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Acta Geodyn. Geomater., Vol. 6, No. 3 (155), 383-390, 2009

ACCURACY OF THE RELATIVE GRAVITY MEASUREMENT

Martin LEDERER

Geodetic Control Section, Special Works Department, Land Survey Office, Pod sdlitm 9, 182 11 Praha 8*Corresponding authors e-mail: [email protected]

(ReceivedDecember 2008, acceptedMarch 2009)

ABSTRACT

Precise relative gravimeters achieve the internal precision about a few Gal 1, even in field conditions. Nevertheless thisprecision is in fact concerned with the instant of measurement and can not be confused with the accuracy of the gravity at thegravity station, which is influenced by other effects. The best approach of these two values is question of high-qualityelimination of instrumental errors and time-variable disturbing effects affecting the relative gravity measurements.

KEYWORDS: accuracy, precision, relative gravimetry

( ) ( ) +++= tZotokkgg IIiIi0 , (2)

where g is the reading in [CU], k the given scale

factor, 0k the additional scale factor, ( ) toIi sum of

corrections caused by external influences (Earth and

ocean tides, air pressure changes and other) and IIio

sum of corrections caused by internal influences

(barometrical effect, mechanicalhysteresis and other).Keeping with the formula (2) the possible errorsources of relative measurements is possible todivided to errors caused by

external influences, internal (instrumental) influences.2.1.EXTERNAL INFLUENCES IN GRAVITY

MEASUREMENT

External influences cause real gravity changesand it is necessary to remove them from themeasurement. These effects are independent on

the used gravimeter.

EARTH AND OCEAN TIDES

Application of tidal corrections is possible withthe accuracy better then 1 Gal. Provided that wehave to well know parameters and of tidalwaves. For the accuracy better then 1 Gal is alsoneedful to measure time with the accuracy 0.5 minuteand a latitude with the accuracy ten arc second.According to (Torge, 1989) the basic term for gravitychanges caused by tidal forces is

( ) ( ) ++= =

n

i iiiiit tAtg 1 cos , (3)

1. INTRODUCTIONA precision of relative gravity measurements on

the level of a few Gal offers application also forgeodynamic purposes. The question remains whetherthe inherent instrument precision (root mean squareerror of one measurement) corresponds to the realaccuracy of the gravity at the station. For instance, the precision for gravimeter Scintrex Autograv CG-5

No. 10125 and gravimeter LaCoste & Romberg GNo. 1068 (LCR) reaches values:

Gravimeter Number PrecisionScintrex Autograv CG-5 10125 2.1 1.1GalLaCoste & Romberg 1068 8.9 4.2Gal

The precision in table means an average rootmean square error for one measurement in one typicalday unit. The final value in the table was computed asan average from four years gravity data (Fig. 1).Preservation of so high precision (Scintrex) needsa consistent application of mathematical corrections

and an elimination of other effects, which can affectthe final accuracy.

2. CORRECTIONS OF GRAVITY MEASUREMENTAccording to (Olejnk and Divi, 2002) the

gravity acceleration g at the station is given by theequation

( ) ++= tZogg ir , (1)

where rg stands for the relative value of the gravity

acceleration, io sum of corrections and ( )tZ thedrift of the gravimeter represented as the timefunction. The relation (1) is specified to the followingform

1 1Gal = 10-8 ms-2

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M. Lederer384

TMOSPHERIC PRESSURE CHANGES

Atmospheric pressure changes produce firstlydirect gravity effect (Lederer, 2004). The pressurevariations change in periods from a few hours to oneyear and reach values about a few hPa. For thecomputation of the correction is commonly usedthe empirical form

( )np ppg = 3.0 [Gal] , (5)

where p [hPa] is the measured pressure and np [hPa]normal atmospheric pressure for the internationalstandard atmosphere (ISA). In accordance with theform (5) the pressure change 3 hPa causes the gravitychange 1 Gal. During the day unit is sometime possible to expect bigger changes of atmospheric pressure then 3 hPa. Neglecting of atmosphericpressure changes can produce an error in the gravityoverhangs 1 Gal.

OTHER EXTERNAL INFLUENCES

Among other less important or difficultlyregistered effects, which can influence the gravityacceleration, belong:

pole motion, mass changes inside the Earths body seismical effects, vertical movement of the Earths crust, seasonal periodical and quasi-periodical

influences (if they are not covered elsewhere).

2.2.INTERNAL INFLUENCES IN GRAVITYMEASUREMENT

2.2.1. INSTRUMENTAL ERRORSInstrumental errors mean changes of the

recorded gravity produced by the technicalimperfection of gravimetrical instruments.

where i is the index of the tidal wave, A symbolizesthe theoretical amplitude, is the circular frequency, the theoretical phase in time 0=t , theamplitude factor and the phase shift.

Computation of tidal corrections is nota restricting factor for the accuracy of relativemeasurements.

HYDROGEOLOGICAL INFLUENCES

A hydrological influence in gravity is quite bigand its magnitude significantly exceeds themeasurement accuracy (measuring over longerperiod). The amount of the groundwater and the soilmoisture correlates with many factors (morphology,geological subsoil and other) and it is almostimpossible to mathematically simulate potentialgravity changes. The global models (dynamic modelsfrom the satellite mission GRACE or globalhydrological models, (Andersen et al., 2005)) can help partly. Unfortunately the gravity change is mostsensitive to changes in the station vicinity (60 % ofthe whole hydrological effect is up to 1 km) andtherefore we need additional information.

Measurements of additional physical quantities(precipitations, soil moisture, groundwater and other)nearby the gravity station bring at least partialinformation about the mass changes. Then we are ableto find a possible correlation of one or more measuredquantities with the gravity change. The correction ispossible expressed in a simplified way in term

( )QEPfgHydro ,,= , (4)

where P is the precipitation, E the water evaporationand Q the water outflow in the given area. The possible gravity changes have a seasonal character,

when the amplitude can reach up to 15 Gal,(Harnisch and Harnisch, 2002).

Fi . 1 RMS evolution durin time eriod 2005 2008.

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ACCURACY OF THE RELATIVE GRAVITY MEASUREMENT 385

PERIODICAL ERRORS OF THE READING SCREW(GRAVIMETERS WITH GEAR BOX)

Formally the periodical errors belong to thecalibration errors, because they negatively impactthe reading conversion to the physical unit.

We assume that the periodical errors are simplyfunctions of the reading screw position. Due toimperfections of the gear box and the screws (we donot consider the backlash now), the handling of thesensor is in another position then the reading screwshows. Each specific effect of the periodical error ofthe expected (known) periodPand then the frequency

Pf 1= is possible to express by the phase shift sine

wave

( )( )++= xfAxX sin , (6)

whereA is the amplitude, the phase of the error,x

the original reading and X its real value. The periodical error omission can produce four times bigger error then is the derived amplitude in theextreme case (Trger et al., 1978). For example, thegravimeter LCR G No. 1068 has the amplitude equal5 Gal (Lederer, 2005) for the period P = 7.33 CU.The amplitudes of some LCR meters exceed value10 Gal (Lederer, 2005; Torge, 1989; Figure 2).

DRIFT

In stationary and field operation, gravimetersexhibit a temporal variation in the display of the zeroposition, which is called the gravimeter drift. The driftis caused by fading of spring tensions and byuncompensated or unshielded external effects. Type

and magnitude of the gravimeter drift are the functionof: type and characteristic of the specific instrument

(quartz systems have larger drifts than metalspring gravimeters),

CALIBRATION FUNCTION

The calibration function serves for convertingthe reading from CU (Counter Unit) to the physicalunit (usually to mGal = 10-5 ms-2). The function isusually derived from the lab measurement performed by the manufacturer. It is used a special additionalequipment to obtain a relative calibration table/curve.The calibration table is unique for each gravimeterand represents the realization of non-linearcomponents of the transformation from CU to thephysical unit (accuracy 10-3. . . 10-4). The calibrationfunction has the following form

( )zFg= ,

wherezis the reading in [CU]. For an approximationis usually used 3rd 5thorder function.

In addition to this transformation an additionallinear factor (scale factor) is determined atgravimetrical baselines every year (Lederer, 2002).The scale factor (SF) changes with time and should be

influenced by other effects, for instant humidity. TheSF accuracy derived from the main gravimetricbaseline is about 1.10-4. It represents possible error 10Gal for the gravity difference g = 100 mGal. Anerror on the level 10-5 should be caused by using justthe linear scale factor model. It can produce the errora few Gal for the gravity difference g= 100 mGal.

New LCR gravimeters are equipped with thecapacitance beam position indicator (CPI). It reducesoperators fatigue and the galvanometer can be set tohave greater sensitivity then the optical system. TheCPI must be regularly calibrated by the reading screw.

With the feedback (magnetic induction orelectrostatic force is used to balance the beam in thezero position) is possible to achieve the most accuratereading. The feedback linearity and the primecalibration are supposed.

Fig. 2 LCR G No. 1068 periodical errors.

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Fig. 3 Gravity transfer scheme (Scintrex CG-5).

Czech Republic the vertical gradient ranges from 2500to 4500 E (Olejnk, 1988).

We consider the gravity measurement withthe gravimeter Scintrex CG-5 (Fig. 3) at the stationwith an extreme vertical gradient ( 4500=zzW E) for

the model example. The common measuredinstruments height is 47. meash cm. The difference

from the normal gradient 1415= zzW E together

with 27.0202.0. = meashh m causes the error

38.2 Gal only by applying the normal verticalgradient instead of the directly measured one.

3. RECOMMENDATIONSThe enumeration of disturbing influences

(Chapter 2) itself shows the complexity of the theme.It takes big demands on the measurement and the post

processing of the relative gravity measurement. Weare not able to eliminate or evaluate all effectssatisfactory yet. Therefore, for the high measurementaccuracy, we must try to eliminate disturbing effectsevery which way.

The recommendations are focused on the relativegravity measurement in the gravity or geodynamicalnetworks, for that reason are not all of them needfulfor gravity prospecting.

3.1.ELIMINATION OF EXTERNAL EFFECTSA good knowledge of the tidal parameters ,

and application of the latest tide potential

development allows us to remove tidal effects withoutan accuracy losses.Removing of the hydrogeological influences

remains the hot issue. The topic is really complexand needs a lot of additional measurements, such asa groundwater level, precipitations, temperature, soilmoisture and so on. In addition to that, the quantitieshave to be measured permanently. It is not possible tofulfill these demands for relative gravitymeasurement. The global gravity and hydrologicalchanges models help partly, but the gravity is mostsensitive to mass changes in the near vicinity. Thehydrogeological changes have a strong seasonal effect

and they may significantly influence the relativegravity measurement performed in stages. For the bestelimination is useful to measure every stage in thesame time of the year. In case we know the time behavior of the gravity change (nearby GGP 33

stations) we can extrapolate seasonal gravityvariations for close relative stations. However we arestill talking about the approximate solutions.

Air pressure changes are not so criticalaccording to the magnitude of the effect. Applying ofthe formula (5) secures the sufficient accuracy (lessthan 1 Gal).

unstable voltage, reading errors, backlash of the measurement screw, shocks (sudden reading change).2.3. VERTICAL GRADIENT

A possible error caused by neglecting of thecorrect vertical gradient is not produced bythe gravimeter imperfections but depends on thegravimeter design.

The sensor of the gravimeter is in the specificheight h above the gravity station (reference height)during the measurement. This distance is typical foreach type of the gravimeter. It should be from a fewcentimeters (gravimeter LaCoste&Romberg withouta plate) to a few tens of centimeters (gravimeterScintrex CG-5 with an original tripod). The needfulcorrection

zzWg is possible calculate by well known

form

,hWg zzWzz = (8)

where zzW is the vertical gradient and h the vertical

distance between the gravity station and thegravimeter sensor (Fig. 3).

A normal vertical gradient for a latitude50= and for the ellipsoid GRS80 is equal to the

value 3085=zzW E2 (GRS80, 1980). The normal

gradient is used for the transfer of the measuredgravity to the reference, using the formula (8), whenthe vertical gradient was not measured directly. In the

21 E = 10-9 s-23 Global Geodynamic Project, stations with absolute and superconducting gravimeters

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M. Lederer388

First reading carried out in the office (springrelaxation after long clamping).

Suitable measuring schedule secures the best driftapproximation. Stable conditions during themeasurement (temperature, humidity) and protection against meteorological conditions(wind, sunshine) help to minimize drift changes.

Identical instrument orientation minimizesmagnetic field influences.

Setting the approximate reading beforeunclamping and symmetrical reading afterunclamping reduces the mechanical hysteresis ofthe sensor.

Usage of directly measured vertical gravitygradient is necessary condition in order toguarantee the best accuracy.

On average, the gently handling with thegravimeter during the measurement and the transporta priori decreases errors.

3.4. GRAVITY CONTROL OPTIONType of the stabilization and the station locality

selection can also influence the final accuracy. Thelocality choice has to fulfill demands, which have toguarantee the best quality of gravity measurements.

It should be chosen an undisturbed place ina sufficient distance from artificial disturbing effects(e.g. big cities, railway and highways). The placessituated nearby possible mass changes 5 such as dams,mines, rivers are also unsuitable. Higher altitudes arenot fit for the relative measurement, if it is notnecessary. A scale factor uncertainty and bigmeteorological changes in mountains can significantlydecrease the expected accuracy.

3.2.INSTRUMENTS EQUIPMENTIt is important to notify that the relative

gravimeters work on the accuracy level 10-8. Sucha high accuracy of the sensor is paid by the sensitivity

to other physical quantities. A manufacturer is tryingto shield the sensor against disturbing effects, but veryoften the protection does not work accurately, howwas proved by many lab tests.

For the precise relative gravity measurementsis possible to use gravimeters Scintrex CG-3M(Hugill, 1988) and Autograv CG-5, gravimeterBurris ZLS (Jentzsch, 2007) and gravimetersLa-Coste&Romberg G and D (imon, 1995). Thesegravimeters should have some errors described in paragraph 2.2, which is needful to eliminate. Fromthis reason is recommended to do:

periodic checking of the meters according tomanufacturers instruction,

annual probing of the scale factor, regular lab tests, periodical errors checking 4 (gravimeters with

gear box).

Derived correlations are necessary tomathematically remove from the realizedmeasurements.

3.3.MEASUREMENTAn appropriate measuring schedule can partly

eliminate the instrumental errors. We can summarizesome recommendation in a few items:

Fig. 4 Gravimeters Scintrex Autograv CG-5 (left)and ZLS Burris (right).

4Turning clockwise with the nulling dial before reading is important for elimination of the backlash5 If it is not the research goal

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for the Scintrex meter is about 6 Gal and for LCRmeter about 11 Gal in ideal field conditions (alleffects are eliminated as good as we can). Thesevalues belong to specific instruments and can not besimply generalized because each meter has its specificqualities. Also is impossible to include some aspectsto the final accuracy (for instance an error from poorknowledge of the scale factor increases with thegravity difference).

In conclusion we can say that the precise relativegravity measurement is possible to use forgeodynamic purposes, but in close cooperation with

the absolute gravity measurement. The unstable fieldconditions, scale factor and drift uncertainty and alsothe necessity of the reference value are the mostimportant effects which have negative influence to thefinal accuracy of relative gravity measurements.Hydrological changes are most important for longer or periodical measurements where they can cause bigdifferences for stages measurements. It is also themain restricting factor in accuracy for precise relativeand also absolute instruments.

Relative gravity measurements should be usedfor extending of absolute measurements, possibly forsmaller local controls, where the aim is focused to

relatively quick and bigger changes of gravity (forinstance fast changes of the river surface, underminedareas and so on). Application of a group ofgravimeters (at minimum 2 instruments) is necessary.

The stabilization itself is very important forgravimetric surveying. A solid rock or a concrete pillar strongly embedded in the ground((Zemmick ad, 2003), type II stabilization)represents good base for the precise gravitymeasurement. The pillar is not suitable for the gravitymeasurement because of possible troubles with thevertical gradient measurement.

3.5.SUMMARYAll disturbing effects described in Chapter 2 are

transparently summarized in Table 1. It is very

complicated to estimate the extent of elimination andit depends on many other parameters (usedgravimeter, topography, morphology, weatherconditions and so on). Therefore the values in theTable 1 (column Elimination) are just the qualifiedestimates.

4. CONCLUSIONIt is hard to determine the achievable accuracy of

relative gravity measurements. There are manyexternal and internal factors which can affect the finalaccuracy. Considering recommendations mentioned inprevious chapters we can try to estimate the accuracy

for gravimeters LaCoste&Romberg No. 1068 andScintrex Autograv CG5 No. 10125. The values fromthe Table 1 and the precision mentioned in Chapter 1are used for the estimation. The estimated accuracy

Table 1 Overview of disturbing effects and its elimination estimation.Effect Magnitude [Gal] Elimination [Gal] Depend on

Earth and Ocean Tides 280 < 1 a accuracy

Hydrogeological influences tens 5 Hydrogeology

Air pressure changes units < 1 Air pressure

Calibration function tens 110/100mGal Gravity difference

Periodical errors tens 15 Gravimeter

Gravimeter drift hundreds 15 Gravimeter

Tilt effect units < 2 Observers care

Barometrical effect tens units Air pressure

Mechanical hysteresis 1..3 < 1 Gravimeter

Temperature effects tens units Temperature

Magnetic field units < 1 Orientation

Other effects units 12 Gravimeter

Vertical gradient error tens 12 Gravimeter, terrain

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Lederer, M.: 2004, Geodynamical influences in the resultsof gravimetrical measurement. (Phd. Thesis, inCzech), VUT, Praha.

Lederer, M.: 2005, Periodical errors of LaCoste & Romberg

gravimeters. Geodetick a kartografick obzor, 51(93), No. 1, 1 8, (in Czech).Olejnk, S.: 1988, Vertical gradients determination on the

stations of the Czech Gravity Network by calculation.Geodetick a kartografick obzor, 34 (76), 5, 112117, (in Czech).

Olejnk, S. and Divi, K.: 2002, Gravity system 1995 on theCzech Republic territory. Geodetick a kartografickobzor, 48(90), No. 8, 145161, (in Czech).

Plink, V., Kapar, P. and Lederer, M.: 2005, Effect of themagnetic field on LCR gravimeters. Cahiers duECGS, 22, 8993.

imon, Z.: 1995, Gravimeters LaCoste Romberg.Geodetick a kartografick obzor, 41 (83), No. 3, 5254, (in Czech).

Trger, L., Divi, K. and Olejnk, S.: 1978, Examination ofperiodical errors by gravimeter WORDEN and CG-2(SHARPE). Geodetick a kartografick obzor, 24(66), No. 4, 8891, (in Czech).

Torge, W.: 1989, Gravimetry. Berlin - New York, Walter deGruyter.

Zemmick ad: 2003, Methodical instruction for worksin the fundamental gravity field, Praha, (in Czech).

Relative gravimetry is also possible to use for regionswhere is possible to detect larger changes than approx.2 - 3 Gal per year (for example Fenno -Scandinavian uplift).

An important role of relative gravimeters is forgravity gradient measurement. The precise gravityvalue transfer needs very well knowledge of thevertical (possibly horizontal) gravity gradient. There isnot another such precise method than the relativegravity measurement.

REFERENCES

Andersen, O.B., Hinderer, J. and Lemoine, F.G.: 2005,Seasonal gravity field variations from grace andhydrological models, GGSM 2004 IAG InternationalSymposium, 129. Springer Verlag, Heidelberg.

Geodetic Reference System: 1980, Bulletin Godsique, 54,

3.Harnisch, M. and Harnisch, G.: 2002, Seasonal variations ofhydrological influences on gravity measurements atWettzell, Bulletin dInformation, No. 137, 1093710951.

Hugill, A.L.: 1988, The New Scintrex CG-3 AutogravGravity Meter, Description and Test Results. PaperPresented at the ASEG/SEG Conference, Adelaide,Australia.

Jentzsch, G.: 2007, The automated Burris gravity meter a new instrument using an old principle. Proc.Symposium on Terrestrial Gravimetry: Static andMobile Measurements, St. Petersburg, Russia, 2023.

Lederer, M.: 2002, Main gravimetric baseline, Geodetick a

kartografick obzor, 48 (90), No. 3, 4552, (inCzech).

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