Errors, Accuracy and Least Squares Adjustment

Surveying accuracy and error types

Accuracy and precision describe different qualities of a set of measurements. Accuracy is the closeness of a measured value, or of the mean of several values, to the true value. Precision is the closeness of repeated readings to one another, judged by their spread. A survey can therefore be precise but inaccurate, for example when the readings agree tightly around a value that carries a constant bias.

The most probable value (MPV) is the best estimate of a quantity obtained from a set of observations by the principle of least squares. It lies nearest to the true value, but it is generally not exactly equal to the true value, because the observations contain random errors. It does not move in a fixed direction whenever the true value changes.

Errors are classified by their cause and behaviour.

• Mistake (blunder) — an error caused by observer carelessness, misunderstanding or poor judgment, such as sighting the wrong target or misreading a scale. It is not a random error. It is avoided by careful work and by checking readings against redundant observations.
• Systematic error — an error that follows a definable pattern and keeps the same sign and size under the same conditions. Its sources are the instrument, the method and natural effects such as refraction or temperature. Systematic errors can be corrected by applying a suitable mathematical model. They must be removed before or during the adjustment, because least squares does not remove them automatically. Removing gross errors does not remove systematic errors either.
• Random (accidental) error — an error that varies unpredictably in sign and size and follows the laws of probability. Its distribution is symmetric about zero, so positive and negative errors of equal size are equally likely.
• Gross error — a large outlier that is detected and rejected separately from the random errors.

Uncertainty is the interval within which the true value of a measured quantity is expected to lie. It is estimated from the random errors and the systematic errors that remain after correction. Gross errors are excluded by repeating or rejecting the faulty observations.

The governing principle of surveying is to work from whole to part. The overall control framework is established first, and the detail is fixed relative to it, which keeps the accumulation of errors under control.

A spheroid (ellipsoid of revolution) is a mathematical surface that approximates the Earth and serves as the basis of geodetic computation. The geoid is the equipotential surface of the gravity field that closely matches mean sea level, and it is the physical reference for heights. The true figure of the Earth is irregular and is represented by the geoid, so it is neither a sphere nor a perfect spheroid. WGS-84 is a single global geodetic datum adopted worldwide, not a datum that changes from country to country.

The WGS-84 reference ellipsoid is an oblate ellipsoid of revolution, flattened at the poles. GPS positioning directly gives the ellipsoidal height h. The orthometric height H, measured above the geoid, is obtained as H = h - N, where N is the geoid undulation. The geoid is not fixed in time, because mass redistribution, sea level change and crustal movement slowly change its shape.

Relative precision is the ratio of the error to the measured quantity, written as 1:n, where n equals the measured length divided by the error. For example, an error of 0.5 m in a 500 m length gives a relative precision of 1:1000.

For a Gaussian error distribution with standard error \sigma, the probable error is 0.6745\sigma, the average error is \sqrt{2/\pi}\,\sigma \approx 0.7979\sigma, and the 90% error is 1.645\sigma. The order is therefore probable error < average error < standard error < 90% error. To convert a standard error into a probable error, multiply by 0.6745. The result is the error bound that is exceeded with probability 0.5.

The standard error of the mean of n independent observations of equal precision is \sigma/\sqrt{n}. Reducing the standard error by a factor k therefore requires k^2 times as many observations. For a sample, the standard error of the mean is s/\sqrt{n}, where s = \sqrt{\sum v^2/(n-1)} is the sample standard deviation of the readings.

The variance of a sum of independent quantities equals the sum of their variances. A length made of n equal independent sections therefore has standard error \sigma\sqrt{n}, not n\sigma.

For a derived quantity such as horizontal distance computed from slope distance and angle, the variance is obtained by first-order propagation. For H = S\cos\theta, \sigma_H^2 = \cos^2\theta\,\sigma_S^2 + (S\sin\theta)^2\sigma_\theta^2, with \sigma_\theta in radians. Balanced accuracy assigns equal shares of the allowable variance to the distance term and the angle term. Averaging n independent angle readings divides the angle variance by n, so the minimum n is found by solving the inequality and rounding up to the next integer.

Gray-level transformations are point operations that map an input pixel value r to an output value s. The negative transformation s = (L-1) - r is a linear mapping with slope -1. A thresholding function s = T(r) gives only two output values and so produces a two-level binary image. The logarithmic transformation s = c\log(1+r) has its steepest slope at dark levels and expands the dark input values. A power law s = c\,r^{\gamma} with 0 < \gamma < 1 also maps dark inputs into a wider output range.

Least squares weighted adjustment

Least squares adjustment estimates the most probable values of the unknowns by minimising the weighted sum of squared residuals, \mathbf{V}^T\mathbf{P}\mathbf{V}. The weight matrix is \mathbf{P} = \sigma_0^2\boldsymbol{\Sigma}^{-1}, so each weight is inversely proportional to the variance of its observation, and a more precise observation controls the solution more strongly.

Under a correct functional model, the least squares estimators are unbiased, so the expected value of the residuals is zero. This is one of the defining properties of the adjustment.

For observation equations \mathbf{A}\mathbf{x} = \mathbf{L}, the normal equations are \mathbf{A}^T\mathbf{P}\mathbf{A}\mathbf{x} = \mathbf{A}^T\mathbf{P}\mathbf{L}. With equal weights they reduce to (\mathbf{A}^T\mathbf{A})\mathbf{x} = \mathbf{A}^T\mathbf{L}. For a small system the 2\times2 inverse is obtained from the determinant, and the residuals must then satisfy \mathbf{A}^T\mathbf{P}\mathbf{V} = \mathbf{0}.