Bayesian methods for estimating multi-segment discharge rating curves

This study explores Bayesian methods for handling compound stage–discharge relationships, a problem which arises in many natural rivers. It is assumed: (1) the stage–discharge relationship in each rating curve segment is a power-law with a location parameter, or zero-plane displacement; (2) the segment transitions are abrupt and continuous; and (3) multiplicative measurement errors are of equal variance. The rating curve fitting procedure is then formulated as a piecewise regression problem where the number of segments and the associated changepoints are assumed unknown. Procedures are developed for describing both global and site-specific prior distributions for all rating curve parameters, including the changepoints. Estimation and uncertainty analysis is evaluated using Markov chain Monte Carlo simulation (MCMC) techniques. The first model explored accounts for parameter and model uncertainties in the interpolated area, i.e. within the range of available stage–discharge measurements. A second model is constructed in an attempt to include the uncertainty in extrapolation, which is necessary when the rating curve is used to estimate discharges beyond the highest or lowest measurement. This is done by assuming that the rate of changepoints both inside and outside the measured area follows a Poisson process. The theory is applied to actual data from Norwegian gauging stations. The MCMC solutions give results that appear sensible and useful for inferential purposes, though the latter model needs further efforts in order to obtain a more efficient simulation scheme.     

Data-snooping procedure applied to errors-in-variables models

The theory of Baarda’s data snooping — normal and F tests respectively based on the known and unknown posteriori variance — is applied to detect blunders in errors-invariables (EIV) models, in which gross errors are in the vector of observations and/or in the coefficient matrix. This work is a follow-up to an earlier work in which we presented the formulation of the weighted total least squares (WTLS) based on the standard least squares theory. This method allows one to directly apply the existing body of knowledge of the least squares theory to the errors-in-variables models. Among those applications, data snooping methods in an EIV model are of particular interest, which is the subject of discussion in the present contribution. This paper generalizes the Baarda’s data snooping procedure of the standard least squares theory to an EIV model. Two empirical examples, a linear regression model and a 2-D affine transformation, using simulated and real data are presented to show the efficacy of the presented formulation. It is highlighted that the method presented is capable of detecting outlying equations (rather than outlying observations) in a straightforward manner. Further, the WTLS method can be used to handle different TLS problems. For example, the WTLS problem for the conditions and mixed models, the WTLS problem subject to constraints and variance component estimation for an EIV model can easily be established. These issues are in progress for future publications.     

The variable projection method for waveform inversion with an unknown source function

This paper compares three alternative algorithms for simultaneously estimating a source wavelet at the same time as an earth model in full‐waveform inversion: (i) simultaneous descent, (ii) alternating descent and (iii) descent with the variable projection method. The latter is a technique for solving separable least‐squares problems that is well‐known in the applied mathematics literature. When applied to full‐waveform inversion, it involves making the source wavelet an implicit function of the earth model via a least‐squares filter‐estimation process. Since the source wavelet becomes purely a function of medium parameters, it no longer needs to be treated as a separate unknown in the inversion. Essentially, the predicted data are projected onto the measured data in a least‐squares sense at every function evaluation, making use of the fact that the filter estimation problem is trivial when compared to the full‐waveform inversion problem. Numerical tests on a simple 1D model indicate that the variable projection method gives the best result; actually producing results in quality that are very similar to control experiments with a known, correct wavelet.     

Parameter Estimation for Groundwater Models under Uncertain Irrigation Data

The success of modeling groundwater is strongly influenced by the accuracy of the model parameters that are used to characterize the subsurface system. However, the presence of uncertainty and possibly bias in groundwater model source/sink terms may lead to biased estimates of model parameters and model predictions when the standard regression‐based inverse modeling techniques are used. This study first quantifies the levels of bias in groundwater model parameters and predictions due to the presence of errors in irrigation data. Then, a new inverse modeling technique called input uncertainty weighted least‐squares (IUWLS) is presented for unbiased estimation of the parameters when pumping and other source/sink data are uncertain. The approach uses the concept of generalized least‐squares method with the weight of the objective function depending on the level of pumping uncertainty and iteratively adjusted during the parameter optimization process. We have conducted both analytical and numerical experiments, using irrigation pumping data from the Republican River Basin in Nebraska, to evaluate the performance of ordinary least‐squares (OLS) and IUWLS calibration methods under different levels of uncertainty of irrigation data and calibration conditions. The result from the OLS method shows the presence of statistically significant (p < 0.05) bias in estimated parameters and model predictions that persist despite calibrating the models to different calibration data and sample sizes. However, by directly accounting for the irrigation pumping uncertainties during the calibration procedures, the proposed IUWLS is able to minimize the bias effectively without adding significant computational burden to the calibration processes.     

A NOTE ON THE ESTIMATION OF THE PARAMETERS IN LOGARITHMIC STAGE-DISCHARGE RELATIONSHIPS WITH ESTIMATES OF THEIR ERROR

Abstract

Under certain assumptions the stage-discharge relationship of a channel cross-section can be approximated by a logarithmic relationship. Observational pairs of stage and discharge plotted on log-log paper often cluster around a straight line and this suggests that the assumptions involved are often approximately satisfied.

In such cases the parameters of the logarithmic relationship are usually estimated graphically from the position and slope of the straight line on the log-log paper. In this paper principles and methods are outlined for the estimation of the parameters with estimates of their standard error, via regression analysis. Because the water level of zero flows is usually one of the unknown parameters, the regression is non-linear and least squares optimal estimates can be obtained by a step-by-step approximation. The variances of the parameter estimates can be obtained from the dispersion matrix of the joint distribution of the least squares estimators via the likelihood function. An estimate of the error in predictions of the discharge depending on the corresponding stage may be obtained.     

Modeling discharge rating curves with Bayesian B-splines

River discharges are traditionally modeled by employing a standard power-law methodology. Recently, the Bayesian approached has successfully been applied to improve the estimates of the standard power-law. In this article, an extension to the standard power-law based on Bayesian B-splines is developed and tested on data sets from 61 different rivers. The extended model is evaluated against the standard power-law using two measures, the Deviance Information Criterion and Bayes factor. The extended model captures deviations in the data from the standard power-law but reduces to the standard power-law when that model is adequate. The standard power-law is inadequate for 26% of the rivers while the extended model provides an adequate fit in all of those cases and for the remaining 74% of the rivers the extended model and the power-law model both give adequate fit with almost identical estimates.     

A fully non-stationary linear coregionalization model for multivariate random fields

This paper introduces an extension of the traditional stationary linear coregionalization model to handle the lack of stationarity. Under the proposed model, coregionalization matrices are spatially dependent, and basic univariate spatial dependence structures are non-stationary. A parameter estimation procedure of the proposed non-stationary linear coregionalization model is developed under the local stationarity framework. The proposed estimation procedure is based on the method of moments and involves a matrix-valued local stationary variogram kernel estimator, a weighted local least squares method in combination with a kernel smoothing technique. Local parameter estimates are knitted together for prediction and simulation purposes. The proposed non-stationary multivariate spatial modeling approach is illustrated using two real bivariate data examples. Prediction performance comparison is carried out with the classical stationary multivariate spatial modeling approach. According to several criteria, the prediction performance of the proposed non-stationary multivariate spatial modeling approach appears to be significantly better.     

Exploring some issues in acoustic full waveform inversion

The least‐squares error measures the difference between observed and modelled seismic data. Because it suffers from local minima, a good initial velocity model is required to avoid convergence to the wrong model when using a gradient‐based minimization method. If a data set mainly contains reflection events, it is difficult to update the velocity model with the least‐squares error because the minimization method easily ends up in the nearest local minimum without ever reaching the global minimum. Several authors observed that the model could be updated by diving waves, requiring a wide‐angle or large‐offset data set. This full waveform tomography is limited to a maximum depth. Here, we use a linear velocity model to obtain estimates for the maximum depth. In addition, we investigate how frequencies should be selected if the seismic data are modelled in the frequency domain. In the presence of noise, the condition to avoid local minima requires more frequencies than needed for sufficient spectral coverage. We also considered acoustic inversion of a synthetic marine data set created by an elastic time‐domain finite‐difference code. This allowed us to validate the estimates made for the linear velocity model. The acoustic approximation leads to a number of problems when using long‐offset data. Nevertheless, we obtained reasonable results. The use of a variable density in the acoustic inversion helped to match the data at the expense of accuracy in the inversion result for the density.     

Tomographic imaging of P- and S-wave velocity structure beneath Costa Rica

45287 P-wave and 26813 S-wave arrival times from the data base of the Costa Rica network have been tomographically inverted to image the structure beneath Costa Rica. A regularized recursive least squares inverse method was used to produce the high resolution and minimum variance model parameter estimates. The first arrival times are calculated using a finite difference technique, which allows for flexible parameterization of the velocity model and easy inclusion of topography and source-receiver geometry. The P wave velocity structure and hypocenters are determined simultaneously, while the S wave velocity structure is determined using the relocated seismicity and an initial model derived from the P wave model assuming an average P to S wave velocity ratio of 1.78. The most prominent features in the inverted model are a low velocity structure under the volcanic chain in the center of the country, which is related to the hot material connected with the active volcanoes; and a high velocity zone in the mantle, which is related to the Cocos plate subducted under Costa Rica.     

Unscented transformation with scaled symmetric sampling strategy for precision estimation of total least squares

The errors-in-variables (EIV) model is a nonlinear model, the parameters of which can be solved by singular value decomposition (SVD) method or the general iterative algorithm. The existing formulae for covariance matrix of total least squares (TLS) parameter estimates don’t fully consider the randomness of quantities in iterative algorithm and the biases of parameter estimates and residuals. In order to reflect more reasonable precision information for TLS adjustment, the derivative-free unscented transformation with scaled symmetric sampling strategy, i.e. scaled unscented transformation (SUT), is introduced and implemented. In this contribution, we firstly discuss the existing various solutions of TLS adjustment and covariance matrices of TLS parameter estimates and derive the general first-order approximate cofactor matrices of random quantities in TLS adjustment. Secondly, based on the combination of TLS iterative algorithm and calculation process of SUT, we design the two SUT algorithms to calculate the biases and the second-order approximate covariance matrices. Finally, the straight line fitting model and plane coordinate transformation model are used to demonstrate that applying SUT for precision estimation of TLS adjustment is feasible and effective.     

A new method to discriminate between a valid IP response and EM coupling effects

The problem of discrimination between a valid induced polarization (IP) response and electromagnetic (EM) coupling effects is considered and an effective solution is provided. First, a finite dimensional approximation to the Cole‐Cole model is investigated. Using the least‐squares approach, the parameters of the approximate model are obtained. Next, based on the analysis of overvoltage, a finite dimensional structure of the IP model is produced. Using this overvoltage‐based structure, a specific finite dimensional approximation of the Cole‐Cole model is proposed. Summarizing the analysis of the finite dimensional IP model, it is concluded that the proposed IP model, which fits the field data much better than the traditional Cole‐Cole model, is essentially an RC‐circuit. From a circuit‐analysis point of view, it is well known that an electromagnetic effect can be described by an RL‐circuit. The simulation results on experimental data support this conception. According to this observation, a new method to discriminate between a valid IP response and EM coupling effects is proposed as follows: (i) use a special finite dimensional model for IP–EM systems; (ii) obtain the parameters for the model using a least‐squares approach; (iii) separate RC‐type terms and RL‐type terms – the first models the IP behaviour, the latter represents the EM part. Simulation on experimental data shows that the method is very simple and effective.     

Recalibrating aeolian sand transport models

A quality‐controlled data set comprising measurements of aeolian sand transport rates obtained at three disparate field sites is used to evaluate six commonly employed transport rate models (those of Bagnold, Kawamura, Zingg, Owen, Hsu, and Lettau and Lettau) and to recalibrate the empirical constants in those models. Shear velocity estimates were obtained using the von Kármán constant and an apparent von Kármán parameter. Models were recalibrated using non‐linear regression and non‐linear regression with least‐squares lines forced through axes origins. Recalibration using the apparent von Kármán parameter and forced regression reduced the empirical constants for all models. The disparity between the predictions from the different models is reduced from about an order of magnitude to about a quarter of an order of magnitude. The recalibrated Lettau and Lettau model provided the greatest statistical agreement between observed and predicted transport rates, with a coefficient of determination of 0·77. Evaluation of the results suggests that our estimations of threshold shear velocity may be too slow, causing errors in predicted transport rates. Copyright © 2012 John Wiley & Sons, Ltd.     

Modelling for registration of remotely sensed imagery when reference control points contain error

1 Introduction The process of remotely sensed data acquisition isaffected by factors such as the rotation of the earth, finite scan rate of some sensors, curvature of the earth, non-ideal sensor, variation in platform altitude, attitude, velocity, etc.[1]. One important procedurewhich should be done prior to analyzing remotely sensed data, is geometric correction (image to map) or registration (image to image) of remotely sensed data. The purpose of geometric correction or registration is to e…     

Iteratively re‐weighted and refined least squares algorithm for robust inversion of geophysical data

A robust metric of data misfit such as the ?₁‐norm is required for geophysical parameter estimation when the data are contaminated by erratic noise. Recently, the iteratively re‐weighted and refined least‐squares algorithm was introduced for efficient solution of geophysical inverse problems in the presence of additive Gaussian noise in the data. We extend the algorithm in two practically important directions to make it applicable to data with non‐Gaussian noise and to make its regularisation parameter tuning more efficient and automatic. The regularisation parameter in iteratively reweighted and refined least‐squares algorithm varies with iteration, allowing the efficient solution of constrained problems. A technique is proposed based on the secant method for root finding to concentrate on finding a solution that satisfies the constraint, either fitting to a target misfit (if a bound on the noise is available) or having a target size (if a bound on the solution is available). This technique leads to an automatic update of the regularisation parameter at each and every iteration. We further propose a simple and efficient scheme that tunes the regularisation parameter without requiring target bounds. This is of great importance for the field data inversion where there is no information about the size of the noise and the solution. Numerical examples from non‐stationary seismic deconvolution and velocity‐stack inversion show that the proposed algorithm is efficient, stable, and robust and outperforms the conventional and state‐of‐the‐art methods.     

EXPERIENCES WITH AUTOMATIC MAGNETIC INTERPRETATION USING THE THICK PLATE MODEL *

Various experiments are described in designing two-dimensional magnetic interpretation algorithms using computer curve fitting techniques. For a single anomaly the position of the anomaly maximum and the half-width of the anomaly give good initial estimates of the plate position and thickness. A nomogram and formulae for improving these estimates is given. Curves and estimates for the effects of finite depth extent of a plate show the limits, when the lower surface of the plate can be neglected in curve fitting. The combined anomaly of two parallel plates can be separated into partial anomalies with no common points using the horizontal derivative of the anomaly. The changes of the anomaly maxima and changes in anomaly half-widths are studied as a function of plate separation. The position of the maxima and the half-widths can be corrected before applying the one-plate procedure for obtaining initial estimates of plate positions and thicknesses. The performance of standard optimization methods of Powell, Davidon, and Marquardt in improving the values of the plate parameters are compared. The Powell method seems to be the most reliable for both single and multi-plate anomalies. All methods become unacceptably slow when the number of plates is greater than 2 or 3. In these cases feasible interpretation times are obtained using the partial anomalies and sequential parabolic search of the parameter values as tailored specially to the thick plate model. Experiments with three different error norms, the classical least squares, weighted least squares and minimax, show that the first norm gives the best overall performance in automatic interpretation. The behaviour of the classical least squares norm as a function of the plate parameters is also briefly described.     

Sea ice as an evaluation target for HEM modelling and inversion

We investigate helicopter electromagnetic (HEM) inversion schemes applied to synthetic and measured HEM sea ice profiling data. Direct HEM data-to-ice-thickness inversion is compared to three different formal, least squares layered earth inversion algorithms.By making several approximations, it is possible to directly invert a single channel measurement (i.e., the in-phase or quadrature component of a single frequency measurement) to obtain an estimate of sea ice thickness. Measurements from multiple input channels, however, can be used in a layered earth inversion to simultaneously recover several model parameters such as sea ice thickness, sea ice conductivity and sub-ice bathymetry. Synthetic data sets for a particular two-frequency HEM system showed that simple least squares inversion algorithms produce reliable estimates of sea ice thickness in cases where the ice is thicker than 3 m. These methods could also recover acceptable estimates of sea ice thickness when a thin, conductive, partially melted sea ice layer was present, and could determine shallow, sub-ice bathymetry in brackish water. As expected, 1D transformations and inversions of synthetic data for a three-dimensional pressure ridge keel structure contained artifacts, notably broadening of the apparent width of the keel.Prior to inverting a field data set acquired over rather thin (~ 0.5 m) Antarctic sea ice, we found it necessary to recalibrate the phase angle of the measurements using a phasor diagram-based method. Direct transformation of a single channel from the recalibrated data set produced more accurate estimates of sea-ice thickness than formal inversion of multi-channel data. We suggest that the least squares inversion methods are inferior in this situation because of the particular characteristics of the two-frequency HEM system used in this evaluation; the extreme differences in sensitivity of high and low frequency data components, the overall low sensitivity to sea ice conductivity (especially for thin ice), and the partially low signal-to noise ratios of the measurements. The data sets used in this study will be made available to the public to allow alternate inversion approaches to be applied and evaluated. It is suggested that inclusion of parameter bounds and other forms of regularization could help to improve the inversion results.     

Tide modeling using partial least squares regression

Ocean Dynamics - This research explores the novel use of the partial least squares regression (PLSR) as an alternative model to the conventional least squares (LS) model for modeling tide levels....     

A Robust Method for Relative Gravity Data Estimation with High Efficiency

When gravimetric data observations have outliers, using standard least squares (LS) estimation will likely give poor accuracies and unreliable parameter estimates. One of the typical approaches to overcome this problem consists of using the robust estimation techniques. In this paper, we modified the robust estimator of Gervini and Yohai (2002) called REWLSE (Robust and Efficient Weighted Least Squares Estimator), which combines simultaneously high statistical efficiency and high breakdown point by replacing the weight function by a new weight function. This method allows reducing the outlier impacts and makes more use of the information provided by the data. In order to adapt this technique to the relative gravity data, weights are computed using the empirical distribution of the residuals obtained initially by the LTS (Least Trimmed Squares) estimator and by minimizing the mean distances relatively to the LS-estimator without outliers. The robustness of the initial estimator is maintained by adapted cut-off values as suggested by the REWLSE method which allows also a reasonable statistical efficiency. Hereafter we give the advantage and the pertinence of REWLSE procedure on real and semi-simulated gravity data by comparing it with conventional LS and other robust approaches like M- and MM-estimators.     

Scattered ground‐roll attenuation using model‐driven interferometry

Scattered ground roll is a type of noise observed in land seismic data that can be particularly difficult to suppress. Typically, this type of noise cannot be removed using conventional velocity‐based filters. In this paper, we discuss a model‐driven form of seismic interferometry that allows suppression of scattered ground‐roll noise in land seismic data. The conventional cross‐correlate and stack interferometry approach results in scattered noise estimates between two receiver locations (i.e. as if one of the receivers had been replaced by a source). For noise suppression, this requires that each source we wish to attenuate the noise from is co‐located with a receiver. The model‐driven form differs, as the use of a simple model in place of one of the inputs for interferometry allows the scattered noise estimate to be made between a source and a receiver. This allows the method to be more flexible, as co‐location of sources and receivers is not required, and the method can be applied to data sets with a variety of different acquisition geometries. A simple plane‐wave model is used, allowing the method to remain relatively data driven, with weighting factors for the plane waves determined using a least‐squares solution. Using a number of both synthetic and real two‐dimensional (2D) and three‐dimensional (3D) land seismic data sets, we show that this model‐driven approach provides effective results, allowing suppression of scattered ground‐roll noise without having an adverse effect on the underlying signal.     

INVERSION OF SURFACE AND BOREHOLE ELECTROMAGNETIC DATA FOR TWO-DIMENSIONAL ELECTRICAL CONDUCTIVITY MODELS*

A method for inverting electromagnetic fields induced by a line source in an earth of two-dimensional conductivity structure is developed. Certain unique features of the finite element method are used to construct an efficient algorithm for the accurate calculation of the Jacobian matrix of partial derivatives, and the resulting linearized equations are solved using the damped least squares method. Case studies of theoretical data generated from a simple model of interest in geophysical prospecting show that, in general, it is impossible to obtain, from surface data alone, accurate estimates of the conductivity of structures buried deeper than 0.2 skin depths under a conducting overburden. The addition of borehole data to the surface data is found to increase the resolving power of the electromagnetic method dramatically. In particular, the borehole data appear to stabilize the inverse when only a poor initial estimate of the likely structure is given.