersects the various areas of our calculated Deviation Z axis data from Image 2 above.
Images 3 & 4 – Cross section line on Area A
Viewing the cross section from Area A on a grid (Image 5) we can see that there exist areas of low cohesion (blue arrow) and yet other areas of high cohesion (red arrow).
Image 5 – Point Cloud cross section of Area A from sample width 0.05m
Min Z, Max Z, Mean Z
Before a volume is calculated, a decision must be made regarding the method of smoothing the surface. This decision as to which method to use to create a surface is a source of potential error when computing volume. In this discussion we will analyze all three methods and calculate the impact each may have upon our volume calculations.
Closer inspection of the area denoted by the red arrow from Image 5 (example of high cohesion), the three methods show strong agreement on the final calculated surface (Image 6). This is the ideal result as there will be minimal difference in calculated volume realized from the different methods utilized. Min Z is the Blue line, Max Z is the Red line, and Mean Z is the Black line. Notice too that the Mean Z (Black line) generally lies between the Min Z and Max Z lines.
Image 6 – Min, Max, Mean Z DEM surfaces at Low ΔZ
Using the area denoted by the blue arrow from Image 5 (example of lower cohesion) the three methods are again displayed in Image 7. This region shows a larger disparity in the resulting calculated surface. It is these regions of low point cloud cohesion where we will see higher deferential in volume calculations introduced between the three methods. Min Z is the Blue line, Max Z is the Red line, and Mean Z is the Black line.
Image 7 – Min, Max, Mean Z DEM surfaces at High ΔZ
It is beyond the scope of this discussion to explain in detail how the various DEM methods (Min, Max, Mean) derive their respective surface. But briefly, the entire point cloud surface is draped with a grid of fixed size (in this example the grid size was 0.15m) and based on the population of points of each individual grid, a Z value is produced that best represents that region based on the method (Min/Max/Mean) chosen. Thus, the algorithm for the chosen method may position the line higher or lower than actual points at a given location.
It is also clear by comparing the Deviation Z ( Image 2) to our Ortho (Image 1) that shadows can have an impact on the derived point cloud (see previous posts on shadows and point cloud solutions), with areas in shadow tending toward lower cohesion in the Z axis. Also, steep slopes can have an impact in point cloud cohesion quality.
Let’s now measure the volume of Areas A and B using the point cloud model derived from our images captured. In the table below the volumes are calculated using the three methods of Min Z, Max Z and Mean Z. An important consideration when calculating volume is the relationship to a reference plane. The calculated volumes shown are relative to a base plane from the identified flat area around the base of each stockpile. Positioning of this base plane can be a source of error. Other factors that should be considered are shape of the stockpile base, elevation delta of the base and whether a signed or unsigned differential is required. These are all valid considerations, but are beyond the scope of this discussion, as the concern here is the potential effects of point cloud quality.
Using a defined base plane that is averaged from the flat surface points around each stockpile, the calculation of volume from points above the plane are computed for each stockpile using the three methods.
Table 1 – Calculated Volumes in Cubic Meters
In both example stockpiles, the calculated values using the Mean Z method trend closer in value to those derived from the Max Z method. Mean Z does not equal median or a midpoint value.
Reviewing the graphic in Image 2 it is evident that we have a point cloud solution that largely has very high cohesion. In the stockpile areas of interest, the cohesion separation is only greater than 0.05m (weaker Δ) in relatively small sections. Area A seemingly shows the largest variance in ΔZ, but this loss of cohesion is mostly at the top of the cone of the stockpile. When considered with the overall low ΔZ, this area represents a small impact on actual volume differential. Point cloud saturation within Area A is approximately 800 ppm2, so the surface point density is considered well saturated.
The point cloud in Area B is likewise well saturated with approximately 800 ppm2. However, this stockpile profile is flatter and therefore the area under the higher ΔZ, while smaller than in Area A, is spread across a larger x/y area and this creates a greater total volume variance between Min Z and Max Z.
While the stockpile in Area A appears to have lower cohesion over a larger area, the reality is that this has a lower impact on the volume variance, 1.5% opposed to 1.9%.
Takeaway / Conclusion
Stockpiles can tricky. Stockpile material, stockpile shape, surrounding ground condition and surface elevation, and lighting conditions are several of the variables that can affect / induce error from photogrammetry. Image quality is key, so camera settings such as appropriate white balance, shutter speed, ISO, aspect ratio and output format are critical. Image overlap and position is also very key to success.
Pilots and mission planners will need to observe their target area / surface and make field accommodations with flight planning and execution to best handle mission goals. Field checks of the image quality is important. Also, field decisions to adjust flight plans and, where necessary, add elements to each mission is encouraged. Do not be afraid to re-fly certain target areas as needed.
Post processing should be QC’d using statistical analysis to validate the point cloud. Where necessary, reprocessing the point cloud using higher resolution outputs may be necessary to achieve more optimal and accurate results.
When calculating stockpile volumes, it is imperative that quality checks be implemented in each step. Some areas and surfaces may require LIDAR as the best solution to meet the customer’s needs. Always set customer expectations so that they have an accurate and thorough understanding of what the capabilities are and any limitations that may exist.
Fly Safe and have Fun!