Gravity Works in Nepal

Western Nepal Topographic Mapping Project (WNTMP) – Phase I
Final Report on Control Surveys
Volume- III Definition of Gravimetric Geoid
HMG Survey Department of Nepal
FM – International FINNMAP
June 1997

This report includes following details regarding gravity observations and data:

List of observations of western Nepal:
Gravity observations were done in GPS points.
Number of gravity observed stations/points: 56 stations.
Number of gravity stations specified as old British and other points: 43 stations.

List of observations of eastern Nepal:
Gravity observations were done in GPS points.
Number of gravity observed stations/points: 43 stations.
Number of gravity stations specified as other gravity stations: 10 stations.

Number of gravity stations specified as old British points: 95 points.

Note: For numbering of stations, latitude, longitude, ellipsoidal height, orthometric height, and gravity values – see Annex J and Annex K of WNTMP report.

“Surface Gravity Information of Nepal and Its Role in Gravimetric Geoid Determination and Refinement” – By Niraj Manandhar published in Nepalese Journal of Geoinformatics -9, 2067.

Gravity base was established in the country during 1981-84 British Military Survey.
The gravity reference system: ISGN 1971
Instruments used: LaCoste Romberg Model G gravimeter
Number of Stations observer: 21 out of 36 (MODUK)

Fundamental gravity base at Tribhuvan International Airport (TIA), Kathmandu was established designated as KATHMANDU J. IN 1981 gravity transfer from ISGN71 station BANGKOK to Kathmandu was made. At this time 45 gravity stations are surveyed at 35 different locations, mostly at the airports and accessible places around the country.
KATHMANDU J = 978661.22+-0.047 mgal

Total 375 gravity detail stations were established till now which includes WNTMP + ENTMP.


First order gravity network of Nepal:
According Niraj sir on his this article:
Number of first order gravity points: 36 points, 25 points in airport and airstrips and remaining 11 in government buildings, army barracks, and police stations.


Absolute gravity
Nagarkot Geodetic Observatory, Nepal
Observations, Corrections, and Results
Gravity ties to Kathmandu and Simara airports.
National Geodetic Survey, USA
HMG Survey Department, Nepal
University of Colorado University, Boulder, USA
March/April 1991

• To establish a reference datum for the local gravity network in Nepal.
• To establish points that may be remeasured to reveal changes of elevation in future years.
The Fundamental Absolute Gravity Stations (FAGS) was established in 1991 at Nagarkot in assistance with above three organizations.
The absolute gravity value measured at Nagarkot (FAGS -1). The corrected value of the FAGS -1 indoor point at ground level is 978494834.7 +-6.7 microgal.
The gravity gradient at floor level (0 to 0.43m) was 4.4194 microgal/cm.
Relative ties were made to three GPS points: Nagarkot, Kathmandu airport, and Simara airport. The relative differences from FAGS-1 to these points are as follows:

Nagarkot FAGS-1 978494834.7+-6.7 microgal
Nagarkot GPS -0.691+-0.002 mgal
KATHMANDU J +166.469+-0.005 mgal
SIMARA J +368.599 +- 0.017 mgal
SIMARA GPS +368.706 +- 0.013 mgal

The ties were undertaken using a pair of Model D LaCoste Romberg gravimeters. For Nagarkot GPS point which is less than 10m from the brick building where GPS measurements were made.
The absolute accuracy of the 1991 measurements is +- 6 microgals or approximately +- 1.5cm in elevation.


Airborne Gravimetry in Nepal
Article Title: “Geoid of Nepal from Airborne Gravity Survey”
Niraj Manandhar, Kalyan Gopal Shrestha, Rene Forsberg
Earth on the Edge: Science for a Sustainable for a Planet, IAG Symposia 139, Springer-Verlag Berlin Heidelberg 2014

An airborne gravity survey of Nepal was carried out December 2010 in cooperation between DTU-Space, Survey Department of Nepal, and National Intelligence Agency- NGA USA.
Primary goals:
• To provide data for a new national geoid model, which will in turn support GPS surveying and national geodetic infrastructure.
• To provide gravity information for future global gravity field EGM 2020.

Secondary goals:
• To provide training of physical geodesy to Nepalese geodesists.
• To make an improved estimate of the geoid at Sagarmatha allowing an independent height determination by GPS.
• To collect airborne data which will provide an independent validation of GOCE and EGM08 in most mountainous part of the Earth.

Airborne gravity survey details:
Aircraft : Beech King Air aircraft by Danish Company COWI.
57 flightlines @ spacing of 6 nautical miles.
13 flight days. Dec 4-17, 2010
Coverage of border regions of the country towards india and china was not possible. The survey operations were a major challenges due to excessive jet streams at altitude as well as occasional excessive mountains waves. Even a reflight attempts did not provide useful data due to persistent wind effects of Annapurna and Mustang valley regions. Surface gravimetry data of those regions should be used. The reported results appeared accurate to few mgal – despite the large 400 mgal + range of gravity anomaly changes from indian plains to Tibetan plateu.

Instrument installed on aircraft: Lacoste Romberg S –type gravimeter which was controlled by Ultrasys Control System with a number of GPS receivers onboard the aircraft and on ground providing the necessary kinematic positioning.

Checkan –AM gravimeter alongside LCR gravimeter with the special goal to augment the LCR gravity data in expected turbulent conditions.
The airborne gravity reference point at Kathmandu airport tied to NOAA absolute gravity measurement at Survey Department and at Geodetic Observatory Nagarkot. The actual gravity values at altitude differ by 1000’s of mgals from reference values due to the height changes and Eotvos corrections.

The overall accuracy of airborne gravity data collected was 3.3 mgal. The preliminary geoid was computed and the accuracy of computed geoid was likely at the 10-20cm in the interior of Nepal but higher near the border due to lack of data in China and India. This preliminary geoid was compared to eight reasonable GPS-levelling precise data in the Kathmandu valley. It is seen that a significant geoid improvement has been obtained relative to the EGM-08 geoid.
The computed height of Sagarmatha which confirms the official Survey Department height 8848 m within range of +- 1m, based on an ellipsoid height of the snow summit of 8821.40 +- 0.03m.

Present Works in Gravity: Refinement of Geoid around Sagarmatha region
There has been planning for the purpose of refinement of geoid around Sagarmatha regions for the computation of orthometric elevation of Sagarmatha from GPS heightening and geoidal undulations. To refine the geoid around Sagarmatha, we need dense/extra surface gravimetry data which will improve the past geoid computed from 2010 airborne gravity geoid. The surface gravimetry observations were planned to take on 5km*5km grids, around that region.


Peculiarity of Gravity Field of Nepal

  • How gravity field behaves around the terrain of Nepal?
  • What are characteristics of gravity field of Nepal?
  • How can we correlate or decorrelate the relationship between geographical variation and gravity value variation moving  60m altitude to 8848m altitude?
  • Our country is rich in Himalayans, we are called land of Himalyas- so gravity field must behave weird here right?
  • What are the statistics/numerical information/quantitative information regarding gravity/gravity field in Nepal. and Where are these statistics?
  • What and where might be the lowest gravity value and what and where might be the largest gravity value?
  • Any possibility of dense mineral deposits within/under the Nepal?

Read this articles:

  1. springer-geoid of Nepal from Airborne Gravity Survey
  2. geoid-of-nepal-from-airborne-gravity-survey
  3. DTU-geoid-of-nepal-from-airborne-gravity-surveyResearch Gate –
  4. Geoid_of_Nepal_from_Airborne_Gravity_Survey

Procedure of Geoid Determination – 1.

There are various methods of geoid determination/geoid undulations determinations, which are as follows:

  1. xxxxx
  2. xxxxx
  3. gravimetric geoid determination
  4. xxxxx
  5. xxxxx

I ll be writing here very short procedure of geoid determination through gravimetry i.e using surface gravity measurement. And this is called the gravimetric approach.

  • Measure surface gravity at points which are well distributed in terms of distance and height. Distribution  and density of points (gravity stations) depends on character of topography –> gravity varies regular and smooth in fairly flat terrain but not so in hill and mountainous area where gravity is not smooth and irregular so points should be distributed accordingly. The observed gravity values are the raw data used to determine geoid.
  • The observed gravity at surface of the topography need to be reduced to the level of geoid. The reduction (corrections) are applied to observed gravity.
    1. Determine/calculate free-air correction. Then calculate corresponding free-air anomaly.
    2. Then determine incomplete (simple) Bouger’s correction. Then calculate corresponding incomplete (simple) Bouger’s anomaly.
    3. Again Determine complete (refined) Bouger’s correction. Calculate complete (refined) Bouger’s anomlay.
  • The complete (refined) Bouger’s anomalies are used; these values are used in Stoke’s integral as an input which gives us the geoidal undulations.
  • Above obtained geoidal undulations are in fact with respect to the surface called co-geoid not with respect to the actual geoid. so determine/calculate the indirect effect, applying indirect effect to above obtained geoidal undulations provides us with N value with respect to actual geoid level.

Airborne Gravimetry and Airborne Gravimetry for Geoid Determination Ep-II


Unified terrain correction and their effect on airborne gravimetry:

  • set of unified formula for computation of topographic attraction/ terrain correction has been developed with gravity measurements taken either in earth’s surface or in space. you can find detailed formula on appendix A and B. the density of earth surface/topography is either constant or varying horizontally. DTM can either be mass-line or mass-prism represenTAtion.
  • /effect of varying densities of austrian alps  on computation of terrain correction .
  • /effect of topographic attraction of canadian rocky mountains on processing of airborne gravity measurements and airborne gravity disturbances.


Terrain correction for vector gravity measurements:

  • to provide gravity reference value at flight level or to downward continue the gravity disturbance to the ground level, terrain correction or topographic gravitational attraction need to be modeled. three components of topographic gravitational attraction is given in Heiskanen and moritz (1967). formula/equation see in dissertation. and this equation involves integration over a area, gravitational constant, coordinates of computation point, distance between points, topographic density and topographic height of variable integration point, height of the reference surface.
  • Terrain correction is divided into two parts:
  1. Bouger correction which represents the gravitational attraction of mass plate with thickness equal to topographic height of a computation point and another
  2. terrain correction which is gravitational attraction of residual mass/terrain with respect to Bouger plate. these corrections are well illustrated in Heiskanen and Moritz 1967.
  • If constant density is assumed, the Bouger plate is considered circular plate of radius R with its center at computation point. THE horizontal component of Bouger plate is equal to zero, and the vertical component is approximated as following equation:
  • with horizontally varying densities all three components of gravitational attraction of Bouger plate have to be computed by integration ecplicitly. for grided DTM, double integration over xy plane is carried out.
  • Effects of two topographic model: mass-line and mass-prism representation on terrain correction is give in Li and Sideris(1994).

The effect of density on terrain correction:

  • numerical examples of effect of topographic density on terrain correction on the ground level can be found Forsberg(1984)/Bhaskara et al (1993).

Topographic effect on processing of airborne gravity measurements:

  • a DTM of 960 by 600km grid under the flight area, was used to compute gravitational attraction on two constant altitude of 5km and 6km.  a constant density of 2.67gm/cm3 was used in computation. the vertical component of topographic attraction at the height of 5km is shown in figure.
  • when performing downward continuation of gravity disturbance to the reference level, at msl level, topographic gravitational attraction must be removed from airborne gravity disturbance before performing downward continuation, then restore it at reference level afterwards. this is called remove-restore process.

The topographic effect on processing of airborne gravity disturbance:

  • whether the topographic attraction need to be removed from airborne gravity disturbance.

Summary of the results:

  • Topographic attraction and terrain corrections for vector gravity measurements can be computed from the formula. topography can be modeled either by mass-line or mass-prism representation. horizontally varying densities can be gridded in same grid of DTM can be used in computation. the computation can be performed either at earth surface or at a constant altitude in space.
  • there is a high correlation between topographic attraction and gravity disturbance, the remove-restore process should be used for downward continuation of airborne gravity data to sea level. however it is not necessary, remove and restore of topographic effect on processing of airborne gravity measurements on geoid determination.


PART 6/CHAPTER 5: Determination of Reference Geoid.

  • to provide reference for geoid undulations computed from airborne gravimetry, gravimetric geoid undulations are computed between latitudes 36 & 66 and longitudes 229 &259 grid.

Geoid determination from stokes integral:

  • original stokes integral and modification of kernel function of stokes integral, is used to compute gravity anomaly and geoidal undulations. see the different raster representation of those geoid in dissertation in corresponding chapter. Effects of stokes kernel modification:
  • gravimetric geoid undulations from stokes kernel modified integral compared with gravimetric geoid undulations derived from GPS/levelling over 209 benchmarks.

Geoid determination from Hotine integral:

Height anomalies determination from molodensky series:

  • high accurate detailed DTM information can be obtained as well as airborne gravity data. Detailed DTM can contribute to geoid determination.

Effect of integration radius on geoid determination:
Summary of results:

  • geoid undulations computed with modified stokes integral used as reference for geoid computed from airborne gravity.

PART 7/CHAPTER 6: The Importance of Downward Continuation.

  • Gravity field upward continuation is necessary to provide the gravity references at flight level. For geoid determination, downward continuation of gravity disturbances from flight level to ground level is necessary. Poisson’s integral is used for downward continuation of gravity field.
  • the downward continued airborne gravity disturbances is used for gravimetric geoid determination. so evaluation of the effect of downward continued airborne measured data is necessary. the accuracies of gravity disturbances collected at flight level is compared with upward continued gravity references and downward continued airborne gravity disturbances is compared with ground level gravity measurements.

The poisson’s equation and its inversion:

  • Poisson’s equation/integral is solution to first geodetic boundary value problem which is Dirichlet problem. The beautifully attracting solution of Dirichlet’s problem is given in dissertation and another most important thing is to thoroughly read Heiskanen and Moritz 1967.
  • gravity anomalies at lower level surface can be estimated from gravity anomalies measured at higher level surface using linear equations.

Regularization of Inverse Poisson Integral:

  • developing a well posed solution to improperly posed problems. truncated singular value decomposition, Tikhonov regularization, stochastic methods, projective methods, and iterative methods.

Numerical examples with synthetic gravity animalies:

PART 8/Chapter 7: The use of Airborne Gravimetry Data in Geoid Determination

  • geoid undulations determinened from airborne gravity disturbances is compared with geoid undulations derived from ground gravity measurements.
  • hotine integral was used to determine geoid undulations from airborne gravity disturbances.
  • how well can the gravimetric geoid be determined from airborne gravity?gravimetry?

Evaluation of local airborne geoid at flight level:
Evaluation of local airborne geoid on the ground:

Evaluation of regional airborne geoid on the ground:

Airborne Gravimetry and Airborne Gravimetry for Geoid Determination Ep-I

Part 1: Tools on Geoid Determination.

There are 5 methods of geoid determination, and they are as follows:

  1. least squares collocation.
  2. spherical harmonic expansion.
  3. intergral formula for solution of GBVP. e.g Stoke’s integral.
  4. Hotine integral.
  5. Molodenskii integral series.

Here, GBVP stands for Geodetic Boundary Value Problem. And there are three GBVP in physical geodesy. They are as follows:

  1. Dirichlet’s Problem.
  2. I ll write next time when it comes to my mind.
  3. Neumann’s Problem.
  • Stoke’s integral computes geoid/ geoid undulations from gravity anomalies.
  • Hotine integral calculates geoid undulations from gravity disturbances.

You could ask what is the difference between gravity anomalies and gravity disturbances. Go through the pages of physical geodesy book: Heiskannen and Moritz (1967) and get the crystal clear concept of those gravity anomaly and gravity disturbance.

Stoke’s integral formula, Hotine integral, and least squares collocation, all these three methods need gravity reductions. And here gravity reduction means reduction of gravity value measured on earth surface to the geoid level. And all three methods must account/remove the topographic mass outside of the geoid computationally.

Stoke’s integral is mostly used in geoid determination/ geoid undulation determination because gravity anomaly is the convinient variable obtained from terrestrial/surface/ground gravity measurements.

Part 2: Speech on Airborne Gravity Measurements

I ll be using acronym AGS which stands for Airborne Gravity System.

AGS measures the special kind of force acting on aircraft which we call –> specific force. and the acceleration of aircraft is called –> dynamic acceleration. AGS determines the gravity disturbance along the flight trajectories by separating the aircraft’s dynamic acceleration from measured specific force.

DGPS component gives the position and velocity of the aircraft. Aircraft’s dynamic acceleration is obtained through differentiation of aircraft’s position or velocity.

PINS –> system of INS and La-Coste Romberg gravimeter–> popular gravimeter. PINS and La-Coste Romberg gravimeter is the most used combination in airborne gravimetry and accuracy obtained is 2 mgal for the resolution of 6 km. But we need the gravity measurement at the microgal accuracy level.  2mgal accuracy is far away poor accuracy than we need. and AG can not give this much of accuracy, AGS is still in early stage/infant stage.

AG measures the gravity data at flight level. Airborne gravity data is reduced from flight level to ground level using gravity field downward continuation. Poisson’s integral gives upward continuation taking inputs of surface gravity and predicting gravity on outer space. Inversion of this formula works for downward continuation in linear combinations.

And we already mentioned somewhere above that the effect of topographic mass outside of the geoid should be accounted/removed computationally. Removal of topographic attraction from observed gravity or/computation of topographic attraction or terrain correction extensively used and well understood by Nagy/ Sideris/Tziavos Li on their books. Where can we get these materials?

We intend to achieve the 1cm geoid accuracy through AGS but till present time 10 cm geoid accuracy have been achieved with 5′ resolution in Europe and America.

Part 3: This part highlights on very short

  • what are the data requirements for precise geoid determination?
  • On stoke’s integral, hotine integral, molodensky integral-GBVP.
  • AG for improvement of geoid in poorly and well surveyed area both.

Part 4: This part highlights on:

  • Principle and potential contribution of AGS.

IMU –> Intertial Measurement Unit. DGPS –> Differential GPS. SISG –> Strapdown xxxxx. RISG–> Rotation Invariant xxxxx.

Part 5: This part discuss on:

  • unified terrain correction formula,and topographic effect on airborne gravity data processing.

Part 6: This part talks about:

  • Reference geoid/ground gravity measurements/detailed topographic model.
  • gravimetric geoid undulations computed from above S/H/M formula compared with gravimetric geoid and GPS/levelling derived geoid.

Part 7: This part deals with:

  • gravity field continuation using Poisson’s integral.
  • Accuracy of gravity disturbances.
  • Inverse poisson’s integral for downward continued gravity data.
  • synthetic gravity anomalous gravity field generated by point mass model.

Part 8: Finally the geoid from AGS:

  • geoid undulations computed from gravity disturbances and compared with reference  geoid obtained from ground gravity measurements.

Part 3/Chapter 2:

Hotine Integral

  • Hotine integral developed in 1969.
  • Hotine integral is solution to second kind of GBVP which uses geoid as a boundary surface and gravity disturbances as boundary condition on the geoid.
  • kernel function???infinite series?closed expression?
  • Using a remove-restore procedure, Hotine integral computes geoid undulations N subscript r, from contribution of residual gravity disturbances.
  • mathematical formula/models/equation is given in dissertation.
  • modification of Hotine integral kernel function?

Molodenskii Integral Series:

Height anomaly determination using molodenskii integral series:

  • Molodenskii series developed in 1960.
  • solution to third kind of GBVP, earth surface is a boundary surface and gravity anomalies delta g on earth surface as data.
  • uses stokes integral as its kernel function.
  • gravity anomalies on telluroid is used.
  • computation point,running element, spherical distance, terrain inclination angle,etc?
  • realtion between molodensky and stokes integral?
  • free-air gravity anomalies linearly correlated with topographic height-its an assumption made by Moritz-1980.

Other Considerations:

  • requirement of wavelength resolution:
  • relation expected accuracy of geoid and wavelength resolution of gravity anomalies?
  • quantitative analysis must be done. point gravity measurements data are needed.
  • data obtained from geodetic division of geomatics canada and KMS of Denmark of both flat and mountainous areas.
  • global geopotential model EGM96?
  • gravity anomalies data table 2.1
  • geoid undulations calculated from stokes integral.
  • table 2.2 statistics of residual geoid undulations.
  • power spectral density of gravity anomalies/hankel transform/covariance model of gravity anomalies?
  • frequency domain/fourier transform?
  • PSD power spectral density.
  • peroidogram/anisotropic.
  • To achieve 1 cm geoid undulations accuracy minimum wavelength of 14 km inflat areas and 5km in mountain areas is required.
  • 10cm-70km flat-40km mountaion.


  • what are the requirements of precise geoid determination?
  • centimeter geoid accuracy-minimum wavelength 14 km in flat areas-5 km in mountainous areas.
  • 10 cm-70 km Flat-14 km Mountainous.
  • kernel functions of above mentioned integral is needed for the integration to be carried  over limited regional area.
  • integration radius???


AGS and its potential contribution.

  • highly technical implementation/mechanization of AGS which contains strapdown inertial navigation system(SINS) and differential global positioning system(DGPS). mechanize?
  • SINS errors and DGPS errors and contribution to airborne gravity measurement errors.
  • potential resolution and accuracy of geoid determined from airborne gravity measurements?
  • plz someone enlighten me about PSD analysis.

System Principle:

  • based on newton’s second law of motion. AGS measures the force and mass is known then acecleration due to gravity can be computed.
  • Explain me Inertial Reference  Frame(i)/position vector of a moving object from origin of inertial reference frame/second derivative of this position vector/specific force vector/all the gravitational attraction acting on a moving object.
  • local-level frame: inertial reference frame to local-level frame.
  • *gravity vector is sum of earth’s gravitational attraction vector and earth’s centripetal attraction acceleration vector due to earth’s rotation wrt inertial reference frame.
  • *derivative of velocity v dot:dynamic accn of aircraft and velocity vector.
  • *the dynamic/kinematic accn, velocity and position of aircraft is obtained using GPS with carrier phase and phase rate measurements.
  • *see the figure/play with the figure.
  • *specific force is measured by set of three accelerometers of an inertial frames. this specific force is transformed to local-level frame from body frame through transformation matrix and parameters for transformation matrix is given by attitude of aircraft roll, pitch, yaw. These roll, pitch, yaw is rotation of the body frame to local level frame sequentially through x,y,z axes.
  • *see the menace transformation matrix formula in that thesis.
  • *attitude angles are obtained by integration of angular velocities from body frame to inertial frame after correcting for earth rotation and aircraft rate.
  • *earth rotates but what is the rate of rotation means what is the angular velocity’s of the earth’s rotation, and another question what is the reference frame to measure the angular velocity of rotation of the earth. what is the angular velocity of local level frame with respect to the earth.  and why do we need meridian and prime vertical radii of curvature of reference ellipsoid and ellipsoidal height of INS centre.
  • *gravity vector is usually expressed as sum of normal gravity and gravity disturbance. normal gravity is function of position and computed from normal gravity formula and if you need better approximation of normal gravity, coefficients of global geopotential model can be used. so the core task/problem of  of AGS is the determination of gravity disturbances. vector gravity disturbance is vector sum of dynamic accn of aircraft, specific force, zazazaaz, and normal gravity.
  • *vector gravimetry? fundamental equations of vector gravimetry? three components of vector gravity disturbance on each axes of cartesean system.
  • *scalar gravity measures the verical gardient /vertical component of gravity only. the deflection of vertical rarely exceeds 30″ and the vertical component of gravity and magnitude of vector gravity has small difference of 0.05 mGal which well above the accuracy for most of the applications.
  • *two mathematical procedure: starapdown inertial scalar gravimetry (SISG) and rotation invariant scalar garvimetry (RISG). figure description is given in dissertation plz see.
  • *error model of airborne gravimetry consists: error in gravity disturbance vector, attitude errors due to initial misalignment and gyro measurement noise, accelerometer noise, errors in computed aircraft’s accn and velocity, error in computed normal gravity due to position error, time sync error between INS and GPS.

Effect of INS errors in AG

  • INS has two errors:attitude errors contributed by gyro errors and specific force error resulted by acceleration error. attitude errors induce errors on each component of gravity disturbance vector when specific force exists.
  • *constant velocity of an aircraft is ideal operational environment which removes the specific force from horizontal channels.
  • *phugoid motion??? what is this?
  • phugoid motion induced gravity error of 0.5 mgal when the horizontal attitude error is 1″ and 2.5 mgal for 5″.
  • *during flight specific forces and dynamic/kinematic accn usually vary.
  • * INS has ring-laser gyro, and its output errors can be approximated/modeled. angular velocities, scale factor error when transforming gyro output impulse to angular velocity, errors from misalignment between gyro physical body frame and system body frame , gyro mounting errors, and gyro bias.
  • *total attitude error can be approximated from constant initial attitude error, a constant gyro drift rate, and all accelerometer bias.
  • *specific force error is approximated from accelerometer bias error, linear and quadratic scale factor errors which is function of specific force, sensor noise. specific force error is modeld as second order Gauss-Markov process.
  • *The gravity error is induced by INS attitude errors and specific force error. perfect flight condition means zero horizontal component of specific force.

Effects of GPS errors in AG

  • *The major function of GPS subsystem in INS/GPS AG is to provide the precise kinematic description of aircraft trajectory.  the observables are carrier phase and pseudo range data typically in a double differenced form. Acceleration is derived from GPS derived velocity  by single differentiation or GPS derived position by double differentiation.
  • *accuracy of airborne DGPS in the range of 50km to 200km is 10cm with high quality receivers and reliable ambiguity resolution. accuracy of aircraft velocity is 1cm/s in hz direction and 2cm/s in vertical direction when using carrier phase observation.
  • *1cm/s velocity error will induce 0.15 mgal gravity error and 10 cm position error will generate 0.03 mgal gravity error.  Expected accuracy of AGS is 1mgal so the effect of velocity and position can be neglected. but GPS-derived acceleration error on airborne gravity must be investigated.
  • *expected tropospheric errors, orbital errors, carrier phase ambiguity fixing have little effect on accn determination. high frequency errors such as receiver phase noise and multipath can be reduced with low pass filter but these errors are amplified by double differentiation which would be critical part which effect achieveable accuracy of AGS.
  • *GPS height measurements error from equation of ellipsoidal height, height from laser altimetry, error from orthometric height, and geoidal undulations error. laser range errors, geoid variations, lake surface dynamics are the causes.
  • *the error in gps positioning extremely depend upon distance between airbornereceiver and and master station at the ground.


Effect of combined GPS/INS errors on AG:

  • estimated gravitational attraction is the difference between INS specific forces and GPS-derived accns.
  • current navigation grade strapdown inertial systems can not measure the horizontal component of gravity components with meaningful accuracy say for example 2~3 mgal.


Effects of other critical factors on Airborne Gravimetry:

  • actual sensors errors, vibration noise, and aircraft dynamics should be considered. if INS and GPS sensors are kept in same location in aircraft most of the errors from vibration noise and aircraft dynamics is eliminated by the differencing process. but putting those sensors on the same location doesnt happen.
  • What is the differencing process?
  • Time synchronization between GPS and INS sensor/system is critical for airborne gravity system. To keep gravity error induced by time synchronization below 1mgal, time sync between GPS and INS system should be better accuracy than 1 ms.
  • Aircraft dynamics have serious effects on performance of airborne gravity system. Aircraft dynamics is divided into longitudinal dynamics and lateral dynamics. longitudinal dynamics is main concern due to motion in vertical direction. the phugoid motion has potential effect on airborne gravimetry.
  • from an operational point of view, aircraft dynamics can be minimized at high flying altitude and high speed of flight during stable atmospheric conditions. but optimal operational parameters must be chosen as …


Potential geoid resolution from airborne gravimetry:

  • continuation of geoidal undulations upward into space is used to investigate the geoidal signal at flight level. geoid undulations in space at the altitude of h can be estimated from equation.
  • aircraft velocity and flight altitudes affect the wavelength ranges within which gravity information can be detected. lower velocity will allow shorter wavelength.  the effect of flight altitude on wavelength/geoid resolution is given in table.
  • Conclusion: airborne gravimetry has potential to give information required for centimeter level geoid determination in both flat and mountainous areas.


Potential geoid accura y fr m airbor e gravimet y:

  • spectrum of geoidal undulations errors propagated fro m airbor e gravi y erro s says that /wavelength upto 100km-geoidal undulation is 1cm or less/for 500km -10cm or less/1000km-30cm or less. geoid accuracy determination is about 0.1 ppm for wavelength upto 100km, 0.2ppm for wavelengths upto 500km, and 0.3 ppm for wavelengths upto 1000km. Now on the basis of these facts we can confidently say that AIrborne gravimetry can contribute to precise geoid determination.
  • Two gravity sattelites GRACE and GOCE.
  • by obtaining gravity signal for minimum wavelength from 5 to 10 km, local precise geoid can be determined of centimeter level accuracy with medium and long wavelength gravity of those area from global geopotential model. GRACE and GOCE sattelites provide gravity information/field upto wavelengths 100 to 200km. by  combining sattelite and airborne gravity data, centimeter level accuracy is achievable globally.
  • In africa and asia gravity field is not well represented in geopotential model, airborne gravity can be used to get relative geoid of accuracy 10cm to 30cm with profile spacing of 20 to 25km.
  • Nyquist theorem.


Airborne Gravimetry Test Flight Description:

  • flight area should be chosen in such a way that there should be variability in gravity field and and availbility of relatively dense gravity field measurements for reference. and the area is canadian rocky moutains.
  • june 1995 test:
  • *east-west profile of 250km/honeywell LASERF III, which is a high performance INS system/GG1342 laser ring gyros/QA2000 acclerometers/two gps receivers with zero baseline in plane and 5 gps receivers at the ground. only L1 frequency was used.
  • *the gravity field along the trajectory is obtained.
  • september 1996 test:
  • *day1 east-west direction at 4350m height/day2 north-south direction at the same height/day 3 east-west direction at the height of 7300m./examining the effect of flight direction on gravity disturbances/ examining the effect of altitude on gravity disturbance/areas was 100km by 100km/height varies from 800 to 3600 m on that terrain/all flights were flown between 12am to 6am of the nights to minimize the air torbulence./average flight speed was 360km/h.


Summary of the results:

  • principle of Airborne gravimetry with strapdown INS and DGPS subsystem is explained and effect of system errors on gravity determination has been analyzed.
  • potential contribution of airborne gravimetry to determine the relative geoid.
  • geoid undulations can be determined with minimum 10km wavele gth in flat areas and 6km in mountaionous areas with flight speed 360km/h and altitude of 2.5km or lower.
  • Two main contribution of airborne gravimetry:
  1. detailed high precision geoid in areas where medium and long wavelengths are well represented by global geopotential model.
  2. global geoid of more uniform accuracy.
  • geoid undulations with accuracy of centimeter level is expected for wavelength range between 5 to 100km.


Physical Geodesy- खेस्रा Notes.

These notes are from “Physical geodesy LN43” by Peter Vanieck.

  • Sir Gabriel Stokes (1819-1903) – lived for 84 years long OMG!, was mathematician and physicist of Cambridge. He did most amazing contribution to physical geodesy and that is he derived a formula – called Stoke’s Formula. and this formula is most importnant formula in physical geodesy. This formula computes the geoidal undulations from observed gravimetric data. or it determines the geoid related to assumed reference ellipsoid from gravimetric data.
  • N-S and W-E components of deflection of vertical can be computed from gravity anomalies known from all over the surface.
  • Three kinds of Geodetic Boundary Value Problem(GBVP). Study/Read/Research/Play in these.
  • Global Geopotential Model EGM96? Research on it.
  • Spectral analysis? Open course/online course on it? on
  • When will you dive into GRACE and GOCE mission.


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