K. P. Schwarz

Digital image georeferencing from a multiple camera system by GPS/INS

In this paper, the development and testing of an airborne fully digital multi-sensor system for digital mapping data acquisition is presented. The system acquires two streams of data, namely, navigation (georeferencing) data and imaging data. The navigation data are obtained by integrating an accurate strapdown inertial navigation system with a differential GPS system (DGPS). The imaging data are acquired by two low-cost digital cameras, configured in such a way so as to reduce their geometric limitations. The two cameras capture strips of overlapping nadir and oblique images. The GPS/INS-derived trajectory contains the full translational and rotational motion of the carrier aircraft. Thus, image exterior orientation information is extracted from the trajectory, during post-processing. This approach eliminates the need for ground control (GCP) when computing 3D positions of objects that appear in the field of view of the system imaging component. Two approaches for calibrating the system are presented, namely, terrestrial calibration and in-flight calibration. Test flights were conducted over the campus of The University of Calgary. Testing the system showed that best ground point positioning accuracy at 1:12,000 average image scale is 0.2 m (RMS) in easting and northing and 0.3 m (RMS) in height. Preliminary results indicate that major applications of such a system in the future are in the field of digital mapping, at scales of 1:5000 and smaller, and in the generation of digital elevation models for engineering applications.

A comparison of stable platform and strapdown airborne gravity

To date, operational airborne gravity results have been obtained using either a damped two-axes stable platform gravimeter systems such as the LaCoste and Romberg (LCR) S-model marine gravimeter or a strapdown inertial navigation system (INS), showing comparable accuracies. In June of 1998 three flight tests were undertaken which tested a LCR gravimeter and a strapdown INS gravity system side-by-side. To our knowledge this was the first time such a comparison flight was undertaken. The flights occurred in Disko Bay, off the west coast of Greenland. Several of the flight lines were partly flown along existing shipborne gravity profiles to allow for an independent source of comparison of the results. This paper presents the results and analysis of these flight tests. The measurement method and error models for both the stable platform and strapdown INS gravity systems are presented and contrasted. The results of the flight tests show that the gravity estimates from the two systems agree at the 2-3 mGal level, after the removal of a linear bias. This near the combined noise levels of the two systems. It appears that a combination of both systems would provide and ideal airborne gravity survey system; combining the excellent bias stability of the LCR gravimeter with the higher dynamic range and increased spatial resolution of the strapdown INS.

What can airborne gravimetry contribute to geoid determination

This paper evaluates the suitability of airborne gravimetry for geoid determination. Spectral analysis of gravity data is used to derive the high-frequency geoid spectrum and to compare it to the error spectrum of the geoid, as derived from an airborne gravity system. Gravity values in various areas with typical topographic features are used for the spectral analysis. The achievable accuracy of airborne gravimetry in geoid determination is studied through an assessment of inertial navigation system (INS) and global positioning system (GPS) errors in the spectral ranges of interest. The results of these computations are presented in a number of graphs which allow the following conclusions. To determine geoid undulations with an accuracy of 1 cm (rms), the minimum wavelength to be resolved is 14 km in flat terrain and 5 km in mountainous terrain. Using a flight speed of 300 km/h at an altitude of 2.5 km or less above ground, current hardware sensitivity is sufficient to resolve the geoid spectrum for such wavelengths. The total GPS/INS induced geoid undulation error (rms) in this case is less than 1 cm for wavelengths between 5 km and 100 km and less than 10 cm for wavelengths between 5 km and 500 km. The cumulative rms geoid error is less than 30 cm for wavelengths up to 1000 km. The accuracy for relative geoid determination is 0.1 ppm for wavelengths up to 100 km and 0.2 ppm for 500 km. The results were verified by analyzing the data collected during an airborne test. The results indicate that airborne gravimetry has the potential to determine geoid undulation with centimeter-level accuracy and to fill the gaps in the global gravity map in an efficient manner and with an accuracy superior to current terrestrial methods.

Solving Molodensky's series by fast Fourier transform techniques

The use of the fast Fourier transform algorithm in the evaluation of the Molodensky series terms is demonstrated in this paper. The solution by analytical continuation to point level has been reformulated to obtain convolution integrals in planar approximation which can be efficiently evaluated in the frequency domain. Preliminary results show that the solution by Faye anomalies is not sufficient for highly accurate deflections of the vertical and height anomalies. The Molodensky solution up to at least the second-order term must be carried out. Part of the unrecovered deflection and height anomaly signal appears to be due to density variations, verifying the essential role of density modelling. A remove-restore technique for the terrain effects can improve the convergence of the series and minimize the interpolation errors.

A study of the contributions of various gravimetric data types on the estimation of gravity field parameters in the mountains

In this paper, new rigorous, simpler, and more efficient formulas are used to estimate the effect of the terrain on geoid undulations and deflections of the vertical. The new formulas are based on the successive application in the spectral domain of Molodenskys vertical derivative operator to powers of heights. A number of numerical tests are carried out in the area of British Columbia (BC) in western Canada. Geoid undulations and deflections of the vertical are computed by combining a geopotential model complete to degree and order 360, mean 5 arc min × 5 arc min free air gravity anomalies and height data on a 1 km × 1 km grid. In geoid undulation computations, the indirect effect of Helmerts second condensation reduction is taken into account. To assess the quality of the predicted deflections of the vertical and geoid undulations, the computed quantities are compared to a set of astronomic deflections of the vertical and geoid undulations derived from a combination of Global Positioning System with leveled orthometric heights. Test results present a precision of about ±2.0 arc sec for deflections, thus fulfilling the requirements for the reduction of geodetic measurements in any case. The achievable precision for geoid undulation differences, close to ±0.50 m, is also acceptable to determine an accurate relative geoid in the test area useful for various geodetic applications.

A consistency test of airborne GPS using multiple monitor stations

In October 1990, several airborne GPS tests were conducted in the Ottawa region by the Canada Centre for Surveying (CCS) and the Canada Centre for Remote Sensing (CCRS). Ashtech XII receivers were located at up to three monitor stations with baseline lengths to the aircraft ranging from 1–200 km. Approximately two hours of airborne data, collected at a 2 Hz rate, were available for each of the three test days. Post-processing of the differential data was done using the University of Calgarys SEMIKIN package which utilizes a Kalman filter algorithm to estimate both the remote receivers position and velocity. Comparisons were made between the aircraft position and velocity determined from each of the monitor stations to assess the consistency of differential GPS when different reference stations are used. Results show that the degree of consistency is dependent upon the distance to the monitor stations. Agreement at the decimetre-level is achieved in position when the baseline lengths are within 100 km. Agreement in velocity is usually better than 1 cm s−1 (RMS).

A framework for modelling kinematic measurements in gravity field applications

The determination of the local gravity field from sensors mounted in a fixed wing aircraft has long been a dream of geodesists and geophysicists. The progress in sensor technology during the last decade has brought its realization within reach and recent tests indicate that results at the level of a fewmGal are possible. To assess different sensor configurations and their effect on the resolution of the gravity field spectrum, a state model for motion in the gravity field of the earth is formulated. The resulting set of differential equations can accommodate first and second order gravity gradients, specific force, kinematic acceleration, vehicle velocity and position as input. It offers therefore a rather general framework for gravity field determination from a variety of kinematic sensors, such as gravity meters, gravity gradiometers, inertial systems, differentialGPS, laser altimeters and others. The derivation of the basic kinematic model and its linearization are given in detail, while sensor error models are discussed in a generic way. A few remarks on the modelling of gravity gradiometer measurements conclude the paper.

A numerical investigation on height anomaly prediction in mountainous areas

This paper provides numerical examples for the prediction of height anomalies by the solution of Molodenskys boundary value problem. Computations are done within two areas in the Canadian Rockies. The data used are on a grid with various grid spacings from 100 m to 5 arc-minutes. Numerical results indicate that the Bouguer or the topographicisostatic gravity anomalies should be used in gravity interpolation. It is feasible to predict height anomalies in mountainous areas with an accuracy of 10 cm (1σ) if sufficiently dense data grids are used. After removing the systematic bias, the differences between the geoid undulations converted from height anomalies and those derived from GPS/levelling on 50 benchmarks is 12 cm (1σ) when the grid spacing is 1km, and 50 cm (1σ) when the grid spacing is 5′. It is not necessary, in most cases, to require a grid spacing finer than 1 km, because the height anomaly changes only by 3 cm (1σ) when the grid spacing is increased from 100 m to 1000 m. Numerical results also indicate that, only the first two terms of the Molodensky series have to be evaluated in all but the extreme cases, since the contributions of the higher order terms are negligible compared to the objective accuracy.

Advances in the numerical solution of the linear molodensky problem

The solution of the linear Molodensky problem by analytical continuation to point level is numerically the most convenient of all the theoretically equivalent solutions. It is obtained by successively applying the same integral operator and it does not depend explicitly on the terrain inclination. However, its dependence on the computation point restricts somehow the computational efficiency. The use of the Fourier transform for the evaluation of the integral operator in planar approximation lessens significantly the burden of computations. Using this spectral approach, the problem has been reformulated and solved in the frequency domain. Moreover, it is shown that the solution can be easily split into two steps : (a) “downward” continuation to sea level, which is independent of the computation point, and (b) “upward” continuation from sea to point level, using the values computed at sea level. Such a treatment not only simplifies the formulas and increases the numerical efficiency but also clarifies the physical interpretation and the theoretical equivalence of the continuation solution with respect to the other solution types. Numerical tests have been performed to investigate which terms in the Molodensky series are of significance for geoid and deflection computations. The practical difficulty of differences in the grid spacings of gravity and height data has been overcome by frequency domain interpolation.

The use of wavelets for the analysis and de-noising of kinematic geodetic measurements

The purpose of this paper is to discuss and demonstrate how wavelets can be used to improve the performance of geodetic measurement systems; more specifically to explore the use of wavelets as a tool for analyzing and de-noising the errors that are inherent in GPS/INS systems. Because the wavelet transform provides a time-resolution representation of a signal, it offers the unique ability to analyze the error characteristics of such systems at different frequencies and to localize them in time. This is something not offered by tools that operate in the time or frequency domains. It is seen that these features can be useful for analyzing and removing the errors in kinematic geodetic systems