FACULTY OF ENGINEERING AND SUSTAINABLE DEVELOPMENT
Department of Computer and Geospatial Sciences
P.G.C.C.Fonseka 2020
Degree project, Advanced level (Master degree, one year), 15 HE Geomatics
Master Programme in Geomatics
Scientific Supervisor: Jonas Ågren External Supervisor: Per-Anders Olsson
Examiner: Mohammad Bagherbandi Co-examiner: Andrew Mercer
Evaluation of airborne and marine gravity data over Kattegat region
Subtitle of your thesis, if any
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Cover picture: ZLS D-13 Marine gravity meter. Photo: Chrishan Fonseka, 2020.
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Preface
This research work would not have been successful in this short without the timely and valuable guidance, inspiration, technical discussion, support, and supervision of Dr. Per-Anders Olsson (Lantmäteriet) and Dr. Jonas Ågren (Högskolan i Gävle).
Their in-depth knowledge, vast experience, noble intellectual, continuous enthusiasm for research has helped me immensely and inspired to work further.
They have also always been accessible, approachable, and willing to help besides their own tight schedules.
Special thanks to Mr.Josefsson Örjan (Lantmäteriet) for his guidance and support in operating the ZLD-D13 dynamic gravimeter owned by Lantmäteriet and Dr.
Mohammad Bagherbandi, the examiner, for his valuable inputs and feedback for this report.
I am very much grateful to all my friends and family, specially to my wife M.I.N.
Fernando, who have given their moral support throughout the journey of life.
Special acknowledgement for the Swedish Institute Scholarships for Global Professionals (SISGP) scholarship program for opening the avenue towards a sustainable future.
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Abstract
The Kattegat sea region between Denmark and Sweden is identified as a region both economically and environmentally sensitive. Statistics indicates that over two thousand vessels per day navigate in the region. Navigation route optimization for the region is vital for efficient transportation. Optimized routes allow a vessel to carry the maximum amount of goods per course leading to efficient fuel
consumption, which can greatly benefit in an economical and environmental aspect.
Such optimization requires a highly accurate and reliable vertical reference surface for efficient transportation. In the Baltic Sea and Kattegat, a geoid is now used as such a surface. For geoid modelling, homogenous and reliable gravity measurements are required over a larger area surrounding the computation point. The Kattegat region consists of gravity data mainly from the Swedish Fyrbyggaren marine campaign 2019, Kattegat airborne campaign 2018 and several older datasets from the Nordic Geodetic Commission (NKG) database. These gravity data over Kattegat region have been measured using different instruments in various time epochs that inherit them with uncertainties depending on the platform type, instrument sensor type, filter type, corrections applied, processing software and many other
parameters. In this study, the data uncertainty of gravity measurements from various sensors in the Kattegat region was studied through statistical and graphical
evaluations. It was found out that the data from Kattegat airborne campaign 2018 deviate systematically with from the more reliable Fyrbyggaren marine campaign 2019 and other marine datasets. The airborne campaign was therefore tentatively corrected by the estimated shift +1.46 mGal before further analysis was made of the other datasets. It is found that NKG publication numbers 29, 42, 44, 610, 611 and 616 from the NKG gravity database have a standard uncertainty of around 2-3 mGal. Which is within the range of allowable uncertainty for future applications.
These datasets may thus positively contribute to NKG database along with data from the Swedish Fyrbyggaren marine campaign 2019 and the shifted Kattegat airborne campaign 2018. These datasets should be used to model the geoid over the region in the future.
Key words: Airborne gravity, Marine gravity, FAMOS, Kattegat Sea, Geoid
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Table of contents
Preface ... 3
Abstract ... 4
Table of contents ... 6
1 Introduction ... 8
1.1 Finalizing Surveys for the Baltic Motorways of the Sea (FAMOS) project ... 10
1.2 Problem statement, aim and objectives ... 11
1.2.1 Main aim ... 12
1.2.2 Individual research objectives ... 12
1.2.3 Limitations ... 12
1.3 Previous studies ... 12
1.4 Previous studies over the study area ... 13
2 Theory and central concepts ... 15
2.1 Gravity ... 15
2.2 Gravity measurements ... 15
2.3 Gravity reduction ... 16
2.3.1 Elevation correction (free air reduction) ... 16
2.3.2 Bouguer and terrain corrections ... 17
2.4 Normal gravity field and free air gravity anomaly ... 18
2.5 Geoid ... 18
2.6 Quasigeoid ... 19
2.7 NKG2015 geoid model and NKG gravity database ... 21
3 Methodology ... 24
3.1 Gravity data ... 24
3.2 Software and programs ... 24
3.3 Data Processing ... 25
3.3.1 ZLS-D13 Dynamic gravimeter ... 25
3.3.2 Processing of ZLS measurements ... 25
3.3.3 iMAR Strapdown gravimeter ... 27
3.4 Workflow ... 27
4 Results and discussion ... 31
4.1 Fyrbyggaren west Kattegat 2019 marine gravity campaign ... 31
4.1.1 Harbor ties (step 3)... 31
4.1.2 Internal crossovers (step 4) ... 32
4.1.3 External crossovers (steps 5, 6 and 7) ... 34
4.1.4 Comparisons with global geopotential models and with the regional NKG2015 free air gravity anomaly grid (step 8) ... 38
4.2 Kattegat airborne campaign (steps 10 and 11) ... 39
4.2.1 Kattegat airborne campaign 2018 (iMAR) as a reference data set (step 11) ... 41
5 Conclusions ... 43
5.1 Future work ... 43
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5.2 Ethical Aspects ... 44
5.3 Aspects for sustainable development ... 44
References ... 46
Appendix A ... 51
Appendix B ... 52
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1 Introduction
The Geodetic Glossary (NGS 2009) defines Geodesy as “The science concerned with determining the size and shape of the Earth”. Fundamentally Geodesy is subdivided into three main disciplines based upon its focuses. According to Vanicek & Krakiwsky (1986) they are:
• Geometrical Geodesy, which is focused on determining locations in a geometrical aspect and defining primary products such as reference and coordinate systems.
• Physical Geodesy, which is concerned with Earth’s gravity field and the mathematical representation of the Earth’s gravity field necessary for defining and computing vertical reference surfaces such as geoid models and deflections of the vertical.
• Satellite Geodesy, which is focused on products of orbiting satellites launched for geodetic purposes.
This study falls under the category of Physical Geodesy, where the focus is the determination of Earth’s gravity field. Gravity has been a hot topic for more than four centuries. The most well-known concepts of gravity among the public are Isaac Newton's law of universal gravitation (1687) and Albert Einstein’s explanation of gravity in the Theory of General Relativity (1915). From a geodetic point of view, gravity is considered as the combination of gravitational and centrifugal forces. In modelling the gravity field of the Earth, the gravity potential is frequently used as it is a scalar quantity. The potential approach makes it less complex to address the issues of finding the gravitational field for a given density distribution and compute the gravitational field outside a defined volume from observations on its boundary surface (Torge, 1989). Gravity measurements are widely used in physical geodesy for the purpose to precisely determine the geoid, which is used as a physically meaningful vertical reference surface. The geoid is defined as the equipotential surface the Earth’s gravity field that best approximates mean sea level over the oceans at a certain epoch (Torge & Müller, 2012). The Swedish national geoid model is SWEN17_RH2000 (Jivall & Ågren, 2016), which is regularly used for height determination in the Swedish national height system RH 2000 using GNSS (Global Navigation Satellite Systems). This model is really a quasigeoid model (see section 2.5), but as the difference between the geoid and quasigeoid is practically zero close to sea level (like everywhere in the Kattegat area), we do not consider the difference between geoid and quasigeoid in this thesis.
Having a highly accurate geoid model over a region have many advantages as it is needed for height and depth measurements making use of GNSS. The Baltic Sea serves as major transportation hub connecting Europe’s North and East parts by marine shipping. One of the fundamental elements of the marine navigation
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infrastructure are charts specifying the water depth. By improving the depth data and their density by hydrographic surveying and mapping, safety is increased for the Baltic Sea region. This region has among the world’s highest marine traffic density, which is crucial for the global and European economy.
As uncharted territories in the Baltic Sea region were charted in the FAMOS project (see Section 1.1 below), depth information will be improved, optimum routes for marine vessel navigation can be designed and old routes can be updated.
Optimization of routes improves efficiency in both environmental and economic aspects mainly by the reduction of fuel consumption and emission. Under Keel Clearance (UKC) awareness allows for safer navigation through shallow waters.
Accurate determination of UKC benefits the economy and the environment through reduced fuel consumption and increasing the amount of goods that can be carried per journey. All these expected outcomes are directly linked in a technical aspect to being able to determine the vertical position of the ship in the used chart datum. As the geoid is now about to be introduced as the reference surface for depths in the Baltic Sea, which is called the Baltic Sea Chart Datum 2000 (Ågren, Liebsch, &
Mononen, 2019), an accurate model of the geoid is of utmost importance for marine transportation in the Baltic Sea. Hence, study on the quality of the gravity data is important to achieve this goal.
In Physical Geodesy, gravity is studied as the force acting on a unit mass. As this is numerically the same as acceleration, the unit of measurement is taken as the same as for acceleration. Usually the unit Gal is used, which is defined as 1 Gal = 0.01 ms-2. Gravity measurements with an uncertainty acceptable for geodesic applications was first made in 1792 by Borda and Cassini using a pendulum (Krynski, 2012). The gravity variation mainly depends upon geographical latitude, height, geology
(density distribution), and topography (Vanicek and Krakiwski, 1986). At present, there are mainly two approaches to measure gravity. One is absolute gravity measurements based upon ballistic phenomena, and the other is relative gravity measurements based on Hooke’s law 1660 (Brown, 2000). A relative gravimeter contains a spring that supports a proof mass. When the surrounding gravity changes, the force acting upon the proof mass will change resulting in a change in the length of the supporting spring. The magnitude of the external force required to bring back the proof mass to its initial position is used to determine the gravity value at the location in question (Torge, 1989). Apart from geoid determination, due to heterogeneity of earth’s structure, gravity can be used for many applications such as standardization and establishment of reference systems (Drewes, 2009), studies on geodynamic motions such as postglacial land uplift (Hill, Davis, Tamisiea, &
Lidberg, 2010;Olsson et al., 2019 ), geophysical applications such as oil and mineral exploration, investigation of buried streambeds or caves (Zhang, Qiao, Zhao, &
Lan, 2019), military applications and many more. The Swedish gravity reference
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frame is RG 2000 (Engfeldt et al., 2019). It has been established based on both absolute and relative gravity measurements.
1.1 Finalizing Surveys for the Baltic Motorways of the Sea (FAMOS) project
The Finalizing Surveys for the Baltic Motorways of the Sea (FAMOS) project was co-funded by the European Union (EU). The project started in 2014 and ended by 2019. It was a collaboration of around 15 organizations in the countries around the Baltic Sea. Besides the Nordic and Baltic countries, Germany also participated. The overall purpose of the project was to improve safety and efficiency of marine
transportation in the Baltic Sea. To achieve these goals several activities were made, hydrographic surveying, improving vessel navigation for the future, surveying infrastructure and data workflow from sounding to chart. The activity most relevant for the current research project is the second one aiming at improving vessel
navigation for the future. The work here mainly focused on improving the marine gravity dataset and gravimetric geoid model over the Baltic Sea region.
This thesis project is concerned with evaluating the gravity data situation in the Kattegat sea region, making use of marine gravity data collected in the FAMOS project. The most important gravity datasets in Kattegat are the following:
• The Swedish Fyrbyggaren west Kattegat marine campaign 2019, which was observed by Lantmäteriet (Swedish mapping, cadastral and land registration authority) using their ZLS D-13 gravimeter (ZLS corporation, 2017) onboard the Swedish Maritime Administration’s vessel M/S Fyrbygaren.
• Kattegat airborne campaign 2018, measured by Lantmäteriet and DTU Space (National Space Institute at Denmark’s Technical University) using the ZLS D-13 and the iMAR strapdown gravimeter owned by DTU Space (Jensen, Olesen, Forsberg, Olsson, & Josefsson, 2019).
• Old gravity datasets in the Nordic Geodetic Commission (NKG) database over the Kattegat region. The NKG gravity database contains gravity data both on land and at sea that have been collected over many years (Märdla et al., 2017; Ågren et al., 2016).
The quality of the old marine NKG data in Kattegat is mainly unknown. It is an important task to evaluate the quality of these old datasets and to investigate how the above three datasets should best be combined. This is the main task of this study.
The study area is Kattegat sea region between Sweden and Denmark; see Figure 1.
The bounding box of the area of interest is:
Latitude: 55.5° to 59.5°
Longitude: 9.5° to 13.5°
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Figure1: Kattegat sea region between Sweden and Denmark
1.2 Problem statement, aim and objectives
The available gravity data over Kattegat region consists of observations from various instruments measured at different times. Hence, the accuracy and quality of the data varies depending on the instrument and method used for observation ultimately leading to data uncertainties. To evaluate these uncertainties, so that the data could
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be merged and usable for creating geoid models and in other applications, it is vital to determine the uncertainty of the different gravity datasets. Low quality data should either be eliminated from the existing databases, corrected, or down weighted in geoid computations.
1.2.1 Main aim
The aim of this study is to estimate the quality and propose suitable gravity datasets (Section 1.1) from various sources over Kattegat region for future applications such as geoid modelling.
1.2.2 Individual research objectives
• Analyze the computation of the latest Swedish Fyrbyggaren marine campaign west 2019 in detail over Kattegat region to estimate its uncertainty in a reliable way.
• Evaluate the Kattegat airborne 2018 campaign mainly by comparing with the above Fyrbyggaren west Kattegat 2019 marine campaign gravity data.
• Evaluate the old marine gravity data sets from Nordic Geodetic Commission (NKG) database mainly by comparing with the above two datasets.
1.2.3 Limitations
When reducing airborne gravity data to the reference surface only the free air downward continuation method is followed in this thesis. Due to the limited time available there is no time to investigate other methods such as the gradient method (LaFehr, 1991) or least square collocation (Märdla et al., 2017). This is left for the future.
1.3 Previous studies
The study done by Strykowski et al. (1996) presents high precision marine gravity data over Greenland waters. Acquisition and processing of marine gravity data collected from 1991 to 1995 is discussed in this study. Just as the ZLS D13 sensor which had been used to acquire gravity data over Kattegat sea this study also uses a LaCoste and Romberg type gravimeter which is literally the same. Though in the era of where this study was conducted, advanced GNSS techniques were not in practice, they have used the P-code of GPS, which is dedicated for military purposes, for position estimation. The study does not provide any information related to position accuracy. The authors claim that the root mean square of the crossover points is less than 0.5 mGal hence the accuracy is high. They also state that the key to obtaining such high accuracy results is slow vessel speed and precise GPS navigation.
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Prince & Forsyth (1984) presents a method minimize crossover errors without modelling the source of the errors. They have taken a least square approach to model the crossover differences. The study proves that in this method the root mean square error (RMS) of the crossover errors had been reduced from 10.3 mGal to 2.9 mGal after segmentation corrections are made. Furthermore, by estimating individual track line segment correction 87% of the variance of the original crossover errors is explained such that 37.5% is because of constant difference between cruises and 49.5% is due to variations associated with each segment.
Denker & Roland (2005) provides an approach to unify and reduce inconsistencies in shipborne gravity data collected from various sources over Europe. This was done by crossover adjustments for individual ship tracks. The results of this study indicate that the RMS of the crossover differences from the original data set is 15.5 mGal, 8.4 mGal from the edited data and 4.7 mGal from the from the final adjusted data.
Sokolove (2011) describes practical issues faced in airborne gravimetry, especially in separating the gravitational and inertial accelerations and how to separate them including improving the accuracy of field measurements. The study presents various techniques and equipments along with processing algorithms supporting the cause.
The author repeatedly states the main advantage of airborne gravimetry as the possibility of measuring gravity over regions physically inaccessible. They further expresses that to improve the measurement resolution, flight altitude in the phase mode must be determined accurately which rely on the accuracy of the satellite measurements, instruments must be improved to reduce instrument dependent errors and methods and procedures for data processing must be improved focusing gravitational and inertial accelerations.
In fields of Geodesy, Geophysics, Navigation, Mineral and hydrocarbon expeditions gravity measurements play a major role. Currently the most common sensor type for such airborne and shipborne gravity measurements is relative sensors. Bidel et al.
(2018) provides somewhat an uncommon approach where they use an absolute gravity meter which uses atom interferometry onboard a ship for gravity measurements. The authors claim that they have achieved a precision below
0.00001 meters per second and the comparison of results with a commercial spring gravimeter have proved a better performance. This study opens a perspective towards new avenue where absolute gravity meters based on atom interferometry are used on dynamic platforms for gravity measurements.
1.4 Previous studies over the study area
Jensen et al. (2019) evaluate the performance of both ZLS D13 and iMAR sensors based on gravity data acquired over Kattegat Sea region between Sweden and Denmark. The two fundamental challenges of airborne gravimetry are
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differentiating the kinematic acceleration induced by aircraft movement and estimating the sensor orientation parameters during aircraft dynamics (Sokolov, 2011). Jensen et al. (2019) study state that the iMAR sensor comes with a temperature stabilization unit for the inertial measurement unit whilst the ZLS sensor contains a spring-type platform-stabilized gravity system. The authors further explain the advantages of having a strap-down configuration such as higher dynamic range and resolution, high resolution navigation solutions, economic advantages and all the drawback of mechanical platform such as smaller size, less power
consumption along with fail rate and many more. Several static recordings have been done at the lab to investigate the performance of iTempStab-AddOn for 42 hours and the temperature variation was found a 0.004°C. And, the error terms quantization noise, velocity random walk, bias instability, acceleration random walk and drift rate ramp were determined. In the data processing step, a bias adjustment and a trend adjustment had been done for the data from iMAR and ZLS for
interpolation purposes. The final results of comparisons are presented in terms of no. of crossings, mean, standard deviation, minimum, maximum, root mean square and root mean square error. In conclusion, the statistics prove an accuracy of 1.0 mGal for the iMAR strap-down system and 1.8 mGal for the ZLS platform system.
Lu et al. (2019) present data processing strategies and preliminary geoid determination tests done over the Baltic Sea region. The authors state that the marine gravity observations are the most accurate among other techniques such as satellite gravimetry, airborne gravimetry, altimetry etc. The sensor used for this study is Chekan-AM which is a double quartz elastic system. The data processing strategies have been decomposed into several steps. Processing of gravity meter recordings, estimation of GNSS kinematic trajectories and vertical accelerations, applying of a low pass filter to eliminate the disturbing high pass signals. The results of this study indicate a 0.76 mGal of a root mean square error between crossover points and an accuracy of 0.5 mGal for the Baltic Sea region
These two studies comprehensively explain the data acquiring and processing methods for airborne and marine gravity campaigns over the study area in order to contribute to the gravity databases.
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2 Theory and central concepts
2.1 Gravity
Gravity is a physical phenomenon that can be expressed as the combination of Earths’ gravitational force and the centrifugal force acting upon a body at rest on the Earths’ surface (Heiskanen & Moritz, 1967 ). If the Earth would have been a
homogeneous perfect sphere without rotation, the gravity at every surface point would be the same. But due to phenomena such as ellipsoidal shape, rotation, irregular surface and internal mass distribution, gravity will vary along the surface of the Earth. The concepts of gravity are mainly described through Isaac Newton's universal law of gravitation and the second law of motion. According to Newton’s universal law of gravitation, the gravitational force between any two point masses is denoted by:
Figure 2: Attractive force acting between two masses
𝐹1 = 𝐹2 = F =G𝑚1𝑚2
r2 (1)
where 𝑚1 and 𝑚2 are the masses of the two bodies, r is the distance between the centres of the two masses, and G is the universal gravitational constant, which is equal to 6.673 x 10-11 m3 kg-1 s-2.
Gravity measurements are expressed in terms of acceleration. The international system of units (SI) defines the measurement unit of gravity as meters per second squared (ms-2). In centimeter gram second (c.g.s) system, the unit of gravity
measurement is referred to as Galileo (Gal), which is defined according to 1 Gal = 1 cms-2. The unit normally used for gravity surveys is milli Gal (mGal). Gal is defined as the magnitude of the gravitational field or unit of acceleration that will act on a mass of 1 g with a force of 1 dyne. Average gravity on the Earth’s surface is 980 Gal and from equator to pole it varies approximately 5.17 Gal (Haldar, 2018).
2.2 Gravity measurements
The main two types of gravity measuring instruments are absolute gravimeters and relative gravimeters. Starting from 35 mGal of uncertainty for the first absolute gravity measurements measured from reversible pendulums, the modern FG5
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absolute gravimeter provides measurements with an uncertainty of around 1 µGal under laboratory conditions (Krynski, 2012). Due to their comparative lack of mobility, absolute gravimeters are used commonly to establish fundamental benchmarks for gravity networks, whilst relative gravity measurements with instruments equipped by elastic springs are more popular for carrying out gravity surveys due to their robustness and mobility (Hwang, Wang, & Lee, 2002). Vehicle based (dynamic) gravity measurements are usually carried out with relative
gravimeters mounted on precision gyro-stabilized platforms. Two main measuring vehicles are typically used, airborne measurements from aircraft and marine gravity measurements form surface vessels (Sokolov, 2011). The airborne method has several advantages such as the accessibility of regions which are hard to approach by terrestrial or marine gravimeters, such as mountain ridges, water, and land
boundaries and polar caps (Johnson, 2009). But when comparing with marine measurements, airborne measurements have larger uncertainties due to platform perturbations from air currents (Mimset al.,2009). A large ship will not have a greater impact by chop and swell on a calm day.
The Kattegat region between Sweden and Denmark, which is the area of interest for this study (see Sections 1.1 and 1.2), is covered by gravity data from both types of platforms. The Kattegat airborne campaign 2018 gravity survey data consists of data acquired by the two different sensors ZLS-D13 and iMAR (see below). Fyrbyggaren west marine campaign 2019 data is acquired by ZLS-D13 sensor on the vessel
Fyrbyggaren. The Nordic Geodetic Commission (NKG) database contains old marine gravity data over Kattegat region from mainly unknown sensors as well as terrestrial gravity data collected by relative land gravimeters like Scintrex CG-5(Reudink, 2014).
2.3 Gravity reduction
A difference exists between gravity measured on the physical surface of the earth, or at flight altitude, and gravity referring to the geoid (or sea level). A reduction is required to transfer gravity down to sea level (Hackney & Featherstone, 2003). The reduction method may differ depending on how the topographic masses above sea level are dealt with. Gravity reduction is essential for determining the geoid, for gravity interpolation/extrapolation and for investigating the Earths’ crust. The main two steps of gravity reduction are removing the topographic masses outside the geoid, or shifting them below sea level, which requires knowledge of the density of the topographic masses, and reducing the gravity station from the Earths’ surface to the geoid (Featherstone & Dentith, 1997; Featherstone & Kirby, 2000).
2.3.1 Elevation correction (free air reduction)
According to Newton’s law in Eq. (1), when the distance increases from the centre
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of the Earth, gravity decreases at the surveyed point. The free air reduction to the geoid means increasing gravity using the normal gravity gradient (+0.3086 mGal per meter above sea level; see e.g. Heiskanen & Moritz, 1967, Märdla et al. 2017) as if there were no topographic masses above the geoid. Accordingly, calculations assume that the space between the observation point and at sea level contains “free air”.
𝛿𝐹 = 0.3086ℎ 𝑚𝐺𝑎𝑙 (2) Where,
𝛿𝐹-free air reduction
ℎ-elevation of the observed gravity point in meters
This correction should also be applied to the measured gravity both if the survey station lies above or below the geoid.
2.3.2 Bouguer and terrain corrections
The effect of gravitational pull associated with the topography are usually removed by two different corrections. The simple Bouguer correction means that the masses above sea level are represented by an infinite horizontal slab whose thickness is the height of the station above the geoid (see Figure 3). The complete Bouguer
reduction is the sum of the simple Bouguer correction and the terrain correction, where the latter models the attraction difference between the real shape of the surrounding terrain and the planar Bouguer plate.The complete Bouguer gravity anomaly helps us to achieve no-topography gravity data. Sometimes, one speaks of the expanded Bouguer reduction when a spherical Bouguer plate and the
corresponding terrain correction is used (Ayala et al., 2016). Because of extra mass beneath an object located at a high elevation, the Bouguer effect (opposite sign to correction) produces a higher amount of gravitational acceleration. The Bouguer effect can even be used as an approximation of gravity value at very high elevations, but it should be noticed that the Bouguer effect is to a larger large extent
compensated by isostatic effects (e.g. Sjöberg & Bagherbandi, 2017). The simple Bouguer correction can be computed using the following equation (Hofmann &
Mortiz, 2006):
𝛿𝑔𝐵 = 2𝜋𝜌𝐺𝐻 (3) Where,
𝛿𝑔𝐵- Bouguer reduction 𝜌- density of the material
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𝐺- constant of gravitation
𝐻- thickness of Bouguer plate in meters
Figure 3: An approximation how the Bouguer shell, ocean, terrain, and the geoid physically exist (Source: Kuhn et al., 2009)
2.4 Normal gravity field and free air gravity anomaly The classical gravity anomaly is defined as the difference between gravity on the geoid and normal gravity on the reference ellipsoid. Normal gravity is the gravity generated by a rotating ellipsoid with the same mass as the Earth and rotating with the same speed. The surface of the ellipsoid is here assumed to be an equipotential surface (Heiskanen & Moritz 1967). The normal gravity field used nowadays is GRS 80 (Moritz 2000). The gravity anomaly is caused mainly by that the mass distribution of the real earth differs from that of the normal gravity field (ellipsoid).
It reaches from –300 mGal to +300 mGal, approximately.
2.5 Geoid
The geoid is defined as the equipotential surface of the Earth’s gravity field, approximately coinciding with the mean sea level of the oceans. It is a complex geometrical figure that is dependent on the gravity field of the Earth and its motion (Li & Gotze, 2014). A geoid model is of importance to engineering and to
geosciences as a physically defined surface for determining orthometric heights. The orthometric height is defined as the distance along the curved plumb line between a surface point and the ellipsoid (Heiskanen & Moritz, 1967). This definition
corresponds to the common understanding of height above sea level. However, because of the curved nature of the plumb line and unknown variations in gravity down along the plumb line, it is not possible to physically observe or compute the true orthometric height (Jekeli, 1973).
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A geoid model is a model of the separation values between an ellipsoid and the geoid for a given area, be it global, regional, or local. The relationship between the
different height systems is described by the following equation (Heiskanen &
Moritz, 1967):
H = h – N (4)
where H is the orthometric height, h is height above ellipsoid and N is the geoid height (geoid to ellipsoid separation).
The most common methods for geoid determinations are as follows:
• Satellite tracking method based on calculations made for changes in satellites orbits and other parameters, e.g. CHAMP, GRACE and GOCE missions (Pail, 2014).
• Gravimetric method based on a combination of terrestrial gravity and satellite derived global gravity models. The main approaches are the Stokes (1851) and Molodensky (1945) approaches. Many methods exist and several corrections need to be applied, for instance topographic reduction and downward continuation corrections. Often the computation is made in a remove-compute-restore way (Huang & Veronneau, 2005). A well-known computation method besides (modified) Stokes’ formula is least squares collocation (Srinivas et al., 2012).
• Geometric method based on point wise determination using measurements with Global Navigation Satellite Systems (GNSS) and spirit levelling
(Benahmed Daho et al., 2006).
• Combined method is a combination of both gravimetric and geometric methods. Today a gravimetric model is typically fitted to the geometric geoid heights using first some simple transformation (usually a shift). Often some kind of smooth residual interpolation is then used to improve the fit to the regional reference systems (Duquenne, 1999).
As gravity is a function of the density of the Earth’s mass, which is not evenly distributed, the shape of the geoid is not regular, nor can it be represented by a regular mathematical expression. As a result, geoid models are often tabulated regular grids, from which terminate values are to be interpolated using for instance bilinear interpolation. Alternatively, spherical harmonic expansions can be used to determine geoid heights.
2.6 Quasigeoid
A quasigeoid can be identifiedd as a geometrical surface where a normal height
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system is referred. Unlike the geoid a quasigeoid is not an equipotential surface of the earth’s gravity field (Sjöberg 2018). Due to this reason though a geoid has a physical meaning, for example the in events of where the water flows on the surface of the earth, the quasigeoid does not have physical meaning. The surface defined by points allocated in a distance where it is equal to the distance between the reference ellipsoid and the quasigeoid below the earth surface is called a telluroid (Heiskanen
& Moritz, 1967). Optical levelling along with the integral mean of normal gravity between the reference ellipsoid and the telluroid can be used to calculate the distance between the ellipsoid and the telluroid (Sadiq et al., 2009). So, the advantage of having a quasigeoid over a geoid is that it can be determined without prior knowledge of the topographic density distribution along the ellipsoidal normal. However, the quasigeoid requires integration over the Earth’s surface (Vaníček 2012) and has a convergence problem. The concepts of quasigeoid and telluroid were introduced by Moledensky et al. (1962) with the boundary value problem in an era where it was challenging to obtain subsurface information.
According to Heiskanen & Moritz (1967) the separation between the geoid and the quasigeoid is given by:
𝑁 − 𝜁 = 𝐻∗− 𝐻 =𝑔̅−𝛾̅
𝛾̅ 𝐻 (5) where,
𝑁-geoid heights 𝜁-quasigeoid height 𝐻*-normal height 𝐻-orthometric height
𝑔̅-mean gravity between the geoid and the Earth’s surface
𝛾̅-mean normal gravity between the reference ellipsoid and telluroid
Since the separation is not readily available it is approximated by the following equation:
𝑁 − 𝜁 ≈Δ𝑔𝑃𝐵
𝛾
̅ 𝐻 (6) where,
Δ𝑔𝑃𝐵-simple Bouguer gravity anomaly at the computation point P. Various studies have been conducted where they modify the above equation to precisely determine the height conversion over the past years. There are several methods such as
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Molodensky’s approach, Remove compute restore technique, Least Squares Modification of Stokes formula with Additive corrections technique to determine the quasi-geoid (Sjöberg 2013). Most European countries use the quasigeoid as the reference surface to determine heights while the rest uses the geoid for the same purpose.
2.7 NKG2015 geoid model and NKG gravity database The Nordic Geodetic Commission (NKG), which was founded in 1953, is an association formed with geodesists from the Nordic region. Its focus is to develop and unify the geodetic infrastructure in the Nordic area. One of the main tasks of NKG is to compute highly accurate (quasi)geoid models over the region to be used as a common vertical reference surface. The NKG2015 geoid model project started in 2011 and ended in 2016 with the release of the NKG2015 (quasi)geoid model (Ågren et al., 2016; Märdla et al., 2017).
The NKG gravity database is maintained for the main purpose to compute NKG geoid models by the countries in question, i.e. Sweden, Denmark, Finland, Norway, Estonia, Latvia, Lithuania, and Iceland (Ågren et al., 2016). The current projects’ study area is over the Kattegat region (Sections 1.1 and 1.2). The following datasets (publications) used to derive the NKG2015 model occur in our study area (quoting from the old metadata file GRADOC used for the NKG gravity database):
• No. 21 Andersen,E.: Gravity Measurements in Själland, Men, Falster, and Lolland by Means of the Askania-Gravimeter, Geodetic Institute Writing (GIS), Ser. 3, Vol. X, 1947.(1300 points within the study area).
• No. 24 Saxov, Svend: Some Gravity Measurements in Thy, Mors, and Vendsyssel, GIS, Ser.3, Vol.XXV, 1956. (720 pointswithin the study area).
• No. 29 Andersen, Ole Bedsted: Surface-Ship Gravity Measurements in the Skagerrak, 1965< 1966. (454 pointswithin the study area).
• No. 30 Saxov, Svend: Gravity Measurements in Central Jylland. GIS, Ser. 3, Vol. XLII, 1976. (5300 pointswithin the study area).
• No. 42 Andersen, Ole Bedsted and Karsten Engsager: Surface-Ship Gravity Measurements in Danish Waters 1970<, Vol. XLIII, 1977.
(1953 pointswithin the study area).
• No. 44Skagerrak aerogravimetric data 1997. Processed by Arne Olesen.
Prepared by Jens Nykjær Larsen 1999. (675 pointswithin the study area).
• No. 343 New gravity data from Sweden. Provided by lars-Ake Haller, 1994. (221 pointswithin the study area).
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• No. 364 Swedish gravity data received from Mikael Lilje,
Landmateriverket, received in connection with NKG2002 geoid. The data replace sources 331, 356 and 357. The data originate from the Geological Survey of Sweden and was received as 3 files. Conversion to 80char format by Gabriel Strykowski, september 2002. (1769 points within the study area).
• No. 610 Gravity data in Skagerrak and the Baltic, AGMASCO and Nordic geoid project. Surveyed by M/V Haakon Mossby, Sep/Oct 1996, processed by D. Solheim, SK. Data from Skagerrak has been edited out and replaced by source 611 (same survey). (884 pointswithin the study area).
• No. 611 Gravity data from Skagerrak, same survey as source 610, but processed by Arne Giskehaug, University of Bergen, September 1997.
Some lines have been edited out due to gravimeter problems, resurveyed by cruise of source 612. (271 pointswithin the study area)
• No. 616 Airborne data from the Baltic Sea processed by Arne Olesen in 1999. (548 pointswithin the study area)
These datasets will be evaluated in this thesis. Note that they are only a subset of the whole NKG gravity database (see Figure 4).
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Figure 4: Old gravity data from the NKG database used in this study
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3 Methodology
3.1 Gravity data
The following datasets with free air gravity anomalies have been evaluated and compared:
• Data from the Swedish Fyrbyggaren west Kattegat 2019 marine campaign observed by Lantmäteriet using the ZLS D-13 gravimeter. This campaign is sometimes abbreviated Fyrbyggaren west 2019.
• Data from Swedish Fyrbyggaren south Baltic Sea 2019 marine campaign in the southern Baltic Sea. This campaign is similar to Fyrbyggaren west above but is used only to check the part of the west campaign that is in the southern Baltic Sea. This one is sometimes abbreviated Fyrbyggaren south 2019.
• Data from the Kattegat airborne campaign 2018 measured by
Lantmäteriet and DTU Space using ZLS D-13 and the iMAR strapdown gravimeter owned by DTU Space. The ZLS D-13 data were processed by DTU Space and Westagard Geo Solutions.
• Old data from the version of the NKG gravity database used for the NKG2015 geoid model both on land and at sea limited to the Kattegat region (datasets/publications as specified in Section 2.7).
• Free air gravity anomalies derived from Earth Gravitational Model EGM2008 (Pavlis et al., 2012), European Improved Gravity models of the Earth by New techniques EIGEN-6C4 (Förste et al., 2014) and the NKG2015 gridded free air gravity model (Ågren et al., 2016).
3.2 Software and programs Mainly the following programs were used,
• In-house programs and scripts for processing of ZLS marine gravimetry developed by Lantmäteriet (Matlab m-files, Fortran 77 programs and Unix c-shell scripts run under Cygwin64).
• Generic Mapping Tools (GMT) for Visualization of results after processing.
• Oasis Montaj (Geosoft) for visualization of observed gravity (filtered or not filtered) and for manual selection of good observations. For instance, when the vessel turns, the gravimeter will start to oscillate, and it will take some time for it to stabilize. The oscillating parts are manually omitted in Oasis Montaj (Geosoft).
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• ArcGIS: Statistical computations for the datasets in the NKG database and generating maps for the overall project.
3.3 Data Processing
3.3.1 ZLS-D13 Dynamic gravimeter
The ZLS-D13 relative gravimeter is a spring type, platform stabilized gravity system that is owned by Lantmäteriet. It is designed for marine use but can be applied for airborne applications as well. The instrument is claimed to be an improved version of beam type gravimeters like the S-type Lacoste & Romberg (Abbasi, 2006) as it designed to eliminate the inherent cross-coupling errors and vibration sensitivity issues associated with the beam type gravimeters (ZLS Cooperation, 2017). In airborne applications, since this instrument does not have a clamping mechanism, during turns of the aircraft the instrument tends to drift too far away resulting in a considerable loss of data. For the Kattegat airborne campaign 2018, this issue was addressed by manually adjusting the spring tension (Jensen et al., 2019) when the aircraft was turning.
3.3.2 Processing of ZLS measurements
When the ZLS gravimeter is used in marine mode, then GNSS is used to compute the horizontal position of the vessel, which is then used to compute the Eötvös correction. This correction applies when the gravity measurements are made from a moving vehicle. The magnitude of the error can be more than 1 Gal if not corrected for airborne measurements. It is smaller in the marine case, but still rather large and requiring a correction. For example, gravity measurements made on a ship
travelling with a speed of one knot at latitude 450 N the Eötvös correction will be approximately 5.4 mGal (Blakely, 2001). The basic formula for the Eötvös correction is given by Glicken (1962):
𝛿𝑔𝐸ö𝑡𝑣ö𝑠 = (𝑅𝜙+𝐻
𝑅𝜙2 ) (2𝑉𝜙𝑉𝑒+𝑉2) (7) where
𝑅𝜙 is the mean radius of the earth at latitude 𝜙,
𝑉𝜙 is the speed of rotation of earth’s surface at latitude 𝜙, 𝑉 is vehicle speed,
𝑉𝑒 is eastward component of the vehicle’s ground speed, and 𝐻 is the height above the geoid.
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Another correction needed for ZLS measurements is the drift correction. Most relative gravimeters drift in time (in a systematic manner) at the same location due to mechanical, thermal, and electrical changes in the instrument. As an example, thermal expansion of the beam due to changes in temperature might cause changes in the tension in the “zero-length” spring. This type of variation with time is denoted as instrument drift (Hwang, Hsiao, & Shih, 2006). By repeated readings at base stations with known gravity values, instrumental drift can be estimated and corrected for. For marine ZLS measurements, Lantmäteriet computes the drift correction using piecewise linear interpolation between pairs of harbor ties. It should further be mentioned that Lantmäteriet low pass filters all marine raw data in a consistent way using the ZLS designed standard filter specified in the manual (ZLS Cooperation, 2017).
When the ZLS gravimeter is run in the airborne mode, then GNSS is utilized also to estimate kinematic accelerations and platform tilt. The processing of ZLS measured airborne data is thus done according to the following equation (Jensen, Olesen, Forsberg, Olsson, & Josefsson, 2019):
𝑔 = 𝑓𝑧− ℏ + 𝛿𝑔𝐸ö𝑡𝑣ö𝑠 + 𝛿𝑔𝑡𝑖𝑙𝑡 + (𝑔𝑡𝑖𝑒 − 𝑔𝑏𝑎𝑠𝑒) (8) where
𝑓𝑧 is measured specific force along the vertical axis, ℏ is kinematic acceleration from GNSS height estimations, 𝛿𝑔𝑡𝑖𝑙𝑡 is the tilt correction,
𝑔𝑡𝑖𝑒 is known gravity tie value, and 𝑔𝑏𝑎𝑠𝑒 is the base ZLS gravity reading.
The platform tilt is given by 𝜙𝑥 = 𝑓𝑥−𝑞𝑥
𝑔 and 𝜙𝑦 = 𝑓𝑦−𝑞𝑦
𝑔 (9)
where 𝑓𝑥, 𝑓𝑦 are the specific forces observed along the horizontal directions of the gravimeter and 𝑞𝑥 and 𝑞𝑦 are the horizontal accelerations derived from the GNSS position estimations.
If the tilt correction is larger than the threshold value, the corresponding measured data will be discarded. For smaller values, the tilt correction is approximated as a linear combination of specific forces, which is given by the following equation.
𝛿𝑔𝑡𝑖𝑙𝑡 = (1 − cos 𝜙𝑥cos 𝜙𝑦)𝑓𝑧+ sin 𝜙𝑥𝑓𝑥 + sin 𝜙𝑦𝑓𝑦 (10)
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The final product is filtered using a six-fold cascaded second order Butterworth filter with a half power point of 170 s both forward and backwards.
3.3.3 iMAR Strapdown gravimeter
This is the second sensor used besides ZLS D-13 in the Kattegat airborne campaign 2018. The iMAR iNAT navigation grade IMU (Inertial Measurement Unit) that is used as a strapdown gravimeter is owned by the DTU Space of Denmark. The unit contains Honeywell QA-2000 accelerometers and GG1320AN ring laser gyroscopes (iMAR Navigation & Control, 2016). Additionally, it contains a temperature
stabilization system that conveniently reduces the accelerometer drift and improves the long wavelength behaviour (Jensen et al., 2019).
3.4 Workflow
The following steps were followed to achieve the overall aim and individual objectives of the thesis given in Section 1.2. (relate with figure 5)
1. The working principles of the ZLS-D13 relative gravimeter was first studied.
Starting from powering up the sensor, mounting the sensor to the gimbled platform, connecting of GNSS receivers and external user interface, sensor calibrations to the final observation sheet received in a digital format. This was done to get familiar with the operational process of the instrument along with the software interface and measurement strategies.
2. Fyrbyggaren west Kattegat 2019 marine campaign data were processed using the standard algorithms applied by Lantmäteriet. These algorithms use for instance the ZLS designed low pass filter as described in Section 3.3.2.
3. From the harbor tie data (known gravity tie value minus base reading) for the Fyrbyggaren west 2019, drift corrections (Section 3.3.2) were
computed using piecewise linear interpolation between each pair of harbor ties. However, before that the standard deviation was also calculated for the harbor ties assuming the same linear drift for the whole campaign.
4. For Fyrbyggaren west 2019 marine campaign, uncertainty parameters (minimum, maximum, range, average, standard deviation, and RMS) were then computed and studied for the internal crossovers. The external crossovers with respect to the Fyrbyggaren south 2019 marine gravity campaign were then calculated and evaluated in order to determine whether the Fyrbyggaren west Kattegat marine gravity campaign 2019 data are good enough to be used as reference to evaluate other campaigns. For these external and internal crossovers, linear interpolation techniques were used.
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Interpolation over 500 m are avoided here. The separation between signal and noise would otherwise have been an impossible task. Other
interpolation techniques might also smooth the measurement values including their errors. Maps were prepared accordingly.
5. The uncertainty parameters for the Fyrbyggaren west 2019 marine campaign in comparison with old gravity data from the NKG database referring to the NKG publications were calculated and evaluated (cf. Section 2.7).
Publication numbers 21, 24, 29, 30, 42, 44, 343, 364, 610, 611 and 616 were having completely or partially common spatial locations with Fyrbyggaren west 2019 campaign and Kattegat 2018 airborne gravity campaigns. The old gravity data from the NKG database are a mixture of airborne, marine and land gravity data that were measured using various sensors and processed with different techniques. Very little metadata are available for the old marine datasets. While some old data are available in ordered lines, some are sporadic (i.e. not ordered in lines).
6. The comparison between the sporadic data and the Fyrbyggaren west Kattegat 2019 marine gravity campaign data could not be made using the standard Lantmäteriet crossover algorithms for data processing as they assume ordered lines. To compare the sporadic old gravity data from the NKG gravity database with the Fyrbyggaren west Kattegat 2019 marine gravity campaign data, the latter was linearly interpolated using Delaunay triangulation interpolation (Dyn, Levin, & Rippa, 1990). Interpolated values were extracted according to the spatial location of the sporadic data given the condition that a sporadic point from the old NKG database can have only a maximum distance of 500 m to the Fyrbyggaren 2019 data contributing to the Triangulated Irregular Network (TIN) surface. If the distance exceeds the threshold value, the point is not considered. In this way, only old sporadic points nearby the Fyrbyggaren west Kattegat 2019 marine gravity campaign are evaluated.
7. Based upon the evaluated results (Chapter 4), it was decided that no correction is required for the Fyrbyggaren west Kattegat 2019 marine gravity campaign data.
8. The uncertainty parameters for the comparisons between Fyrbyggaren west Kattegat 2019 marine gravity campaign data and free air gravity anomalies derived from global models EGM 2008 and EIGEN-6C4 and the Nordic free air gravity anomaly grid from NKG2015 were calculated and evaluated.
9. Kattegat airborne campaign 2018 was measured using two different types of sensors (Jensen, Olesen, Forsberg, Olsson, & Josefsson, 2019). One is the
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ZLS-D13 gravimeter (Section 3.3.1) and the other one is iMAR (Section 3.3.3). Each gravimeter has its own inherent noise and approximations. The data were first downward continued from flight altitude to sea level using the classical free air reduction introduced in Section 2.3.1 (see also Hwang, Hsiao, & Shih, 2006). Other stricter methods such as Least squares
collocation (Ågren, 2016) or downward continuation using the vertical gradient of the gravity anomaly (e.g. Märdla et al., 2017) require good and sufficiently dense gravity data to prevent measurement errors from getting enhanced along with the downward continuation. This is always a problem with downward continuation. As the gravity data here is yet of unknown quality, we limit ourselves to the most robust method in this thesis, i.e. to the free air reduction. Uncertainty parameters were calculated and evaluated for the internal crossovers in Kattegat airborne campaign 2018 for both iMAR and ZLS observed data and the results were satisfactory to continue for the external crossovers in step 10.
10. Uncertainty parameters of the Kattegat airborne campaign 2018 with both the iMAR and ZLS sensors were calculated for the internal crossovers and in comparison with Fyrbyggaren west Kattegat 2019 marine gravity campaign data, old gravity data from the NKG database (the same method explained in step 4 was done for sporadic data of the old NKG gravity database), and free air gravity anomalies derived from global models, EGM 2008 and EIGEN- 6C4 and the Nordic grid-based free air gravity anomalies of NKG2015 and evaluated. Maps were prepared accordingly.
11. The Fyrbyggaren west Kattegat 2019 marine gravity campaign data were used as a reference to estimate a constant correction value for the Kattegat airborne campaign 2018 iMAR gravity data. The corrected Kattegat airborne campaign 2018 gravity data in question were then used to evaluate the old gravity data from the NKG gravity database once again. The reason for using the corrected Kattegat airborne campaign 2018 gravity data to evaluate the old gravity data from the NKG gravity data is it covers a larger spatial extent over the area of interest with a denser amount of data than the Fyrbyggaren west Kattegat 2019 (see Figure 15).
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Figure 5: Flow chart of the workflow
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4 Results and discussion
4.1 Fyrbyggaren west Kattegat 2019 marine gravity campaign
4.1.1 Harbor ties (step 3)
The standard deviation of the harbor tie values themselves is 0.40 mGal (with respect to the mean value). This indicates that the instrument was in good working condition throughout the campaign with a very low drift. A common linear drift was then estimated. Figure 6 indicates that a common drift exists even though it is very small. The standard deviation of the residuals is 0.27 mGal and seems to behave as white noise (see Figure 7). Even though the residuals in Figure 7 are very small in the present case, the final drift corrections are nevertheless computed using piecewise linear interpolation according to the standard method of Lantmäteriet.
Figure 6: Black dots represent harbor tie values in ports as the difference between land base gravity and marine gravity values
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Figure 7: Harbor tie residuals from a linear regression line (difference from the blue line in Figure 6) 4.1.2 Internal crossovers (step 4)
Uncertainty parameters are presented in Table 1 for all crossover evaluations made with the Fyrbyggaren west Kattegat 2019 marine gravity campaign. As can be seen, the RMS (Root Mean Square) value for the internal crossover differences is 0.63 mGal, which is less than 1 mGal. The average of the internal crossover differences is also very low, 0.01 mGal, but a low value here is of course expected in an internal cross over comparison like this (see Figure 8). The low RMS value proves that the data from the campaign is reliable and that the instrument behaved very well throughout the campaign. Figure 9 illustrates this.
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Table 1: Uncertainty parameters of Fyrbyggaren west Kattegat 2019 marine gravity campaign estimated in comparison with all the other available data presented in mGal. The internal crossovers are given in the row
“Fyrbyggaren West”.
Figure 8: The variation in averages and standard deviations calculated using crossover points with Fyrbyggaren west Kattegat 2019 and different publications of the NKG database
Campaign/Publication Minimum
difference Maximum
difference Range of
Differences Average of
difference SD of
difference RMS
EGM2008 -19.96 11.92 31.88 -1.83 3.43 3.89
EIGEN-64C -20.45 13.75 34.20 -1.69 3.45 3.85
NKG2015 -19.18 14.18 33.35 -0.45 3.19 3.22
Airborne Campaign ZLS -2.67 7.02 9.69 1.75 1.90 2.57
Airborne Campaign iMAR -10.01 5.41 15.42 1.46 1.81 2.32
Fyrbyggaren West -1.94 2.83 4.77 0.01 0.63 0.63
Fyrbyggaren South -2.44 1.28 3.72 0.08 0.75 0.74
NKG Publication no.21 No Comparison Points
NKG Publication no.24 -4.55 -0.40 4.15 -2.25 1.30 2.57
NKG Publication no.29 -2.65 0.79 3.44 -0.99 1.54 1.70
NKG Publication no.30 No Comparison Points
NKG Publication no.42 -3.77 9.10 12.87 0.30 2.75 2.74
NKG Publication no.44 No Comparison Points
NKG Publication no.343 No Comparison Points
NKG Publication no.364 No Comparison Points
NKG Publication no.610 -2.42 1.55 3.97 -0.08 0.78 0.78
NKG Publication no.611 0.99 1.83 0.84 1.43 0.34 1.46
NKG Publication no.616 No Comparison Points
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Figure 9: Visualization of internal crossover gravity differences for the Fyrbyggaren west Kattegat 2019 marine gravity campaign
4.1.3 External crossovers (steps 5, 6 and 7)
The RMS of the external crossover differences to the Fyrbyggaren south 2019 campaign is 0.74 mGal and the average is 0.08 mGal (Table 1), which is quite low.
Figure 10 shows the crossover differences on a map. Based on the uncertainty parameters calculated from the harbor ties, internal crossovers and external crossovers, it was decided that Fyrbyggaren west Kattegat 2019 marine gravity campaign is very accurate and trustworthy and may be used as a reference gravity dataset to evaluate the other datasets.
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Figure 10: Visualization of external crossover gravity differences (Fyrbyggaren West 2019 – Fyrbyggaren South 2019).
For NKG publication 610 (i.e. Håkon Mossby marine campaign) the standard deviation of external crossovers with Fyrbyggaren 2019 west is 0.78 mGal the average of differences is -0.08 mGal (Table 1) indicating that the data set agrees well with the Fyrbyggaren west Kattegat 2019 marine gravity campaign data (Figure 12).
For NKG publication 42 the standard deviation is 2.57 mGal while the average of the differences is 0.30 mGal (Table 1 and Figure 12). This is not as good as NKG publication 610(Figure 11) but lies within an acceptable level. The sample sizes are not large enough to make comparisons with the other old publications.
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Figure 11: Visualization of external crossover gravity differences (Fyrbyggaren West 2019 – NKG publication 610).
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Figure 12: Visualization of external crossover gravity differences (Fyrbyggaren West 2019 – NKG publication 42).
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4.1.4 Comparisons with global geopotential models and with the regional NKG2015 free air gravity anomaly grid (step 8)
Comparisons with the global models EGM2008 and EIGEN-6C4 and with the NKG 2015 free air gravity anomaly grids are visualized in Figures 13 and 14. According to the standard deviations in Table 1 and the visualizations in Figure 14, EGM2008 and EIGEN-6C4 behaves almost the same when compared to marine data. Figure 14 also indicates that satellite altimetry data is worse in coastal areas as the coastal areas are systematically coloured in red meaning that the differences are more than 5 mGal negative. Having a standard deviation of 3.19 mGal, the NKG2015 free air anomaly grid is slightly better than the EGM2008 and EIGEN-6C4 models (Figure 13). The NKG2015 model has been computed based on the old NKG gravity datasets with publication numbers 21, 24, 29, 30, 42, 44, 343, 364, 610, 611 and 616 in the Kattegat area. The red areas in Kattegat in Figure 12 are most likely due to the lack of observations in these regions when the NKG2015 grid was produced. These areas have now been measured in the Fyrbyggaren 2019 west campaign.
Figure 13: Visualization of differences between measured gravity values of Fyrbyggaren west Kattegat 2019 marine gravity campaign and the NKG2015 free air gravity anomaly grid (Fyrbyggaren West 2019 –
NKG2015).
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Figure 14: Visualization of differences between measured gravity values of Fyrbyggaren west Kattegat 2019 marine gravity campaign and the EGM2008 and EIGEN-6C4 global models (Fyrbyggaren West –
EGM2008/EIGEN-6C4).
4.2 Kattegat airborne campaign (steps 10 and 11) The 2018 airborne data from both sensors slightly deviates from each other but iMAR is slightly better than the ZLS. iMAR has a standard deviation of 1.81 mGal while ZLS has a standard deviation of 1.90 mGal (Table 1) in comparison with the Fyrbyggaren west Kattegat 2019 marine gravity campaign data. As ZLS and the iMAR sensors were on the same platform during the gravity survey it was decided to proceed primarily with the iMAR data as it is denser and have a larger coverage. As visualized in Figure 15 (being more bluish throughout the region), it was found out that the Kattegat airborne campaign 2018 gravity data is systematically lower than the Fyrbyggaren west Kattegat 2019 marine gravity campaign data. This deviation is approximately a constant value throughout the region. So, it was decided to shift (add) the iMAR data set based upon the average value between the external cross over differences with Fyrbyggaren west Kattegat 2019 marine gravity campaign data that, which is +1.46 mGal (Table 1). There can be several reasons for this
systematic deviation such as a shift due to a blunder made when establishing the gravity reference point at Roskilde, a drift in both the instruments, the downward continuation method used etc. It must be noted that Jensen et al. 2019 have
performed several experiments on both the ZLS and iMAR sensors to prove that the instruments was in good working condition before the survey. If the downward continuation method used is to be challenged it must be compared with other methods which is not possible within this scope of study. (see section 1.2.3). This
40
issue must be further investigated.
Figure 15: Visualization of external crossover gravity differences (Kattegat Airborne 2018 (iMar) – Fyrbyggaren West 2019).
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Another important finding is the following: Initially it was assumed that the Fyrbyggaren marine campaigns were acquiring data lower than the actual value as the marine gravimeter was not being levelled properly all the time due to the perturbations at sea. (This is not a problem for the airborne data since they are corrected for tilt by means of GNSS-observations – see Section 2.2.2). But the comparisons made with the airborne gravity data indicates that this is not the case.
All the comparisons which were made with the Fyrbyggaren west Kattegat 2019 marine gravity campaign were also made with the Kattegat airborne campaign 2018 data for both ZLS and iMAR sensors (step 10) but we do not present the statistics here as they are intermediate products hence will be presented in the Appendix A and Appendix B.
4.2.1 Kattegat airborne campaign 2018 (iMAR) as a reference data set (step 11)
From statistical analysis made in steps 9 and 10 it was found that the gravity data from Kattegat airborne campaign 2018 (iMAR) was systematically 1.46 mGal too low. A shift correction was added to the whole Kattegat airborne campaign 2018 (iMAR) dataset and the corrected data was used to evaluate the old NKG 2015 database.
Table 2: Uncertainty parameters for the difference between the corrected Kattegat airborne campaign 2018 (iMAR sensor) in comparison with all the other available datasets, presented in units of mGal (iMAR- campaign/model)
Campaign/Publication Minimum
difference Maximum
difference Range of
Differences Average of
difference SD of
difference RMS
EGM2008 -18.24 6.64 24.87 -2.99 2.32 3.78
EIGEN-64C -16.66 8.30 24.96 -2.13 2.54 3.31
NKG2015 -10.53 7.95 18.47 -1.31 2.15 2.52
NKG Publication no.21 -7.26 2.19 9.44 -0.98 1.90 2.12
NKG Publication no.24 -4.83 -0.21 4.62 -2.42 1.03 2.62
NKG Publication no.29 -4.72 7.37 12.08 -0.97 2.85 2.95
NKG Publication no.30 -5.16 1.38 6.55 -1.77 1.33 2.21
NKG Publication no.42 -6.33 8.01 14.35 -1.16 2.87 3.09
NKG Publication no.44 -5.07 2.99 8.06 -0.71 2.67 2.63
NKG Publication no.343 -2.11 2.49 4.60 0.18 1.97 1.71
NKG Publication no.364 -7.97 2.16 10.13 -1.88 2.83 3.36
NKG Publication no.610 -6.52 6.40 12.92 -0.90 2.63 2.74
NKG Publication no.611 -1.99 -0.05 1.94 -0.69 0.89 1.03
NKG Publication no.616 -6.75 0.62 7.37 -1.88 1.75 2.55
