New concepts for airborne gravity measurement

Aerosp. Sci. Technol. 5 (2001) 413–424  2001 Éditions scientifiques et médicales Elsevier SAS. All rights reserved S1270-9638(01)01114-2/FLA

New concepts for airborne gravity measurement Fethi Abdelmoula ∗ Institute of Flight Guidance, Technical University of Braunschweig, Hermann-Blenk-Straße 27, 38108 Braunschweig, Germany Received 10 October 2000; revised and accepted 13 July 2001

Abstract

A fundamental survey of vehicle mounted gravimeters shows the superposition from the gravity and the kinematical acceleration due to item modification. The particular problems of airborne gravimetry lie in the presence of relatively strong disturbances due to air turbulence. The vehicle acceleration caused in this manner is from high dynamics and up to a factor of 106 greater than the anomaly signal (local changes of acceleration due to the earth’s gravitation). Thus, the technically scientific problem of utilizing gravimeters in airplanes lies in a suitable compensation for the vehicle’s acceleration. The measurement methods applied reduce the high frequency disturbance signal with the aid of a low-pass filter. These methods are restricted only for the localization of very large anomalies. The required resolution for an air gravimeter cannot be reached only through a direct low-pass or band-pass filtering, since the frequency ranges of the disturbance and the anomaly overlap. A clear improvement of the resolution requires completely new concepts. For separating gravity and kinematical acceleration, there must be further sensors that are not influenced by gravity, e.g. altimeter and the Global Positioning System (GPS). An overall system, into which the individual sensors are integrated and associated via suitable complementary filtering with each other, allows it to register gravity with a higher resolution than is presently possible.  2001 Éditions scientifiques et médicales Elsevier SAS airborne gravimetry / flight measurement / gravity sensor / model-based filters

Zusammenfassung

Neue Konzepte zur luftgestützten Gravitationsmessung. Grundsätzlich wird von einem Gravimetersensor die Überlagerung aus der Gravitation und der kinematischen Beschleunigung infolge Positionsänderungen erfasst. Die besondere Problematik der Fluggravimetrie liegt in den vergleichsweise starken Störungen infolge von Luftturbulenz. Die hierdurch hervorgerufenen Fahrzeugbeschleunigungen sind von hoher Dynamik und bis um den Faktor 106 größer als das Nutzsignal (lokale Variation der Erdgravitation). Das technisch wissenschaftliche Problem beim Einsatz von Gravimetern in Flugzeugen liegt damit in einer geeigneten Kompensation der Fahrzeugbeschleunigungen. Die bislang in Flugzeugen angewandten Messverfahren, bei denen die höherfrequenten Störsignale mit Hilfe eines Tiefpassfilters reduziert werden, eignen sich nur zur Lokalisierung sehr großer Anomalien. Das für ein Luftgravimeter geforderte Auflösungsvermögen kann durch direkte Tiefpass- oder Bandpassfilterung allein nicht erreicht werden, da sich die Frequenzbereiche von Störund Nutzsignal überdecken. Eine deutliche Verbesserung der Auflösung erfordert völlig neue Konzepte. Zur Trennung von Schwere- und kinematischer Beschleunigung müssen deshalb weitere Sensoren hinzugezogen werden, die nicht von der Schwere beeinflusst sind, wie z.B. Höhenmesser und das Globale Positionierungssystem GPS. Ein Gesamtsystem, in dem die einzelnen Sensoren integriert und über eine geeignete komplementäre Filterung miteinander verknüpft sind, ermöglicht es, die Gravitation mit einer höheren Auflösung zu erfassen als bisher.  2001 Éditions scientifiques et médicales Elsevier SAS Fluggravimetrie / Flugmesstechnik / Gravimeter / modellbasierte Filter

∗ Present address: Aerodata Systems GmbH, Dept. Avionics Systems & Components, Hermann-Blenk-Straße 36, 38108 Braunschweig, Germany.

E-mail address: [email protected] (F. Abdelmoula).

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1. Introduction Gravity measurements provide a powerful tool in the fields of geodesy, geodynamics and geophysical exploration. In the big oil companies’ race to develop new oilrich areas, which started shortly after the first World War, the approaches at this time (e.g. torsion balance) were too slow and not comprehensive enough. Conventional gravity measurements, by ships at sea or by stationary measurements on land, are time consuming and expensive as well as logistically difficult in many regions of the world. The goal was not only to locate and survey new deposits but more importantly was to recognize quickly and extensively the overall geologic structure of a larger area. These reasons form the motivation for increasing efforts, mainly during the second half of the 20th century, to develop efficient gravimeters. In particular, interested private industry has invested much to reach this goal. The results of experiments on land or at sea represent the brilliant success of modern precision measurement technology. The practical geophysical research group received a static gravimeter for the reason of working on groundbreaking problems. The application, mounted aboard a carrier vehicle, made it possible to extend the field of application to the so far only roughly explored gravity over sea and to previously inaccessible areas such as mountains, primeval forests, and deserts. An airborne gravity system provides an attractive alternative to conventional terrestrial methods, because of the relatively high speed and low cost at which measurements can be made. Over land, an aircraft provides access to difficult terrain and uniform sampling. The success in land and sea gravimetry cannot be transferred to airborne gravimetry easily. In the beginning, a number of difficulties caused the aircraft application to seem to be not very successful. The difficulties present a big challenge on flight measurement technology and filtering techniques. These factors have driven the research program in airborne gravimetry at the Institute of Flight Guidance (IFF) since 1986. The focus of the airborne gravimetry program at IFF has been on the development, improvement, and application of longrange surveying techniques. The improvement of the airborne gravimetry technique to increase accuracy and resolution is of immediate concern. We are also involved in the development of a gravimetry measurement system using an inertial platform, gravimeter sensor, barometric sensor, and kinematical GPS positioning. 2. Basic knowledge of the airborne gravity survey The gravity disturbance signal may be obtained from the time-synchronized difference between two measurements, which are a gyro-stabilized precise inertial accelerometer (gravimeter sensor) signal and altimeter signal (GPS differential carrier phase, barometric sensor). In a simple interpretation, an inertial sensor provides a measure of the specific force required to counteract the pull

due to the gravitational field and the motion of the aircraft. In effect, this provides a measure of the sum of the accelerations due to gravity and the motion of the vehicle. Free-air gravity anomalies can be derived from airborne gravimetry using the standard expression as a function of position [18]: δgG (h) = g(h∗ ) − azg (h∗ ) + δgF (h∗ ) + δgE (h∗ ) − γ (h∗ ) (1) with the components: δgG (h)

free-air anomaly;

g(h∗ )

observed acceleration at h∗ ;

azg (h∗ )

vertical disturbing acceleration caused by aircraft motion, wind turbulences;

δgF (h∗ )

free-air correction;

(h∗ )

Eötvös correction;

δgE

γ (h∗ )

normal gravity computed from GPS position results.

Another contribution to equation (1), called the Eötvös correction δgE , can be modeled in terms of the velocity vector; in detail it can be described as the magnitude of the horizontal components of the vector and the azimuth of the aircraft motion. In spherical coordinates the Eötvös correction becomes [18]: δgE = 2ΩE v sin (χ) cos (ϕ) +

v2 R

(2)

with ΩE

earth rotation;

v

magnitude of the aircraft’s horizontal velocity;

χ

azimuth of the flight path;

R

mean radius of the spherical earth.

GPS measurements are used to obtain an estimate of only the first, as well as to locate the measurements. The critical problem to solve for the gravity measurement lies in the separation of the kinematical acceleration from the gravitational effect. The difference is the sum of the gravity disturbance and system noise and errors. Both system outputs are affected by large noise with different frequency components. Consequently, the gravity disturbance signal is buried in considerable noise and its frequency resolution will depend on the signal to noise ratio in different frequency bands. Due to the extremely small signal to noise ratio, the extracting of the gravity disturbance from the residual is very difficult. Indeed, while the gravity disturbance amplitude would typically have a variance of 30 mGal and usually not exceed 100 mGal over distances of about 100 km, the noise level of an airborne gravity system is much higher [6]. Although a large

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part of this noise is white noise and can be eliminated by low-pass filtering, a considerable part is correlated noise either due to the time-dependent system errors or aircraft dynamics. So the problem is to distinguish the gravity disturbance signal from the system measurements, which is dominated by the system white noise and amplified by aircraft vibrations. The objective is to derive the most appropriate filtering scheme to recover the gravity disturbance signal from the noisy system measurements. To achieve this objective, a new filtering approach will be considered here. 3. Detailed task description The main problem of airborne gravimetry consists of correctly separating the anomaly signal from the overlaid acceleration due to the motion of the aircraft. This task is based on the existence of suitable locating sensors to measure position and altitude. The historic development of airborne gravimetry [4] has shown that satisfying solutions of the task strongly depend upon the progress in the field of locating sensor development. From this, questions regarding a suitable system integration of the measurement instrumentation to be used and data processing follow immediately. Due to the rapid progress in navigation, mainly resulting from the spread of satellite navigation, the main field of investigation at the present time is the implementation of suitable filter approaches. 3.1. State of the art The general basic approach to separate the gravity variation from the measured vertical acceleration is shown in figure 1. Nearly all groups working in the field of airborne gravimetry use it in this form. The elimination of the normal gravity and the Eötvös correction form the analogue to a sea borne survey. At this point, the precision of the applied sensors directly affects the determination of the

Figure 1. Simplified evaluation principle for airborne gravity survey.

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normal gravity and Eötvös terms. The specific addition in airborne gravimetry is the integration of altitude measurement over a separate way. From this the correction of the free air anomaly is derived. To minimize phase shifts between both signal lines, every sensor is followed by a dynamics model of the other sensor. This measurement for error elimination is not applied by most of the groups currently working in the field of airborne gravimetry, due to the absence of information about sensor dynamics as well as the assumption of ideal sensors. As investigations have shown [8,12,16,17,2], these assumptions cause substantial errors in determining gravitational anomalies. Within the filter the data combine, which is actually used to separate the gravity disturbance signal from the residual signals. The algorithms, generally summarized as ‘filter or estimator’, applied by the different project groups differ in their form and structure. Therefore, the circumstances at the time in this field of airborne gravimetry will be explained more in detail. The explanations will be divided into two sections: section 3.1.1 deals with the international state of the art and in section 3.1.2 the concept of airborne gravimetry at the Institute of Flight Guidance and Control up to immediately before this work will be introduced. 3.1.1. The external state of the art At this time, all applied concepts determine the vertical disturbance acceleration by differentiating the altimeter signal twice. A low-pass filter with a very large time constant then smoothes the noise in the differentiated signal. To avoid phase shift between gravimeter signal and the twice-differentiated altimeter signal the gravimeter signal must also be sent through a low-pass filter of the same time constant. The gravity anomaly signal is then obtained from the time-synchronized difference between the gravimeter signal and the calculated vertical acceleration. The residuals of this difference contain the variation of gravity and the system noise. Due to the previous low-pass filtering, the possible resolution concerning small anomaly length is decreased. The system noise consists of the measurement noise of both signal lines, which can widely be considered as a white random process, and the correlated parts resulting from aircraft dynamics and vibrations. The noise level is much greater than the anomaly amplitudes which causes an extremely adverse signal to noise ratio in the entire frequency spectrum. Suitable filtering approaches are necessary to extract the gravity disturbance signal from the residuals. Due to their filtering strategy, they can be divided into two categories: non model-based and model-based filters [6]. Non model-based filters are low-pass filters (frequency based approaches). The system measurement is treated as a broad frequency band signal. The useful signal, which is the gravity anomaly, is on the low frequency segment. When a small characteristic or lower frequency is chosen, it can be expected that the resulting signal generally describes the variation of gravity. Therefore this approach

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of low-pass filtering is very successfully applied in sea borne gravimetry, because there the disturbances lie in the upper frequency band. In airborne gravimetry, the frequency parts of disturbance signals and the anomaly signal overlap in the same frequency band. Therefore, depending on the cut frequency of the low-pass filter, either higher frequency parts of the gravity information vanish in the filtered signal or lower frequency parts of the disturbance signal remain in the signal. Both cause an erroneous interpretation of the gravity anomaly. This is predominantly the main disadvantage of non modelbased filtering. The quality of filters mainly depends on their slope steepness. This is why only digital architectures, as the FIR-(Finite Impulse Response)-Filter and the IIR(Infinite Impulse Response)-Filter, are used [6]. FIRFilters are not recursive; in other words, the filter output only depends on the weighted sum of the input signals. Those filters have a frequency independent phase shift and therefore no time delay. There are essentially three classes of design techniques for linear phase finite impulse response filters. These are the window method, the frequency sampling method, and the optimal filter design method. The window method provides a relatively straightforward means of truncating the desired filter impulse response coefficients. These coefficients are infinite in extent, so that the resulting filter response is as close as possible to the ideal response. By giving either the impulse response coefficients or equivalently the discrete Fourier-Transform coefficients, the frequency sampling method can uniquely specify a finite impulse response filter. It is known that the discrete Fourier-Transform samples for a finite impulse response filter sequence can be considered as the filter’s z-transform evaluated at N points equally spaced around the unit circle. Consequently, this is performed in order to approximate a desired continuous frequency response. A frequency sampling at N equidistant points around the unit circle and an evaluation of the approximating continuous frequency response as an interpolation of the desired sampled frequency response are sufficient. The approximation error would then be zero at the sampling frequencies and finite between them. An optimal low-pass filter design consists of finding the set of impulse response coefficients that minimize an error function over the frequency bands in which the approximation is made. Usually, the error function is the weighted difference between the ideal response and the real response. IIR-Filters, in addition, have a recursive part which causes a higher slope steepness but also a frequency dependent phase shift. Model-based filters use the fact that often information about the behavior of a process, in this case the behavior of gravitational anomalies, is known. The theory of Kalman Filtering is an often-applied method for a modelbased basic approach. The filter greatly minimizes the error between the observed gravitational anomalies and the estimated value taking into account the measurement

errors [7,11]. The dynamical behavior of the gravitational field as a model – stochastic or deterministic – is part of the filter. Deterministic model filtering is known for its deterministic treatment of the system model dynamics, based on the fact that the gravity disturbance exhibits very slow variations over short distances. Indeed, if the whole data span is divided into a series of small data spans, then the assumption of a constant model for the gravity disturbance dynamics becomes reasonable. Consequently, this approach uses a piecewise continuous function for each estimation cycle. The fundamental condition that has to be met is that for a short time interval, the gravity disturbance can be considered as constant [14]. An alternative approach consists of a two-step estimation procedure [14]. The first step estimates the initial state. The second step smoothes the measurement noise by using the gravity disturbance estimates obtained from the first estimation step. Consequently, the measurement noise can no longer be considered as white. Another method is the so-called ‘wave approach’ [6, 15] which differs from the so far considered standard Kalman Filters. The basic approach of this method is to approximate the input signal as a linear combination of known basic functions fi (t) with the unknown coefficients ci : U (t) = c1 · f1 (t) + c2 · f2 (t) + · · · + cn · fn (t).

(3)

It is assumed that the coefficients ci are constant within a certain time interval. Therefore the estimation period is divided into smaller time intervals in which the system model shall be deterministic. The method as a whole can be considered as semi-deterministic. Three different classes of filtering techniques for the estimation of the gravity disturbance from airborne measurements have been presented. In terms of mathematical structure, all three filtering techniques are essentially equivalent to a low-pass filtering process in the frequency domain [6]. In terms of numerical performance, the specific theoretical layout of the filter as well as the algorithm implementation have significant impacts on the filter performance. However the main advantage of model-based filtering also always displays a deficit of this technology: On the one hand they show a higher ‘intelligence’ due to the included knowledge about the physical behavior of the process to filter, but on the other hand, this also causes an error source due to the finite accuracy of the models used. Another class of filtering is based on adaptive filters [5]. The difference between time-invariant filtering and adaptive filtering of a discrete-time signal is the time variability of the digital systems. Consider the transfer function B(z) of an FIR system that adapts according to signal conditions so that at time tk it is represented by Bk (z). The system is designed such that with noisy data as input, Bk (z) will adapt to minimize the effects of noise

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on the output. Specifically, the coefficients of Bk (z) are adjusted in order to keep the error at a minimum. 3.1.2. Preliminary work If the gravimeter signal is integrated twice, which is not the usual method to obtain the kinematical vertical acceleration, the low-pass effects of the integrators cause a decreasing noise part in the gravimeter signal without negatively influencing the possible resolution. The difference of both signals, with the dimension of a height, is a direct measure of the gravitational anomaly. This concept (figure 2) was developed about 15 years ago at the Institute of Flight Guidance and Control (patent-no. 40 13 570). For the hypothetical case that the sensors have no dynamics of their own and the measurement data are free from errors, such a system works exactly, without errors, also at strong air turbulence and flight maneuvers. The gravitational anomaly to measure can be reproduced exactly. This concept has been tested in extensive simulation investigations for different aircraft types, air speeds, gust speeds and gust wavelengths [8,12,16,17,1,2]. A disadvantage of this concept is the fact that errors of the gravimeter, as bias or scaling errors, are integrated. The low-frequency errors of the altitude measurement, which were eliminated due to a double differentiation, remain in the signal. However, investigations have shown [8,12, 16] that these kinds of errors have no influence on the determination of the gravitational anomaly. If the advantage of this approach (higher resolution) is not counterbalanced by the disadvantage of higher error sensitivity, suitable error models are necessary. The influence of dynamical sensor behavior can also be compensated completely, connecting in series a model of the complementary sensor with the original sensor (see figure 2). It is impressive that small differences between the sensor and its dynamical model cause serious errors in the gravity measurements. For typical sensor dynamics, an error in the characteristic time constant of the complementary model

Figure 2. Filter concept of the Institute of Flight Guidance and Control for airborne gravity survey.

417

of more than 1% results in useless measurements [2,10, 12,16,17]. 3.2. Improvement potentials The introduced concept (section 3.1.2) of the complementary estimation filter to determine the gravity anomalies was iteratively performed during airborne gravimetry tests. In order to reach acceptable results, there is a great potential for refinement of the complete concept (sensor dynamics and error models, estimation filter). This is the subject of this work and can be divided into two main tasks, which will be summarized next. 3.2.1. Sensor dynamics and error models A great potential for improvements lies in the development and refinement of sensor dynamics and error models for the gravimeter and the altimeter sensors, as well as in the identification of the relevant parameters. Since the accuracy requirements are extremely high, they can be reached only with non-linear, time variant models. In addition to the sensors and errors, also the data transfer interfaces, as well as the computer algorithms must be modeled. The technical problem to be solved is the time related raw data collection of the sensors, since GPS and the inertial sensors have different time relations and must therefore be synchronized. The problem of time related data acquisition is one of the most important to solve. 3.2.2. Estimation filter Crucial in the introduced filter concept (figure 2) is the influence of inaccuracies in the developed sensor dynamic and error models, as well as measurement and system noise (in the complete spectrum) which regardless affect the filter inputs. While correlated parts of both signal lines become eliminated due to the comparison, after two integrations and multiplication with the gain k1 the uncorrelated errors from the gravimeter reach the estimated anomaly value. Uncorrelated errors of the altimeter measurement provoke relatively large errors (direct proportional to k1 ) in the estimated gravity anomalies. For example, a measurement error of just 1 mm in the altimeter signal, using a feedback coefficient of k1 = 1 s−2 , causes an invalidation of the anomaly signals of 100 mGal. Realistically, this estimation filter gives a noisy signal of 5000 mGal and more. Therefore the feedback coefficients k1 and k2 must be varied depending on the quality of the signal in such a way that faulty parts of the signal have preferentially a small influence on the anomaly signal estimated by the filter. Due to the continuous adaptation of the filter, the feedback coefficients become non-linear and time variant. The goal is to develop a model-based filter, which shows the explained properties and separates the gravity information from the disturbances in this way. This model-based filter uses the prehistory of all input signals (gravimeter sensor and altimeter sensor signal) to

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improve its dynamics and accuracy; it works in a selflearning manner. For more details see sections 4 and 5.

4. The method applied During the investigations of the estimation filter explained in subsection 3.1.2, it was found that deviations from the ideal case cause non-acceptable results. The deviations due to inaccurate modeling of sensor dynamics and errors, inaccurate synchronization of the measurement signals, as well as quantization and sensor noise provoke a measurement error by determining the gravitation anomaly in the entire measurement spectrum. The measurement signals leave the sensors in digital-code and analogue form, but also as video signals (gravimeter sensor [4]). Even if high-speed computers are utilized, a noticeable amount of time is required for the sensor reaction (sensor dynamics), the data transfer, the decoding and the digitalization of the analogue signals. Therefore, the results of the calculations are available as individual measurement points, or discrete measurement point series, respectively, with different sampling frequencies. So every calculated point will contain characteristic errors. These errors in the system could be divided into rough errors, systematic errors and random errors. Rough errors are e.g. reading errors, wrong device conditions etc., which should be avoided easily. Systematic errors, as e.g. sensor drift, could widely, as far as they were recognized, be taken into account and compensated in the evaluation. The remaining errors are, in general, unpredictable random errors, which can only be taken into account using statistic methods. One part of these errors cannot be explicitly modeled and, therefore, not completely eliminated. However, the influence of these errors can be minimized due to the frequency dependent feedback coefficients. In other words, the difference between altitude measurement and twice integrated vertical acceleration will be divided in a useful (i.e. gravity anomaly) and a disturbance signal based on models using the prehistory of the measurement signals. The improved estimation filter collects instantaneous information about flight trajectory dynamics of the aircraft and the structure of changes in gravity, which are necessary to determine the weighting coefficients between preceding measurements and the actual measurement to obtain good filter dynamics, and accurate gravitation anomalies. In addition, the filter uses information about the quality of the height measurement for determining the magnitude of the feedback coefficients; therefore, it works adaptively. In case the height measurements get worse, higher frequency parts of the signal are eliminated as errors. It can be extremely hard to determine gravity anomalies under real circumstances because of non-uniform changes of gravitation (e.g. due to topography), which are nearly undistinguishable from measurement errors. As a

consequence, the prehistory of a signal can be included into the actual measurement with a specific weighting. Depending on the distribution of the weightings, either the prehistory or the actual measurement can be dominant. If the influence of the prehistory is dominant, the dynamics of the system will be reduced because the actual change in the measurement affects the indicated values very slowly. In the other case, where the actual measurement is dominant, the dynamics are very good but the indication also follows measurement errors up to outliers and it starts rippling. Finding an optimal compromise needs a precise parameter optimization for filter adjusting [4]. However, the filter adjustment has an essential influence on the accuracy and usefulness of the obtained gravity anomalies. Modeling of the dynamical behavior of the gravimeter sensor described in subsection 6.1 results in a non-linear ordinary differential equation. The coefficients are crucial in this equation, which describe the damping and natural frequency as a function of temperature and several different inner structure properties of the sensor. Starting with a second order non-linear differential equation, the solution can be found numerically. It allows a good analysis of the physical behavior of the dynamical processes. Besides formulating the mathematical model, a parameter identification to describe the characteristic change with temperature of density and viscosity of the fluid and of the elasticity characteristic of the quartz threads is also necessary. In contrast to density and viscosity, the elasticity coefficient of the quartz threads increases with temperature. The most important result of sensor modeling and parameter identification is the approximation of the sensor behavior as a first-order delay element [4]. This approximation shows a very good agreement between the model and the measurement results. The non-linearity of the highly damped system causes only small errors in the entire measurement range and was therefore neglected. The determined sensor drift (1.5 mGal/day) is highly dependent on the inner temperature.

5. The estimation filter To determine the gravity anomalies a model-based filter will be applied. For this the anomalies are interpreted as acceleration errors and estimated by the filter. This filter uses a height error as a measurement due to the coupling of the altimeter and the gravimeter sensor. It will be shown in the following how error modeling and feedback of the estimates, which were determined by a cascadetype filter, are applied to extract the gravitation anomaly signal. The height error mentioned above is the difference between the height and the measured height hRe f . The height h was obtained after the double integration of the vertical acceleration. This acceleration was measured using the gravimeter sensor (where all known disturbance

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419

Figure 3. Structure of the improved estimation filter.

accelerations, e.g. Eötvös effect and free air anomaly, are compensated for according to figure 2). The determination of the height difference (z = h − hRe f ), i.e. the height error, is shown in figure 3. From this figure the equation for the height error follows:
t
t z=



 δgd (τ ) − bm (τ ) dτ dt − hRe f

0 0


t

−


t c1

0

Figure 4. Filter feedback gains.

 δg dτ + c2 δ h˙ dt − c3 δh

6. Airborne gravity measurement (4)

0

= (h − hRe f ). t is the observation time interval. The height error z will be used as a measurement for the estimation filter. The height hRe f , which is the output of an altimeter sensor, consists of the true height and the measurement noise. The described filter has been successfully proven in airborne gravimetry tests. The behavior of the feedback coefficients of the estimation filter, on a logarithmic scale as a function of time, is presented in figure 4. It is mainly influenced by the covariance of the height error. With increasing covariance, the feedback coefficients become smaller and eventually approach zero as the height errors increase. Consequently in this case, the actual measurement values are slightly weighted, or can be ignored completely. For comparison, the results of two different test flights are shown. The upper figure presents the feedback coefficients for a very calm height measurement with relatively constant accuracy. The lower figure instead shows the behavior during a varying accuracy in the height measurement. The number of satellites in the current field of view, the geometry of the current satellite constellation, and the influence of the distance between aircraft and reference station cause the inaccuracy.

6.1. Gravimeter system A Russian AeroGravimeter ‘Chekan-A’ forms the basis of the Dornier Do 128-6 airborne gravimetry system, which was originally designed to be installed in submarines. It was developed at the Central Scientific and Research Institute ‘Elektropribor’. The Chekan-A includes highly precise sensors, which are installed in a gyro platform. It implements a two-axis platform equipped with two-axis floated gyroscopes, a gravimeter sensor and three floated accelerometers. However in order to apply the submarine system for airborne surveying, the original hardware such as the electronics and interface were considerably modified. In order to create an accurate-leveling platform, a highly sensitive digital loop was designed and utilized [17]. The gravimeter sensor consists of two identical quartz systems unwrapped from each other by 180 degrees. Each system consists of a quartz framework with a quartz torsion fiber, to which a pendulum with trial mass and mirror are welded. Torsion fibers are previously twisted so that the positions of the pendulums are close to horizontal. The torsion angle variation is a measure of the acceleration increment. Use of the double system with high identity of parameters and exact mutual orientation practically excludes the cross-coupling effect. This effect is inherent in the pendulums due to horizontal and oscil-

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lating acceleration. The internal casing of the gravimeter sensor is filled with a ‘polymethylsiloxane’ liquid. The necessary damping of the pendulums is provided by a choice of viscosity of the liquid (for more details see [4, 17]). 6.2. Test results and analysis Flight experiments in October 1999 (over Magdeburg) were carried out with a twin-engine turbo-prop Dornier 128-6 research aircraft, which belongs to the Institute of Flight Guidance and Control of the Technical University of Braunschweig. The flight profiles were selected on the criteria that high variability of the gravity disturbance was probable. Ground gravity in the area is well known (figure 9). The tests were designed to assess repeatability and accuracy of the airborne gravimetry. East-west and north-south profiles of approximately 100 km across Magdeburg were chosen. The flying altitude was roughly 150–300 m and the average flying speed was about 210 km/h. For the flights, four GPS-NovAtel receivers were used. Two of them were installed in the aircraft and two were deployed on the ground to serve as base stations. The duration of one flight was about three hours. 83 flight hours were performed in 1999. Before processing the flight data, the computation of the kinematical GPS data using code and phase measurements simultaneously in a differential mode was carried out. In the second step, the Eötvös term, normal gravity and the systematic errors are removed from the sensor’s raw data. The filtering process itself (figure 3) eliminates the kinematical accelerations and gives an output signal which will be corrected from the errors due to the influence of the topography and the Bouguer’s plate [6]. The specific system errors, which are difficult to determine, and the random parts remain uncorrected. These errors are essentially due to the noise of the sensors used to construct the system measurement. In spite of not taking into account the last mentioned errors, the corrected signal primarily presents the gravity anomalies searched for (figure 5). The graphic shows the behavior of an anomaly

Figure 5. Comparison of the estimated anomalies for two flight lines in opposite direction (Flight Line N1, see figure 9).

along a specific route partly flown once in north-south direction and flown once in south-north direction. A phase shift results, which is caused by the filter algorithm. If the quality of height measurement remains constant, the phase shift also remains constant. The value of the phase shift here is approx. 0.06◦ or 6.5 km respectively (the total phase shift between both flight lines equals two times the system time constant, approx. 0.12◦ or 13 km). If the phase shift is taken into account a good reproduction of the anomaly results. The higher frequent parts shown in the figure are caused by error influences. These errors also include kinematical residues (Phygoide), which cannot be removed even when using high quality GPS carrier phase evaluation. It should be noted that the controlled aircraft has a Phygoide wavelength of 900–1000 m [3]. There are three types of errors: gravimeter sensor errors, altimeter errors and system integration errors. Thus, the error model for scalar gravimetry can be simplified to: gErr = gGrav + gAlt + gSync .

(5)

Each term of this equation (5) has a distinctive spectral signature. In other words, the largest effects of the different error terms occur in different frequency bands. The first term accounts for the gravimeter sensor error. This term represents the effect of attitude errors on the gravity disturbance error. Attitude errors are the combination of initial alignment errors and integrated gyro errors. The gyro errors include non-orthogonality of the gyro axes, scale factor errors, linear and nonlinear gyro drift. The total error is the sum of the attitude errors, correlated random errors, quantization errors and outliers. Other sources of error are mainly caused by gravimeter sensor bias and scale factor. The gravimeter sensor bias slowly changes in time and therefore its frequency spectrum will considerably overlap into the gravity disturbance frequency spectrum. The second term in equation (5) represents the altimeter errors. By use of GPS carrier phase measurements the main sources of error are measurement noise, multipath effects and some systematic effects due to the residual atmosphere and orbital errors. Thus the dominant errors affecting the GPS measurements are short-term errors due to the multi-path effects and GPS measurement noise. These two error sources have different spectral characteristics and they effect the gravity disturbance determination. Therefore it is dependent upon bandwidth. Long-term errors do not have a significant effect on the accuracy of gravity disturbance determination. It is common knowledge that gravimeter sensor errors tend to increase with time but have short-term stability. In contrast to these, the GPS error does not grow with time and is characterized by a low updating rate and relatively large short-term errors. In other words, the specific force error, due to the gravimeter sensor, dominates the lower end of the frequency band while GPS error dominates the higher one.

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Figure 6. Standard deviation of the estimation accuracy.

Figure 7. Filter output signal with reduced resolution.

The last term in equation (5) represents the effects of time synchronization misalignment between the gravimeter sensor data stream and the altimeter data stream. This term illustrates that both translation and rotational motion contribute to the error. As a consequence, it is the interaction between the size of the time synchronization error and the aircraft dynamics that reflects the real effect of time synchronization misalignment on the gravity disturbance estimation. The estimation filter not only gives the obtained anomaly but it also gives information about the accuracy of the individual estimated values. The information is given as a covariance (expectation value of the estimation error). In figure 6 (upper) the standard deviation of the anomaly, which is also shown graphically in figure 5, is presented. In figure 6 (lower) the behavior of the standard deviation of the obtained anomaly signal for the complete flight as a function of time is shown. On average, the standard deviation of the estimated value is about 3 mGal with a resolution of 2 km. If the resolution of the wavelength is decreased to 5–6 km the standard deviation can be improved to 1 mGal (see figure 7). The verification of the presented methods will be completed using a stationary measured reference of the investigated area. In addition, the methods were applied to data from another research group and compared with their results. The investigated estimation filter shows results that are more encouraging [4].

Figure 8. Comparison of the obtained anomaly with stationary values from Gravity Map of Germany, Bouguer Anomalies, Sheet North, Issue 27 [13] (flight route N1).

Compensating for the phase shift from figure 5, the behavior of the anomaly ensues that approximately represents the local mean value of the measured behaviors. The part of the flight route (N1) discussed here is plotted into the gravity map (figure 9). The anomaly values along this flight route were compared with the estimation filter results (figure 8). The averaged anomaly agrees well with the stationary measured values. Figures 10 and 11 show, analogous to figure 8, the obtained and averaged anomaly behavior for other flight routes of the surveyed area. The agreement with the

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stationary obtained reference can be confirmed here as well. Within this field of study, there are numerous problems which remain. The following mentions areas containing many smaller, less extensive troubles.

Figure 9. Section taken from Gravity Map of Germany, Bouguer Anomalies, Sheet North, Issue 27 [13].

Figure 10. Comparison of the obtained anomaly with stationary values (flight route W1 and W2).

Figure 11. Comparison of the obtained anomaly with stationary values (flight route N2, N3, N4 and N5).

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6.2.1. Height measurement Despite all efforts, the barometric sensor (statoscope) could not be applied successfully in airborne gravimetry. The error identification and error modeling that were carried out aim to introduce this sensor to the mentioned accuracy. The modeling of the thermodynamics of the reference pressure is difficult, as well as the calibration of the barometric pressure due to the flow around the aircraft and the pressure fluctuation of atmosphere. The questions regarding the functional context between the atmospheric pressure, the actual flight height and the necessary identification of the corresponding parameters are completely unsolved. Furthermore, the power spectral density analysis shows that GPS carrier phase measurement and statoscope have an almost identical behavior at frequencies smaller than 0.1 Hz. Airborne gravimetry as a tool for local gravity measurements is primarily based on differential carrier phase measurement of GPS. If a flight route has a length of more than 100 km, multiple reference ground stations are necessary to maintain the differential carrier phase measurement. This in turn causes additional requirements regarding the processing of the reference data. 6.2.2. Sensor dynamics The dynamical properties of the sensors have the greatest influence on the measurement of gravity anomalies. So the dynamical models developed should be refined using flight maneuvers. The greatest problem of dynamical modeling of the gravity sensor is the so-called ageing process of the sensor components. Due to the change of spring and fluid properties, the time constant of the sensor changes constantly. 6.2.3. Interaction of the sensors The major problem of the interaction of the sensors as a part of the multi-sensor system gravimeter is the communication of all components of the measurement line with each other. This includes the dynamical behavior of all hardware components, especially A/D and D/A converter. The discretization processes and the induced noise must be modeled and filtered. The high precision sampling of individual measurement values and the following time synchronization are basic requirements for the described measurement approach. 7. Conclusion In the present work we have tried to make a contribution to the determination of gravity anomalies on board an aircraft. Primarily the view was directed to technical requirements and boundary conditions with the goal of developing a suitable filter structure and sensor error models which do justice to flight dynamics. Further fields of problems connected with airborne gravimetry

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could be exposed. From this it could be shown that in order to increase accuracy and resolution of airborne gravimeter new approaches have to be developed. Instead of low-pass or band-pass filters, methods of complementary filtering must be applied. For this, a modelbased filter approach was developed, where the signal of the gravimeter sensor was integrated twice and compared with the height measurement signal. The difference is a direct measure, for the gravity anomaly was contaminated by system errors. First investigations also showed that, by using ideal sensors, the influence of air turbulence and flight maneuvers on the anomaly signal could be completely removed. In addition, the investigations were extended to the application of realistic sensors. The filter shows good dynamics and its operation in the low frequent range can be understood as a steep sloped lowpass filter. With the developed concept, the determination of gravity anomalies of a wavelength below 20 km could be improved. Parallel to the theoretical work, a Russian sea gravimeter was acquired, modified and successfully put in operation in the aircraft. The flight tests carried out gave concrete results about gravity anomalies and confirmed the correctness of the presented approach. Also statements could be made about the accuracy obtainable with presently available sensors. The investigations have given information regarding which minimum requirements for the single components of the measurement system are necessary to guarantee measurement accuracy. Due to the complexity of the measurement system (aircraft and sensors) non-linear system modeling was applied. One difficulty that was encountered was the process of combining the sensor signals mentioned above while the aircraft was in a highly dynamic motion. Already in the case of small time shifts of the signals relative to each other, the accuracy of the results got decisively worse. Therefore, especially the acceleration and height signals must be collected at the same time and with a high scanning frequencies (> 10 Hz). When combining both signals a precise knowledge of the respective sensor dynamics is necessary. In addition to the mentioned difficulties and influences on measuring gravity anomalies, the location of the sensors inside the aircraft must be determined precisely. The application of the developed concepts in flight tests has shown good agreements between the results and a stationary reference. Also the reproduction has been proved to be correct. The resolution of the determined anomaly length reached at present lies in the range of 5–6 km with a standard deviation of 1 mGal. At a resolution of 2 km the standard deviation is about 3 mGal. Also a direct comparison with methods of other working groups confirms the high potential of the presented approach [4]. The investigation of the sensor errors led to some interesting conclusions. The height sensor shows a much more complicated error behavior than the gravimeter sensor. The investigations to date have shown that, essentially, the problem of achieving highly accurate altitude mea-

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surements has not been satisfactorily solved. Further need of research exists concerning the interaction and the coordination of the individual sensors. References [1] Abdelmoula F., Probleme und Lösungsansätze bei der Fluggravimetrie. 1. Braunschweiger Symposium für Flugmesstechnik, Braunschweig, 1998. [2] Abdelmoula F., Neue Konzepte und Lösungsansätze bei der Fluggravimetrie, Geodätische Woche 98, Kaiserslautern, 1998. [3] Abdelmoula F., Design of an open-loop gust alleviation control system for airborne gravimetry, Aerosp. Sci. Technol. 3 (6) (1999) 379–389. [4] Abdelmoula F., Ein Beitrag zur Bestimmung der Erdbeschleunigungsanomalien an Bord eines Flugzeuges, Verlag Shaker, 2000. [5] Bruton A.M., Schwarz K.P., Airborne Gravity Estimation Using Adaptive Filters, KIS97, Banff, Canada, 1997. [6] Hammada Y., Schwarz K.P., Airborne Gravimetry ModelBased versus Frequency-Domain Filtering Approaches, KIS97, Banff, Canada, 1997. [7] Hammada Y., A Comparison of Filtering Techniques for Airborne Gravimetry, MSc Thesis, Departement of Geomatics Engineering, University of Calgary, 1996. [8] Huck V., Untersuchungen zur Bestimmbarkeit von Erdbeschleunigungsanomalien aus Messungen der Vertikalbeschleunigung in einem Flächenflugzeug mit Hilfe eines Schätzfilters zur Systemnachbildung unter Berücksichtigung einer Realitätsnahen Systemsimulation, Diplomarbeit, Institut für Flugführung, 1988.

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