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## AGS procedure evaluation

The information on this page augments Graham, Rice and Reid (2005) (available from the publications page).

Contents of this page:

### Radial lens correction

The use of a complete lens-distorition model, usual in photogrammetric applications, is not feasible for compact cameras with a zoom lens. However, it is possible to derive an approximate correction for the radial distortion, which is by far the most significant of the lens distortions for non-metric cameras (Dymond and Trotter 1997; Wolf and Dewitt 2000; Zhang 2000). Radial distortions vary as a function of distance from the image centre and result in barrel (where points are displaced towards the image centre), pincushion (where points are displaced towards the image edge) or a composite distortion. The radial displacement associated with such distortions may be represented by a polynomial of the form:

*displacement = r _{d} - r = k_{1}r + k_{2}r^{3} + k_{3}r^{5} + ... + k_{n}r^{2n-1}*

where *r _{d}* is the distance from the principal point in the uncorrected image,

*r*is the equivalent distance in the corrected image and

*k*to

_{1}*k*are fitted coefficients (Schenk 1999). For practical applications only three terms are required. It is assumed that the principal point is coincident with the image centre. Provided that a consistent area is photographed from approximately the same height, so keeping the focal length approximately constant, this displacment function may be used to derive a suitable correction for a particular camera.

_{n}The procedure for deriving the lens correction coefficients is as follows. The camera is set up in the laboratory to replicate the configuration used in the field (i.e. at the same height with approximately the same focal length). A tape is laid across the field of view, from corner to corner, and photographed. Using the tape in the image, the coordinates of points equally spaced from the image centre are determined and a least-squares fit used to determine the coefficients. These are then used to transform each field-collected image with a bilinear interpolation. The laboratory work and calculation of these coefficients takes about 15 minutes, but only needs to be applied once for a particular camera and field configuration (patch size photographed from a similar height, i.e. approximately the same focal length).

For the camera and setup used in this study, radial distortion results in objects in the corners of uncorrected images appearing approximately 4% smaller than those at the center. Assuming a random distribution of grain sizes throughout each patch, the effect of not correcting for radial distortion should be small. To test this, all of the images were reprocessed without applying the correction. As expected, the effects are small. The irreducible random error is increased slightly (<0.006 ψ, <8.3%) for each site (Table 1). The pattern of percentile biases is complex (Figure 1) and most standard errors are slightly increased (Figure 2). None of the differences are significant. These results are specific to the lens and setup used in this study and may not be applicable to other configurations. Indeed, the camera height was specifically chosen to minimize the effects of radial lens distortion by avoiding the use of a very short focal length. Given that the correction is straightforward and easily obtained, it is recommended that it should be applied.

Note: all comparisons undertaken using area-by-number data.

**Figure 1: Percentile bias for seven selected percentiles at the three rivers, with and without the application of a radial lens correction. See below for the key.**

**Figure 2: Standard error for seven selected percentiles at the three rivers, with and without the application of a radial lens correction. See below for the key.**

**Key to Figures 1 and 2. **

Mean square error (psi) | Bias (psi) | Irreducible random error (psi) | ||||

with lens correction | without lens correction | with lens correction | without lens correction | with lens correction | without lens correction | |

River Lune | 0.0045 | 0.0050 | 0.0069 | 0.0004 | 0.0667 | 0.0708 |

Afon Ystwyth | 0.0059 | 0.0066 | 0.0336 | 0.0314 | 0.0691 | 0.0748 |

Ettrick Water | 0.0080 | 0.0080 | 0.0082 | 0.0057 | 0.0890 | 0.0891 |

**References**

Dymond JR and Trotter CM. 1997. Directional reflectance of vegetation measured by a calibrated digital camera, *Applied Optics ***36**(18): 4314-4319.

Schenk T. 1999. *Digital Photogrammetry. *Vol. 1. 1st ed., 428 pp., TerraScience, Laurelville, Ohio.

Wolf PR and Dewitt BA.2000. *Elements of Photogrammetry, *3rd ed., 608 pp., McGraw-Hill, Boston.

Zhang Z. 200. A flexible new technique for camera calibration, *IEEE Transanctions on Pattern Analysis and Machine Intelligence* **22**(11): 1330-1334.

### Use of a single flatness index

If, during the AGS procedure, grain sizes are converted to sieve-equivalent sizes, a characteristic grain-flatness value is required. The results presented in Graham, Rice and Reid (in press) have been based on the use of flatness values generated for individual sites. The collection of grain shape information for an adequate number of grains at each site will reduce the benefits associated with applying an automated procedure by increasing field time. The effect of applying a single ‘average’ flatness index on the quality of the derived grain-size distribution is therefore of interest.

The use of a single value is found to reduce slightly the percentile bias for most percentiles at the Afon Ystwyth, whilst increasing the standard error (Figures 3 and 4). For Ettrick Water, the percentile bias is increased slightly for most percentiles, as is the standard error (Figures 3 and 4). None of the differences are significant. Note: the flatness index at the River Lune was equal to the average of the three sites so the use of the average flatness index makes no difference to the results obtained for this site..

These results are reassuring because they indicate that the derived grain-size distribution is relatively insensitive to the flatness index, even when there is a visually significant difference in grain shape. This suggests that a single average flatness index is likely to have a small effect on the grain-size distribution for most lithologies. It should be necessary to survey the flatness index only if the sediment is characterized by exceptionally platy or blocky grains. An exception to this may be where there is a marked variation in grain shape with size. This was not the case at the three sites examined here, but may result in size-dependent errors in the grain-size distribution if present.

Note: all comparisons undertaken using area-by-number data.

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**Figure 3: Percentile bias for seven selected percentiles at the three rivers, with and without the application of a radial lens correction. See below for the key.**

**Figure 4: Standard error for seven selected percentiles at the three rivers, with a site-specific and average flatness index. See below for the key.**

**Key to Figures 3 and 4. **

Mean square error (psi) | Bias (psi) | Irreducible random error (psi) | ||||

site specific flatness | average flatness | site specific flatness | average flatness | site specific flatness | average flatness | |

River Lune | 0.0045 | N/A | 0.0069 | N/A | 0.0667 | N/A |

Afon Ystwyth | 0.0059 | 0.0048 | 0.0336 | 0.0226 | 0.0691 | 0.0658 |

Ettrick Water | 0.0080 | 0.0094 | 0.0082 | 0.0178 | 0.0890 | 0.0953 |

### Linear vs spline interpolation

The traditional method of determining grain size percentiles was to: (i) measure the proportion of grains in each of several size classes; (ii) determine the cumulative percent finer than each size-class boundary; (iii) plot these points on probability paper; (iv) draw a smooth line through them by eye or with a French curve; and (v) read off the percentiles from the graph. Today, it is more usual to use a computer to calculate the precentiles by applying a linear interpolation between the class boundaries. The use of a linear interpolation will result in a systematic over- or under-estimation of the true percentiles, the size of the error being related to the number of size classes employed. Assuming that the true cumulative grain-size distribution curve for the sampled population is smooth and that the population is adequately sampled, Figure 5 demonstrates that linear interpolation will result in an overestimation of the percentiles where the cumulative curve is convex and an underestimation where it is concave. The errors may be greatest at extreme percentiles (close to 0 and 100), because the interpolated grain-size distribution curve must always reach 0 and 100 percent at a class boundary.

**Figure 5: Illustration of the error in derived percentiles associated with the use of a linear interpolation between class boundaries. The interpolated distribution (blue line) deviates from the true (smooth) grain-size distribution (red line). **

A more precise estimate of the true percentile values will result from the application of a smooth interpolation function, such as a spline interpolant. Because the AGS procedure provides information about every grain, it is possible to generate a cumulative grain-size distribution curve without the need for any interpolation. The output from the AGS procedure therefore provides an opportunity to evaluate the errors associated with the application of both linear and spline interpolations to realistic data sets, using the AGS derived sizes as 'definative' control data. Figure 6 is an example of such an evaluation, showing how the grain sizes associated with each integer percentile (1, 2, ..., 99) deviate from the 'true' values when calculated using a spline and linear interpolation. The interpolated values are based on 0.5 psi classes. For this particular example, spline interpolation (solid line) is shown to lie closer to the true value than the linear interpolation (dashed line) for most percentiles. The error associated with both interpolation methods is small (< +/-0.05 psi) for most percentiles below 90. The large deviations above the 90th percentile result because the interpolated distribution curve must always reach 100 percent at a class boundary, whereas the real distribution does not.

**Figure 6: Illustration of the typical errors in grain size for integer percentiles associated with the use of linear and spline interpolation on classified size data compared with the grain-by-grain (AGS) derived size distribution. The inset shows a schematic representation of part of the cumulative grain-size distribution curve between two class boundaries. Linear interpolation generally results in overestimation of percentile values where the cumulative curve is convex. This effect is minimised by the use of spline interpolation.**