Land uplift and its implications on the geoid in Fennoscandia

91

Tectonophysics, 91 (1983) 97- 101 Elsevier Science Publishers

B.V., Amsterdam

- Printed

in The Netherlands

LAND UPLIFT AND ITS IMPLICATIONS

ON THE GEOID IN

FENNOSCANDIA

LARS.

E. SJGBERG

Department of Geodesy, Institute of Geophysics, The Universiiy of Uppsala, Hiillby, S-755

90 Uppsala

(Sweden) and National Lund Suroey, S - 801 I2 Giivle (Sweden)

(Received

by Editor

May 31, 1982; received by Publisher

March

10, 1983)

ABSTRACT

Sjoberg,

L.E., 1983. Land uplift and its implications

Wassef and R. Green (Editors), The rate of change observed deloading

Recent

on the geoid in Fennoscandia.

Crustal

Movements,

of the geoid in Fennoscandia

land uplift is accompanied

In: P. VyskoEil, A.M.

1982. Tectonophysics, 97: 97-101.

has been determined

on the assumption

that the

by a viscous flow of mantle below the crust. The model includes

of water masses due to the uplift. The result is compared

with other numerical

the

studies.

BACKGROUND

The land uplift in Fennoscandia is well documented from studies of ancient shore lines and more recently from repeated precise levellings and tide gauge observations. A compilation depicted

of the observed

land uplift

in Fig. 1. It shows a maximum

from various

sources

uplift of 9 mm/year

the Baltic. The observed uplift is based on the instantaneous

(Ekman,

in the northern

1977) is part of

mean sea level and is therefore

dependent on the variations of this surface. Thus the total or absolute land uplift is the sum of the observed uplift, the eustatic raise of mean sea level and the change of the geoid as a consequence of the change of the geopotential. In 1977 Strang van Hees presented a formula for estimating the change of the geoid as implied by land uplift and change of gravity. In essence his formula is the derivative of Stokes’ formula with respect to time. The formula was applied by Dietrich (1979) on Scandinavian land uplift data and Honkasalo’s (1968) gravity change estimates. However, to date no significant estimates of the change of gravity in Scandinavia have been recorded, and this fact limits the use of the previous formula.

004%1951/83/$03.00

0 1983 Elsevier Science Publishers

B.V.

9X

Fig. 1. The observed land uplift in Fennoscandia.

(Compilation

by M. Ekman.) Unit: mm/year.

99

DERIVATION

OF FORMULA

1n the present study we assume that the observed land uplift is accompanied by a viscous flow of mantle mass of density p below the crust of thickness t. Over the continents

the mass shift within

the infinitesimal

total rate of change of the geoid (N) according

solid angle da, contributes

to the

to:

GpR&da,

6iq =

(1)

Y&i

R = mean Earth radius,

where gravity,

ir, = absolute

uplift,

G = Newton’s

constant

of gravitation,

y0 = mean

and:

GJj being the geocentric angle between the points P,and I$ The absolute land uplift is the sum of the observed land uplift ( fi), the rate of change of the geoid due to the change of the geopotential

(k)

and the eustatic

raise of mean sea level (I$,):

it=&+ir+l;v

(2)

At sea we get accordingly

where p = p,/p,

the contribution:

fij = the uplift of the ocean bottom,

and

L,,= /qm.

last term in (3) is caused by the deloading of water masses of density Integrating formulas (1) and (3) we arrive at: fi -

GRp

.I

Yo

~ + ir,, Jl{( 0

1

if

+ ~i)i)/lji

The p,.

- SiEL~,/L/,}dU,

(4)

where: s, =

Pibelongs to a sea area if P,belongs to a continental

i 0

This formula approximation.

area

(without the last term) was applied by Ekman (1977) in a planar The formula will be used in a numerical study of the rate of change

of the geoid in Fennoscandia below. Neglecting the last term of (4) the variance

of gj due to uncorrelated

random

errors of the land uplift data becomes: u$ = CC$~ ( Ac+/~,;)~

(5)

i

where c = GRp/y,,ai= standard error of h (assumed element. Further details are given in Sjoberg (1982).

constant),

and Au, = surface

Fig. 2. Estimated rate of change of geoid in mm/year.

101

NUMERICAL

RESULTS

The numerical eustatic

study was based

on the observed

( fi) of Fig. 1. The

land uplift

change of mean sea level (i?r,) was set to + 1.0 mm/year.

following

were used: G = 6.67. low8 cm3/(g X s2),

constants

Gal, p = 3.27 g/cm3, Since formula integrated

p, = 1.03 g/cm3,

(4) is a linear

straight

forward.

Furthermore,

the

R = 6370 km, y0 = 980

t = 30 km.

integral

equation

However,

using

of the second

approximate

kind & cannot

be

for fi under

the

values

integral sign the formula converges well in an iterative procedure. In our application & was within 0.01 mm/year after three steps of iteration. The numerical result is shown in Fig. 2. The maximum rate of change of the geoid due to mass shifts is 0.68 mm/year. The error estimation with formula (5) yielded errors within 5 pm/year based on oh = 0.3 mm/year. CONCLUDING

For further

details we refer to Sjoberg (1982).

REMARKS

We have found that the maximum rate of change of the geoid in Fennoscandia is + 0.68 mm/year. Our estimates are on the order of 70% of those by Dietrich (1979) and slightly smaller than Ekman’s (1977) estimates. REFERENCES

Dietrich,

R., 1979. Isostatische

Erdkrustenbewegungen Ekman,

M.,

1977. Berakning

Fennoscandia. English.) Honkasalo,

(Postglacial

Profess.

Ausgleichsvorgange,

in Europa.

av geoidens land

Pap. 1977/5,

T., 1966. Gravity

Niveauanderungen

Vermessungstechnik, uplift

National

r&else

i samband

in Fennoscandia

und die Ableitung

von vertikalen

27 (6). med

den postglaciala

and movement

landhojningen

of the geoid.

Summary

Land Survey, Gavle.

and land upheaval

in Fennoscandia.

Ann. Acad. Sci. Fenn., Ser. A., III: 90,

139-141. Sjoberg,

L.E.,

Department Strang

1982. Studies of Geodesy,

van Hees,

102(10).

G.L.,

on the land University

1977. Zur

i in

uplift

of Uppsala,

zeitlichen

and

its implications

on the geoid

in Fennoscandia.

Rep. No. 14.

Anderung

von Schwere

und

Hohe.

‘2. Vermessungsw.,