First International School on“Geoid Modelling, Gravity Inversion and its Applications”

The school will be arranged at the University of Gävle, Gävle, Sweden, from 9 to 13 of September, 2019.

After the successful experiences in the determinations and evaluations of precise local geoid models in different countries as well as the very well met International Geoid Schools at:

  • Yildiz Technical University, Istanbul, 2010
  • Universiti Teknologi Malaysia (UTM), Johor Bahru, 2012
  • Royal Institute of Technology (KTH), Stockholm, 2014
  • Universiti Teknologi Malaysia (UTM), Johor Bahru, 2016

we plan to arrange another International School but now with a broader program on

  1. Geoid determination
  2. Gravity Inversion.

The school will be organized with theoretical lectures in the mornings followed by computer exercises in the afternoons, where the available software will be used. Since the School has a full-week intensive program, it can be counted as an external full graduate course.

The school is primarily offered only for university students and personnel from public organizations, and the software package is made available only for training of students and scientific works.

The school will be arranged at the University of Gävle (HiG), Gävle, Sweden, from 9 to 13 of September, 2019.

The contents comprise the following topics

Geoid modelling

Geoid modelling

The Least Squares Modification of Stokes Formula with Additive corrections (LSMSA) approach to geoid determination is unique in the sense that it uses least squares technique in the spectral domain to combine the data in an optimum way by considering the errors of the Earth Gravitational Model (EGM), the gravity data and the truncation of Stokes’ integral to a cap around the computation point. Another feature that distinguishes the LSMSA method from other approaches is the way corrections for topography, atmosphere and ellipsoidal shape of the earth are applied: all corrections are added as separate additive corrections. This method was successfully applied in the determination of several regional geoid models: over Sweden, the Baltic countries, Greece, Iran, Sudan, Zambia, Ethiopia, Tanzania, Serbia, Moldova, part of Turkey, and it was used by the Nordic geodetic Commission in the NKG geoid model 2015. The official geoid models of Sweden and Estonia are based on the LSMSA technique. Finally, it has been performed very well in a number of comparisons with other geoid determination techniques used by various groups. See e.g. Yildiz et al. (2012) and Sjöberg and Bagherbandi (2017, Chapter 6).

The lecture notes will be prepared on a CD, which contains also exercises, data sets and software. Each student will receive a copy of the CD at the start of the school.

All lectures are followed by computer exercises, where the software available will be used. The participants should bring their own laptop for the exercises.

This training course provides a good opportunity for the student to familiarize himself with the latest developments in geoid determination, as well as to enhance the international collaboration in gravity field modelling by building contacts to the professionals dealing with geoid determination in various countries.

Why LSMSA approach?

Many different methods have been proposed through the years for regional geoid determination by gravimetric data, each based on its own technique and philosophy. Today, all such methods combine long-wavelength EGMs with local gravity data, and they mainly differ in the way they combine these data sets. The LSMSA approach is unique in the sense that a) it uses least squares technique in the spectral domain to combine the data, and b) all corrections to the approximate modified Stokes solution from uncorrected data are added as combined or total effects on topography, atmosphere and ellipsoid. We show that these additive effects are advantageous vs. the traditional direct and indirect effects/corrections. For instance, any of the additive corrections can be added afterward at any time when better data are available for its improvement (without the need to repeat all the computations).

The LSMSA method is an accurate, simple and practical method of determining the geoid that has been developed since 1984 to present mainly by and under the supervision of Prof. Lars E. Sjöberg. (See numerous papers, e.g. in J. of Geodesy and also Sjöberg and Bagherbandi 2017). The method has been successfully applied in the determination of several high-resolution regional geoid models in different areas. Through the LSMSA approach, various data, such as a Global Geopotential Model, gravity anomalies and a high-resolution photogrammetric/SRTM Digital Elevation Model data are combined to a gravimetric geoid model, and the method can be (and usually is) designed to match with GPS/levelling data by using the least-squares principle. Several of the successful applications are reported in M.Sc. and Ph.D. theses at KTH Notable among these studies are the applications in rough topographic areas and in several developing countries with only limited gravity anomaly data. The results of comparisons clearly show that the LSMSA is advantageous to other methods.

Practical Applications of Gravity Inversion

Practical Applications of Gravity Inversion

In this part of the school, we will present gravity inversion for various geophysical, geodetic and geodynamic applications. After introducing some basic geophysical concepts, various isostatic models and their use in determining crustal depth by gravity is described at length and compared and combined with seismic models of crustal depth. The preferred isostatic model is based on Vening Meinesz-Moritz’s hypothesis with a global isostatic compensation of Bouguer gravity disturbances rather than Bouguer gravity anomalies. There are also applications of gravity for estimating tectonic stress in the mantle and viscosity in the mantle.

In addition, we will present and learn about gradiometry for different regional geophysical interpretations and applications.

During the workshop, a software package for determining the crustal thickness and processing of the gradiometry data based on spherical harmonics will be presented.

Modelling of crustal thickness based on isostatic theory

In geophysics isostasy is essential mainly for studying geodynamic processes in the crust and upper mantle, and in geology it helps in explaining various topographic and geologic features around the world. For example, using an isostatic hypothesis the geological interpretation for compensation of topography and studying lithosphere structures can be inferred.

Why is crustal thickness modelling using gravity data important?

Seismic surveys are expensive, and in many areas seismic information feasible for depth estimation of the crust is therefore sparse or lacking, inferring poor crustal thickness models. This is especially the case in oceanic regions. Today gravity surveys and satellite gravimetry missions are much more cost-effective, allowing crustal thickness to be estimated by gravity inversion under the assumption of some kind of isostatic model (e.g., the models of Airy, Pratt, Vening Meinesz or Vening Meinesz-Mortiz), where the gravity anomaly is mainly assumed to be the effect of variations in the crustal thickness.

In this workshop, we will present different techniques to model the Moho depth (or crustal thickness). Over the years, different methods for estimating the Moho depth have been proposed. Here the purpose is to use gravity data for estimating the Moho depth. One of the most recent gravimetric methods is the generalized Vening Meinesz’ isostatic hypothesis (Mortiz 1980), here called Vening Meinesz-Moritz’ (VMM) model. The VMM isostatic (flexural) model represents a more realistic assumption of the global compensation mechanism described for the Earth’s homogenous crust. The main idea is simple, but the theoretical modelling is somewhat difficult, because the mass distribution of the Earth’s crust is complicated, and also many geophysical phenomena should be considered. Sjöberg (2009) formulated this problem as that of solving a non-linear Fredholm integral equation of the first kind, and he presented some solutions for the crustal thickness and Moho density contrast (MDC) that were published by Sjöberg and Bagherbandi (2011). The most important issues that will be discussed during the workshop are:

  • Classical isostatic hypotheses
  • A new gravimetric method to model the Moho parameters (i.e. the depth and Moho density contrast)
  • The Bouguer gravity anomaly or gravity disturbance: which is better for determining the Moho parameters?
  • Additive gravity corrections: To solve the gravimetric problems of isostasy for finding the Moho parameters, the gravitational contributions of all other (disturbing) mass density contrasts within the Earth’s crust should be modeled and subsequently removed from observed gravity data (i.e. the gravity disturbances/anomalies should be reduced for the gravitational contributions of topography and mass density contrasts of ocean, ice and sediments, etc).
  • A combined seismic and gravimetric Moho model will be outlined.
  • Geodynamical applications of Moho i.e. sub-crustal stress due to mantle convection.

Gradiometry with Geodynamical applications

Satellite gradiometry can be used as a complementary tool to gravity and geoid information in interpreting the general geophysical and geodynamical features of the Earth. Due to the high altitude of the satellite, the effects of the topography and the internal masses of the Earth are strongly damped. However, the gradiometer data, which are nothing else than the second order spatial derivatives of the gravity potential or gravity gradients, efficiently counteract signal attenuation at the low and medium frequencies (Kiamehr et al. 2008).

The gradiometry data provide important contributions to geophysical exploration when terrestrial instrumentation like the torsion balance is employed. Satellite gravity gradiometry is an exciting new technology, which can be used to explore the general patterns of crust. Gravity gradiometry components show spatial rates of change in the gravity field. They can capture the near-surface lateral density variations better than conventional vertical gravity field parameters (gravity anomalies). This is possible, because the strength of gradiometer signal falls off with the cube of the distance to the target (Hammond and Murphy 2003), in contrast to the conventional vertical gravity signal, which decays with the square of the distance. Also, as gradient components have shown to be drift free, they are very suitable for time-laps measurements (Talwani and Schaeffin 2001), e.g., in the study of crustal motions.



Time: 09:00-12:00

Time: 13:00-16:00


Opening of the school

Lecture 1
- Basic Physical Geodesy
- Modification of Stokes’ formula

Lecturer Prof. Lars Sjöberg

Data preparation
- Gravity data snooping and gridding
- Gravity field determination by global geopotential models.
- Digital Elevation Models and Geoid

Lecturer: Dr. Ramin Kiamehr


Lecture 2
- Additive corrections

Lecturer Prof. Lars Sjöberg

KTH GEOLAB Software (Part1)

Lecturer: Dr. Ramin Kiamehr


KTH GEOLAB Software (Part2)

Lecturer: Dr. Ramin Kiamehr

Lecture 3
- LSMSA vs. the RCR-Technique
- Some practical experiences (e.g., from recent Ph.D. theses)

Lecturers: Prof. Lars Sjöberg and Dr. Ramin Kiamehr


Lecture 4

- KTH GEOLAB Software (Part3)
- Fitting the Gravimetric Geoid to GPS on Benchmarks. (Including Exercises)

Lecturer: Dr. Ramin Kiamehr

Lecturer 5
- Theory of isostasy
- Gravimetric Moho/crustal thickness determination
- Additive corrections
- Combined Moho modelling

Lecturer: Prof. Mohammad Bagherbandi


- Other studies and models
- MohoLAB software
- Geodynamical applications of Moho
(e.g. sub-crustal horizontal stress determination, plate tectonics relation
with crustal thickness)
Lecturer: Prof. Mohammad Bagherbandi

 Lecture 6
- Gradiometry with Geodynamical applications
- EGMLAB software
Closing the school

Lecturers: Prof. Lars Sjöberg, Ramin Kiamehr and Mohammad Bagherband


The school is led by

Lars E. Sjöberg, Professor,Head of School

Mohammad Bagherbandi, Professor, Local Organizer


Mohammad Bagherbandi, Professor
Telephone: 0046 700 67 69 48

Published by: Zara Lindahl Page responsible: Gunilla Mårtensson Updated: 2018-11-22
Högskolan i Gävle
Box 801 76 GÄVLE
026-64 85 00 (växel)