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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Modern observation technology has verified that measurement errors can be proportional to the true values of measurements such as GPS, VLBI baselines and LiDAR. Observational models of this type are called multiplicative error models. This paper is to extend the work of Xu and Shimada published in 2000 on multiplicative error models to analytical error analysis of quantities of practical interest and estimates of the variance of unit weight. We analytically derive the variance-covariance matrices of the three least squares (LS) adjustments, the adjusted measurements and the corrections of measurements in multiplicative error models. For quality evaluation, we construct five estimators for the variance of unit weight in association of the three LS adjustment methods. Although LiDAR measurements are contaminated with multiplicative random errors, LiDAR-based digital elevation models (DEM) have been constructed as if they were of additive random errors. We will simulate a model landslide, which is assumed to be surveyed with LiDAR, and investigate the effect of LiDAR-type multiplicative error measurements on DEM construction and its effect on the estimate of landslide mass volume from the constructed DEM.

Theory and methods of adjustment have been developed, both with the advance of measurement technology and with our deepened understanding of measurement errors. Although manufacturers of surveying instruments would always provide accuracy specifications [

Almost all theory and methods of adjustment have been developed on the basis of the following mathematical or functional model:
_{L}_{a}_{m}^{2}^{2}, respectively. _{m}_{a}_{ij}

The observational model

A digital elevation model (DEM) is a numerical or digital representation of the topography of the Earth. A number of mathematical interpolators have been proposed for the construction of a DEM on the (implicit) assumption that the measurements are contaminated with additive random errors (see e.g., [

Although the errors of GPS, EDM and VLBI baselines and measurements of InSAR and LiDAR have been shown to be of multiplicative nature, almost nothing can be found in the geodetic and DEM literature, except for Xu

As in the case of adjustment with additive random errors, given a set of measurements with multiplicative errors, we have the corresponding adjustment model as follows:
_{i}_{i}_{i}_{i}_{i}_{i}_{i}^{2}, where _{i}_{1},⋯, _{n}^{T}_{y}

When the ordinary LS method is applied to the linear model _{LS}

If the random errors _{i}_{i}_{y}_{y}_{1} are the functions of the parameters _{y}

As a result of

Differentiating _{WLS}_{y}_{y}_{WLS}

The weighted LS estimate _{WLS}_{y}_{WLS}_{WLS}_{WLS}_{WLS}_{WLS}_{y}_{i}_{WLS}_{i}_{bc}

We should note that formula

Since Xu and Shimada [

Applying the error propagation law to the ordinary LS estimate _{LS}

Unlike the ordinary LS estimate _{LS}_{WLS}_{bc}_{WLS}_{bc}_{WLS}_{bc}_{WLS}_{WLS}

In order to derive the bias of the weighted LS estimate _{WLS}_{WLS}_{β}

We now apply the error propagation law to _{WLS}

Denoting the bias of _{WLS}_{β}_{β}_{WLS}_{bc}_{bc}_{bc}

By using the ordinary LS estimate _{LS}

In the similar manner, based on the weighted LS estimate and the bias-corrected weighted LS estimate and by using the formulae

As is well known, the corrections of measurements are important quantities in practical data processing and quality assessment/control. We denote the corrections of measurements as follows:

By linearizing

Because the weighted LS estimate is biased, the corresponding corrections of measurements should also be biased. The bias of _{WLS}_{WLS}_{WLS}_{WLS}

In practical data processing, we are also interested in computing the cross-covariance matrices of the original measurements, the estimated parameters, the adjusted measurements and the corrections of measurements. With the three LS-based methods in hand, if we follow the above lines of thought for the variance-covariance matrices of these quantities, we can then easily derive and obtain their covariances. If we further limit ourselves to the linear approximation in terms of the random errors

The variance of unit weight is one of the most important quantities in statistical quality evaluation and hypothesis testing. Since the measurements are not equally weighted, the conventional estimator of the variance of unit weight by using the ordinary LS residuals of measurements and the redundant number of (^{2} and then estimate ^{2}.In what follows, we will derive the estimates of the variance of unit weight in association with the three LS-based methods.

From the formula _{LSl}

If we use the weighted sum of square of the corrections of measurements in _{LS}_{2} of measurements is given by

For the weighted LS and bias-corrected weighted LS methods, if the bias of the weighted LS estimate is not significant, both methods should lead to almost the same estimates of the variance of unit weight. In the similar manner to the derivation of the estimate of the variance of unit weight in association with the ordinary LS method, we can readily use _{β}

DEM has been playing an increasing role in hazard assessment (see e.g., [

In this section, we will follow the website

To start with the simulation, we first assume a simple mountain of trapezoidal prism, with a slope of 50°, a height of 700 m and the length of 500 m. We then assume that a landslide occurs along the slope of the mountain, say due to a large earthquake. The design of landslide examples may be illustrated in ^{3}.

We now assume that DEMs are constructed by using airborne LiDAR before and after the landslide occurs, respectively. The height of flight is fixed at 900 m and the trajectory of the flight is assumed to be free of errors. For simplicity of simulation, we may also assume that the slope plane

With the simulated LiDAR data at hand, we can then estimate the surface _{sm} of the landslide, the superscript j stands for each of the three estimation methods, namely, the ordinary LS, bias-corrected weighted LS and weighted LS methods. V_{0} and the coefficients _{kl}_{2} in the domain

Listed in ^{3} from the true value and performs better than the ordinary and weighted LS methods. The difference between the true volume and that estimated from the ordinary LS method is 80,790 m^{3}, about the double of the difference by using the bias-corrected weighted LS method; a landslide with a size of this difference by itself is sufficiently large to cause severe damaging, when compared to the large landslides of the 20th and 21st centuries listed in the USGS website

In order to evaluate the errors of the estimated volume by each of the three methods, we apply the results of error analysis in Section 3 and the estimates of variance of unit weight in Section 4 to the estimate of volume ^{T} A^{−1} ^{2}

More precisely, to estimate the accuracy of the estimated volume of the landslide, we first estimate the variance of unit weight by using

Adjustment has been founded on the basis of observational models with additive random errors to process data in geoscience. The most important feature of an observational model with additive errors is that random errors do not change with the size of a signal or a measurement. However, with the advance of modern space observation technology, we realize that some of random errors change with the sizes of measurements, which may be attributed to the random nature of physical media along the path of observation. For example, GPS, VLBI and EDM baselines, InSAR and LiDAR measurements all show the feature of multiplicative random errors in the sense that their accuracy always contains one term that is proportional to the length of the baseline or the strength of signal. Theoretically speaking, the theory and methods developed on the basis of observational models with additive errors cannot meet the need of models with multiplicative errors. New theory and methods have to be developed accordingly. Xu and Shimada [

This paper has substantially extended the work by Xu and Shimada [

This work is supported by the Chinese National Science Foundation (Projects Nos. 41204006, 41374016, 41231174) and a Grant-in-Aid for Scientific Research (C25400449). The authors thank two reviewers very much for their constructive comments, which help further improve the paper.

The authors declare no conflict of interest.

_{2}lidar signal returns from remote targets

Illustrated model of landslide of rotational type. The left and right panels stand for the model mountains before and after the landslide, respectively.

Accuracy improvement of the bias-corrected weighted LS method over the ordinary LS method (per cent) in the 500 independent random experiments.

Cross-covariance matrices among the measurements, the parameter estimates, the adjusted measurements and the corrections of measurements by the LS and bias-corrected weighted LS methods.

^{T}^{−1}^{T}_{y}σ^{2} |
| |

^{T}^{−1}^{T}_{y}σ^{2} |
| |

[^{T}^{−1}^{T}_{y}σ^{2} |
| |

^{T}^{−1}^{T}_{y}^{T}^{−1}^{2} |
| |

[^{T}^{−1}^{T}_{y}^{T}^{−1}^{2} |
||

[^{T}^{−1}^{T}_{y}^{T}^{−1}^{T}σ^{2} |

The estimates of the volume of the simulated landslide mass. Also listed in this table are the roots of the MSE of the estimated volumes and the estimates of the standard deviation of unit weight.

^{3}) |
2.45686 × 10^{7} |
2.46494 × 10^{7} |
2.46073 × 10^{7} |
2.46105 × 10^{7} |

^{3}) |
1.640 × 10^{5} |
1.582 × 10^{5} |
1.583 × 10^{5} | |

σ (m) | 1.268 | 1.251 | 1.245 | 1.245 |

The abbreviations