Geophys. J. R. astr. Soc. (1984) 17, 385 -413
Flexural models of continental lithosphere based on
the long-term erosional decay of topography
Randell
S t e p h e n S O n R e s e a ~ c hSchool of Earth Sciences, AustralianNatinnal
University, Institute of Advanced Studies, PO Box 4 , Canberra, ACT 2601, Australia
Received 1983 June 29; in original form 1983 Pebruary 25
Summary. The most easily observed geophysical characteristic of continents
is their surface topography. Topographic power in the spectral range
1000-1 00 km has been computed for North American geological provinces
and is found t o decrease systematically as the tectonic age of the underlying
crust increases. These data have been quantitatively modelled in terms of
topographic erosion and stress relaxation within the continental lithosphere
through geological time spans. The lithosphere is treated as a thin (Maxwell)
viscoelastic plate which deforms flexurally in response to the vertical surface
loads of topography and its erosion. Effective rheological model parameters
are flexural rigidity D , a measure of the lithosphere’s elastic strength, and
viscous relaxation time constant r , a measure of its viscosity. Erosion is
assumed to occur at a rate linearly proportional to the height of topography
at any given time, the proportionality being determined by the erosion time
constant u. The models are constructed in space and time frequency domains
using simple linear systems theory.
Two loading schemes are considered. Model 1 assumes that a flexurally
competent lithosphere is suddenly loaded by topography which subsequently
erodes whereas model 2 ignores the emplacement and erosion of topography
during orogenesis when the lithosphere is thermally activated and considers
only the load effected by erosion beginning some time later when the lithosphere has cooled and strengthened.
Both models predict that the remnant topography of very old regions such
as the Canadian Shield can be maintained, in fact is favoured, by a viscoelastic
lithosphere with r as small as 1-10Ma. The continued existence of such
topography does not necessarily imply significant long-term lithospheric
strength. Model 1 results indicate low values of the flexural rigidity D ,
10”-1022Nm, which are interpreted to characterize not so much presentday D but that which was effective when topography was formed. Model 2
more successfully reproduces the data and constrains the product rD to
values, -loz4 -loz5Nm Ma, which compare very favourably to those found
-
*Now at: Institute of Sedimentary and Petroleum
3303. 33rd Street, NW Calgary, Alberta T2L 2A7, Car.ada.
Geology.
Geological Survey
of Canada
3 86
R. Stephenson
from studies of individual loads on continental lithosphere. The model results
show that the erosion time constant u is probably wavenumber-dependent
and falls in the range 200-400Ma for the topographic wavelengths under
consideration.
1 Introduction
The Earth’s mechanical behaviour in reply to stresses at high strain rates is revealed by its
seismic wave response. The strength of the Earth’s crust and mantle in such a model appears
to be very great. The ability of the Earth to resist permanent deformation by shear stresses
applied over much longer lengths of time is certainly less. Similarly, the mechanics of the
deformation induced by such stresses will most likely be different. The strain rates of
interest here pertain to processes which occur over millions of years; such processes are
sometimes referred to, in the temporal sense, as ‘geological’ processes.
Historically, studies of the ‘geological’ state of stress and mechanical behaviour of the
Earth have been based on the existence of gravity anomalies (e.g. Jeffreys 1976). Gravity
anomalies are indicators of lateral mass heterogeneities, and therefore non-hydrostatic
stresses, in the Earth’s interior. Long-wavelength gravity anomalies may be related to geodynamic processes such as themal convection in the mantle (e.g. McKenzie 1977), itself
perhaps the mechanism of lithosphere plate movements (e.g. Davies & Runcorn 1980) and
the ultimate source of the tectonic forces responsible for mountain building. In turn, the
crustal topography and internal density anomalies formed during mountain building episodes
are responsible for much of the higher-frequency gravitational variation observed across continental regions. Having been established by tectonic forces, such gravity anomalies, and their
implicit crustal mass heterogeneities, apparently persist in geologically very old regions such
as Precambrian shields. This observation has sometimes been used to argue that the outermost portion of the Earth must possess a finite strength, at least within such a time frame,
only above which can deviatoric stresses induce mechanical failure. This apparently rigid part
of the Earth, consisting of the crust and perhaps some of the upper mantle, is conformable
with the plate tectonic concept of a rigid lithosphere, with its continental loads, passively
drifting across the Earth’s surface above an effectively inviscid, convecting mantle. Such a
mechanically defined lithosphere may be different, particularly in terms of its apparent
thickness, than ‘lithospheres’ modelled on the basis of seismological or thermal observations.
Geophysicists have normally modelled this mechanical lithosphere as a thin, infinitely
planar, plate for which the physical behaviour in response to transverse and in-plane forces is
reasonably well understood (eg. Nadai 1963). There are four general rheological classes, in
the geological time frame, in terms of which such a lithosphere can be considered. They are:
( 1 ) The lithosphere is elastic and will indefinitely sustain shearing stresses with no
apparent time-dependent effects.
( 2 ) The lithosphere is characterized by an elastic-plastic rheology . This type of rheoid
possesses a finite yield strength which may be depth-dependent. The upper part of the
ljthosphere likely deforms cataclastically when the yield strength is surpassed whereas the
lower part likely deforms ductilely . The demarcation between the two failure regimes would
be related to confining or lithostatic pressure (Turcotte, McAdoo & Caldwell 1978;
Beaumont 1979). The purely elastic model (1) can be considered to be a special case of the
elastic-plastic class of models which has a yield strength which is never exceeded during the
geological processes under consideration.
( 3 ) The mechanical lithosphere is characterized by linear viscoelastic (Maxwell) rheology .
A Maxwell body deforms in response to an instantaneous stress change such that there is
immediate elastic strain followed by viscous flow at a constant rate. There is no yield
Flexural models of continental lithosphere
387
mength below which the viscous relaxation fails t o occur. In this respect, the viscoelastic
class of models is fundamentally different from the elastic-plastic class. The elastic model (1)
can be considered to be one case of the viscoelastic class of models in which the time
constant characterizing the viscous relaxation is too great to allow significant relaxation to
take place within the upper bound of the geological time frame (-1000Ma).
(4) The lithosphere is a non-linear rheoid which has pressure-, compositional- and
temperature-dependent properties.
The true intrinsic rheological structure of the mechanical lithosphere almost certainly
falls into class (4) at least with respect to pressure and temperature dependence. Models in
class 4 are analytically difficult, whereas geophysical applications of those in classes ( l ) , ( 2 )
and ( 3 ) have been numerous and reasonably successful (cf Forsyth 1979). In such cases,
rheological properties are usually considered to be depth-integrated such that they are
described by effective model parameters. Nevertheless, there is no general agreement on
which of the three more analytically tractable types of rheologies best represents the largescale deformational behaviour of the mechanical lithosphere.
The general approach of geophysicists has been to compare theoretical deformations,
predicted by the thin plate model having one of the aforementioned rheologies, to observed
deformations associated with known crustal loads. The observations have either been made
directly, at the surface of the crust, in cases such as the flexural bulging up of the oceanic
lithosphere seaward of trenches due to the bending load associated with subduction (e.g.
Hanks 1971) and such as the symmetric downwarping and uplifting of the crust to form
bathymetric moats and uplifts circumscribing seamounts (Walcott 1970a; McNutt & Menard
1978; Lambeck 1981) or indirectly, measuring deformations at the base of the crust and
elsewhere by means of gravity and geoid anomalies (e.g. Watts & Cochran 1974; Watts,
Cochran & Selzer 1975 ;Cazenave et al. 1980; Lambeck 1981 ; Lambeck et al. 1984).
Ocean crustal loads such as single or linear groupings of seamounts are usually assumed to
be formed essentially instantaneously and to experience little modification subsequently.
The deformational response of an elastic lithosphere to such a load is time-invariant, its
regionality being dependent upon the lithosphere’s flexural strength. On the other hand, the
response of a viscoelastic lithosphere changes with time as the initial elastic (flexural) stresses
are viscously relaxed, allowing the seamount to approach a state of local isostatic equilibrium. However, there is an additional temporal component involved in the analysis of
seamount loads: the elastic response of the lithosphere depends on the lithosphere’s effective
thickness which is known to increase with its age (spatially, as it is further removed from its
locus of formation, the mid-ocean ridges), Partly because of these temporal complications
many geophysicists have tended to analyse seamount loads in terms of the elastic class (1)
of models, ascribing perceived time variations to the structure and age of the local lithosphere. Recently, however, an extensive examination of seamounts in the Pacific Ocean,
carefully analysed with consideration to factors such as ocean crustal and seamount ages and
densities has led to the adoption of a viscoelastic class (3) model by Lambeck and others
(e.g. Lambeck & Nakiboglu 1981; Lambeck et al. 1984).
Analogous studies of the mechanical response of the continental lithosphere have been far
fewer owing to difficulty in the discrimination of discrete surface loads. Mechanical analysis
of intracratonic basins requires knowledge of or, more usually, assumptions regarding
initiating loads since the weight of the sediments themselves is normally insufficient to
account for the observed degree of basin subsidence. Occasionally the presence of a distinct
gravity anomaly, without reference to the origin of the presumed governing crustal load, has
been used to study the mechanical behaviour of the continental lithosphere (Walcott
1970b).
3 88
R . Stephenson
In general, the greatest existing vertical load acting upon the continental lithosphere, and
therefore inducing some form of deformational response, has been thought to be the distributed load of the surface topography, The deformation of the lithosphere thus associated
can be indirectly measured by the (wavenumber domain) admittance (generally known as
the isostatic response function) between observed topography and gravity anomalies derived
from geographically large regions (Dorman & Lewis 1970).
Dorman & Lewis were interested in inverting observed admittances in order to reveal
changes in the density structure of the crust related to topographic uplift. They did so
(Lewis & Dorman 1970; Dorman & Lewis 1972), using United States gravity and topography data, in terms of a generalized local isostatic compensation model. Later, their data
were reconsidered in terms of a regional compensation model by Banks, Parker & Huestis
(1977) who developed a method of geophysical inversion based on linear programming
techniques. Similar techniques and models were used by Banks & Swain (1978), McNutt
& Parker (1978), Stephenson (1978), and McNutt (1980) t o interpret respectively isostatic
response functions from East Africa, Australia, the Canadian Shield and the Phanerozoic
orogens of the United States. The basic techniques originarly developed by Dorman & Lewis
have also been extensively applied to data from oceanic regions of various ages and tectonic
character (McKenzie & Bowin 1976; Watts 1978; Cochran 1979; Detrick & Watts 1979;
McNutt 1979; Sandwell & Poehls 1980; Louden 1981 ; Watts & Ribe 1982).
Estimates of the flexural rigidity of continental lithosphere based on elastic isostatic
response models are up to four orders of magnitude less than those based on models of the
isostatic compensation of individual continental features (e.g. Walcott 1970c; Haxby,
Turcotte & Bird 1976; Beaumont 1981). McNutt & Parker (1978) argued that this was a
result of stress relaxation in the lithosphere beneath old topography but, in general, the
application of the response function method to continental studies has not been universally
accepted as successful (cf Forsyth 1979; Cochran 1980). One reason for this failure is the
requirement that topography and gravity data encompass large geographical regions with the
result that the areas which have been studied are geologically and tectonically heterogeneous.
Further problems are found in the complex nature of the origin of continental topography
and in its subsequent evolution and erosion. The effect of these complications is that, unlike
in oceans, a considerably percentage of the surface gravity field occurring in continental
regions is not correlatable to topographic variations; in fact, the coherence between topography and gravity in regions other than very young orogenic belts is often so low throughout much of the spectral range for which observations are available, that the validity of any
computed transfer function between them must be questionable (Stephenson 198 I ) . These
problems have been discussed before. Both Forsyth and Cochran have pointed out that the
model of topography being loaded on to a flexurally competent lithosphere is probably an
unrealistic one; in general, continental crust and presumably its topography are generated
in thermally weakened zones of orogenesis. Both authors suggest that low values of flexural
rigidity based on analyses of continental isostatic response functions reflect the strength of
the lithosphere during the tectonically active stage when topography was formed.
The implication is that continental lithosphere, like its oceanic counterpart, thickens
and grows more rigid as it ages and cools. Heat flow (Pollack & Chapman 1977; Sclater,
Jaupart & Galson 1980) and seismological data (Kono & Amano 1978) support such a
model. If this is the case, however, it becomes necessary t o take into account other timedependent processes as well. The isostatic unloading effects of erosion occurring throughout
the period during which the lithosphere achieves greater flexural strength, for example, may
have to be explicitly modelled. Only if the lithosphere behaves as an elastic plate with timeinvariant properties, as in the class (1) of models listed earlier and as has been the case in the
Flexural models of continental lithosphere
389
continental response function studies to date, is erosion an inconsequential event. In this
kind of model the isostatic response function is not a function of time, the reaction of the
crust to erosion being instantaneous; at any cbservation time it is the remaining topography
alone which is flexurally compensated. If the rheology of the lithosphere falls into one of
the other three general classes of rheological models (elastic-plastic, viscoelastic and nonlinear depth dependent), however, then it is necessary in considering the isostatic response
of continents to incorporate the influence of the progressive decrease in the topographic
load resulting from erosion.
A premise of the present work is that in continental regions the greatest vertical load
isostatically deforming the lithosphere is not existing topography but rather the topography
which no longer exists: that it is not the loading that controls the stress state, but the
unloading. A further premise is that the mechanical behaviour of the continental lithosphere, because its topography erodes and the erosion is isostatically compensated, can be
studied by direct observation of topographic heights of continental regions, in a manner
somewhat analogous to directly observing crustal downwarping adjacent to seamount loads
in oceans. In this way the problem arising from the significant incoherency between
continental gravity and topography is avoided; similarly, the need for model assumptions
regarding the depths in the crust of density perturbations in order to interpret gravity
anomalies is also negated. Eventually, of course, the models must also be consistent with the
observed gravity.
These advantages are partly offset by the need to postulate some modelof erosion suitable
for large-scale topography and long periods of time, until now a virtually untouched subject.
In order to maintain a high degree of mathematical simplicity, given the first-order nature of
the investigation, a linear erosion model is adopted. This allows explicit analytical modelling
to be carried out in terms of the third, or viscoelastic, class of rheological models. The purely
elastic class of models is treated as a subset. The models are used to calculate the form of the
decay of continental topography through geological time spans and are compared to the
observed power spectra of topography of North American geological provinces of vastly
different tectonic ages. Particular emphasis in the present work is placed on whether or not
the results support or refute the presence of viscous relaxation of elastic stresses and, by
implication, the existence of a finite yield strength in the continental lithosphere during
geologically observable lengths of time; or, in other words, whether the lithosphere in such a
time frame is best represented by an elastic, elastic-plastic, or viscoelastic rheoid.
2 Theoretical framework
Erosion can be thought of as a feedback mechanism. There are two competing isostatic
effects to consider. (1) It is known from geological inferences (e.g. Ambrose 1964) and
from direct observations (e.g. Adams 1980) that regions of high topographic relief erode
more rapidly than regions of low relief. But it is recognized, from the existence of pediplains
(e.g. Pugh 1955) and from stratigraphic analysis (e.g. Van Houten 1981) that eroded regions
are isostatically rejuvenated. That is, as topography is worn away and the rate of erosion
diminishes, the action of isostasy is one of uplift and enhanced erosion. (2) If the rheology
of the lithosphere allows the stresses resulting from topographic loads to be relaxed in some
time-dependent manner, the topographic height will decrease through time independently
of erosion as a result of ‘sinking’ into the relaxing lithosphere. The action of this aspect of
isostasy is one of reduced erosion rate.
The notion of erosion acting as a kind of feedback suggests that its isostatic effects may
be quantitatively modelled in terms of a linear filter network in which the feedback compo-
R . Stephenson
390
nent processes measured topography into an eroded remnant. The forward component o f
the network would describe the deformation response of the lithosphere to the initial
topography less the erosion as feedback. The ’+.seof linear systems theory results in mathematical simplicity but limits the choice of rheological and erosional models to those which
are linear. Whether more complex models as required to reproduce the observed isostatic
behaviour of the lithosphere satisfactorily can be determined by the success or failure of the
linear approach.
Models of the isostatic behaviour of a viscoelastic lithosphere subject to a diminishing
surface load are developed using three structural elements: (1) the deformation of a thin
viscoelastic plate in response to a general surface load function; (2) a mass wasting principle,
basically stating that the only material available to be eroded is that which originally overlies
the plate; and (3) an erosion law describing the rate at which the surface load diminishes, a
function of that part of it remaining above sea-level through time. For convenience (e.g.
Dorman & Lewis 1970), the models are constructed in the wavenumber domain.
2.1
DEFORMATION O F A THIN VISCOELASTIC PLATE BY A HARMONIC LOAD
The time derivative of the deformation w(r, t ) , at a point r = x i + y j where i and j are
orthogonal unit vectors, produced by a load p(r, t ) on a thin Maxwell viscoelastic plate
overlying an incompressible fluid substratum is given by the solution of (Nadai 1963)
1
DV4W(r, t ) = b ( r , t ) + -p(r, t )
(li)
7
where P(r, t ) is the time derivative of p(r, t). D is the plate’s elastic flexural rigidity, a
function of its thickness TL and the elastic moduli of its constituent material, Young’s
modulus E and Poisson’s ratio U :
D = ETZf 12( 1 -vz);
(1 ii)
7 is the viscous relaxation time constant. If the plate is assumed to be incompressible
(v = O S ) , then D = ETL/9 and r = 3q/E where 17 is the Newtonian viscosity. In the limit as
~ + m
(and therefore ~ + m ) , equation (li) describes the behaviour of an elastic thin plate.
At any point r the plate is assumed to have a uniform rate of vertical deformation; that is,
w(r, t ) is not a function of the vertical coordinate z and therefore
+
v4W(r, t ) = [ v Z I 2 w ( r , t ) =
$1
2w(r, t).
The total load p ( r , t ) is assumed to consist of (1) forces acting on the surface of the plate
because of overlying material of density p o and height Z(r, t ) and (2) buoyancy forces acting
on the base of the plate caused by the displacement of the fluid substratum of density ,om
by the plate deflection w(r, t). Thus,
(2)
~ ( r t,) = Pogl(r, t ) - P m W ( r , t )
where g is the gravitational acceleration and z is positive downwards. Therefore, for a
positive, downward-directed load p ( r , t ) and resulting deflection w(r, t ) ,Z(r, t ) is a negative
quantity.
The surface load Z(r, t ) can be expressed in terms of its harmonic components by its twodimensional Fourier transform F [ Z(r, t ) ] = L (k, t ) where wavenumber k = k , i + k , j ; the
deformation of the plate in response to L ( k , ’) is therefore found by combining (li), (2)
and (2) and taking the Fourier transform:
&(k, t ) +
+ ( k )W(k, t )
~
r
Po
= - - $(k)
Pm
391
Flexural models of continental lithosphere
where
(3ii)
is known as the flexural response function (cf. Walcott 1976).
Equation (3i) describes a linear system in which an input function L ( k , f ) produces an
output signal W(k, t). The transfer function of the system can be found by taking the
Laplace transform L [ ~ ( t =
) ]F(s):
si?(k, S)
-
W(k, 0+) +
9)
@(k,
S)
=-
7
Pm
$(k) [i(k,
S)
- L ( k , 0+)
+ !z(k,
7
S)
]
(4)
for a plate loaded when t > 0 and L(k, 0+) =Iim[t-+OJL ( k , t ) ,simiiady for W(k, O+), and
$ ( k ) and 7 are assumed to be independent of time. The response of the viscoelastic plate at
t = O+ may be considered to be purely elastic, there having been insufficient time for viscous
flow, and, therefore (cf.Walcott 1976; Banks et al. 1977),
Po
W(k, 0+) = - - ~ ( kL(k,
) O+);
Pm
thus, equation (4) simplifies to
W(k, S )
=
TI(k, 3) Z(k, S)
where
(5 ii)
and TI ( k , s) is the transfer function describing the deformational effect on a thin viscoelastic
plate of a surface harmonic load.
2.2
EROSION A N D THE EFFECTIVE PLATE LOAD
The effective surface load, L(k, t ) , on the viscoelastic plate at any time t , may be thought of
as equal to some assumed applied loading function L o ( k ,t ) modified by erosion such that
L(k, t ) =Lo(k, t ) + E ( k , t )
(6)
where E(K, t ) represents the amount of erosion by time t of harmonic topography of wavenumber k . In turn, L(k, t ) consists of (1) a portion remaining above the undeformed plate
surface, measurable as topography H(k, t ) at the time of observation and (2) a portion which
occupies the plate deflection W(k, t ) at that time:
L (k, t ) = H(k, t ) -. W(k, t ) ;
(7)
therefore, from equations (6) and (7)
Lo(k, t ) = H(k, t ) - W(k,t ) -E(k, t ) .
(8i)
If the plate is assumed to be suddenly loaded at t = 0, subsequent to which the topography
is modified only by erosion, then Lo(k, f ) has the form of a Heaviside step function:
(8ii)
Equation (8i) is schematically illustrated in Fig. 1 ; it shows that eroded material does not
continue to load the plate in some rearranged fashion but rather is removed from the system
and deposited elsewhere, presumably at the continental margin (cf Menard 1961).
The input function of the linear network under construction is L [Lo(k, t ) ]=L(k, s)
which is Lo(k)/s by equation (8ii).
3 92
2.3
R. Stephenson
EROSION O F SPECTRAL TOPOGRAPHY
It is assumed that topography in the spectral domain erodes at a rate which is proportional
to its height. Thus
E(k,t)= - [ ~ ( k -‘H(k,t)
)]
+N,(k,t)
(9)
where k(k, t) is the erosion rate of topography H(k, t) at time t and the ‘noise’ term
N , ( k , t ) accounts for the effects on k(k, t ) related to local changes in lithology, vegetation
and climate. [N,(k, t ) is hereafter ignored on the basis that only long-wavelength (> 100 km)
topography which persists for geological lengths of time shall be considered.] The negative
sign arises from the fact that the loading effects of erosion are opposite to those of topography. It is assumed that o(k), which has dimensions of time, is dependent neither on time
nor on wavenumber direction. It is expected that erosion rate should somehow be related to
topographic gradient (Penck 1923; Luke 1972); in the present spectral model, assuming
(a) Model
I
+O
(b)Model 2
tC
Ah
_------ ----\
\
/
u
\
I
‘.-/’
Figure 1 . (a) Model I , schematic drawing of the erosion E of harmonic topograp .i i f iccurring between
W ;the lithosphere is stippled. The
cffcctive surfacc lwad at any time consists of H-W (equation 7); the sum H - W - E is constant in time
(equations 8). (b) Model 2, schematic drawing of the erosion E of locally compensated topography H
between t , and t , ; the original, locally compensating lithosphere is stippled. The effective surface load at
any time consists of E and results in lithospheric deformation W (equation 27); the sum H- W-E is
constant in time (equation 22).
I , and t , showing the resulting flexural rebound of the lithosphere
Flexural models of continental lithosphere
393
linearity in which each harmonic of topography can be considered independently, this kind of
behaviour transforms to a wavenumber dependence in a(k). Thus, in general,
o(k) = wk-';
a(k)> 0
(10)
where w and E are constants and neither is negative. When k+O, the case of no harmonic
topographic gradient, it is evident from equations (9) and (10) that no erosion would occur.
The limiting case e = 0 (i.e. u is constant in k ) is equivalent to no spatial dependence such
that erosion rate e at any point r would be dependent only upon the height of r and sharpedged peaks would erode no more quickly than plateaux of equal elevation. An erosion rate
dependent upon the second horizontal derivative of spatial topography, as suggested by
theoretical geomorphologists (e.g. Culling 1960), would imply E = 2 although this value may
not be directly relevant at the continental length and geological time-scales under consideration in the present work. a ( k ) is referred to as the erosion time constant spectrum,
terminology which is derived from the solution of (9) independently of isostatic adjustment
to erosion.
The Laplace transform of equation (9) gives, noting that no erosion has taken placed by
t=O,
E(k, S) = TE(k, S) H(k, S)
(1 li)
where
TE(k, s) = -\sc(k)]-';
( 1 iii)
T E ( ~s), is the transfer function describing the erosionaI effect of topography.
The numerical form of the erosion time constant spectrum a ( k ) is treated as an unknown
in the ensuing analysis although its range, for the wavelengths of topography under
consideration (102-103km), can reasonably be expected to lie in the order 102-103Ma.
For example, England & Richardson (1980) considered average denudation rates of young
orogenic belts based on palaeobarometric observations in conjunction with crustal thicknesses beneath present-day mountain ranges and suggested that the erosion of orogens takes
place with a time constant in the range 50-200Ma. The relationship between this estimate
of an erosion time constant and o(k), however, may not be obvious. First, it implicitly
includes the amplification of erosion which may occur as a result of isostatic readjustment;
secondly, England & Richardson's time constant of erosion refers to topography consisting
of many spectral components.
3 Loading models
3.1
MODEL I
Consider the network illustrated in Fig. 2(a) which has input T,(k, s) and output H(k, s).
A network of this configuration has a closed loop network transfer function such that
(Doetsch 1974, p. 85)
According to Fig. 2(a)
@(k, s) = A
s) [Lo(k,s) + E(k, s)]
(13)
and
E(k, s) = B l ( k , s) H(k, s).
(14)
R . Stephenson
394
(a) Model I
(b)Model 2
Figure 2. (a) Model 1 linear filter network with feedback. Filter A , determines the deformational
response of the lithosphere (and thererore the height of topography H remaining above sea-level) to the
initial surface load Lo less erosion E ; filter B , determines the erosion of H . (b) Model 2 linear filter
network with feedback. Filter characteristics are explained in the text.
First, L [(7)] is
t ( k , s) = H(k, s) - W(k, s)
which is used in conjunction with (Si) to write the transfer function between the measured
topography H(k, s) and the effective load z ( k , s):
H(k, S)
but
= [1+
Tl(k, s)] L ( k , S)
L(k, s) = Lo(k, s) + E(k, s)
(15)
from L[(6)] and therefore, comparing equations (13) and
(19,
A l ( k , s) = 1 + TI@, s).
Secondly, comparing equations (14) and (1 1i),
Bl(k, $1= TE(k, s).
Equation (12) can therefore be written
Making substitutions for q ( k , s), T E ( ~s), and L o @ , s) (equations (Sii), (llii) and L[(8ii)]
respectively), (16) becomes
(17i)
(1 7ii)
(1 7iii)
Flexural models of continental lithosphere
395
and
(1 7iv)
Solving in terms of the plate deflection W ( k ,s) gives
If no erosion occurs (i.e. u ( k ) - + m ) and E ( k , t ) =0, then (18) reduces to a form
equivalent to equation (59, the inverse Laplace transform (L-') of which is (e.g. Roberts &
Kaufman 1 9 6 6 , ~181)
.
PO
W(k,t ) = --
Pm
11 + [$ ( k ) - 1I exp [- t$ (k)/7I) Lo(k).
Note that when t + 0 or 7 + m , this equation be equivalent to the elastic plate solution.
Beaumont (1 978) derived a more general result for the response of a viscoelastic plate under
a constant load in terms of space-time Heaviside-Green functions. McNutt & Parker (1978)
derived an expression for W(k, t ) of a viscoelastic plate but required the measurable topography rather than the total effective load to be held constant in time, thus necessitating the
assumption that after the formation of topography by a mountain building episode its
elevation was 'maintained by subsequent minor rejuvenating pulses' (p. 774). Their
expression for W ( k , t ) varies only slightly in form compared to that given above: however,
the notion that the effective load increases through time is an unrealistic one.
Solution for an elastic plate with erosion are found from L-' [(17i)] and L-' [( 1 8)l with
T+CQ and are (Roberts & Kaufman 1966, p. 189)
( 1 9i)
( I 9ii)
Because $ ( k ) < 1 [cf: equation (3ii)l and pm > p o , 01, is always positive and therefore as
t - t m both H,(k, t ) and W,(k, t ) vanish because of erosion. Note that there is no time
dependence in the ratio We(k,t)/H,(k, t ) because the exponential terms cancel one another
and therefore the isostatic response function of an elastic plate with an eroding topographic
load will be constant in time as expected.
Complete expressions for the eroding topography and resulting deformation of the
viscoelastic plate are given by L-' [(17i)] and L-' [(18)] (e.g. Roberts & Kaufman 1966,
p. 200) and are
(Y2
14
f 4p
R. Stephenson
396
and
I
exp[-t01/2] Lo(k);
a2 = 40
where -rl and -r2 are the roots of s2 + as + 0 = 0. H(k, t) and W(k, t ) are real for real or
complex roots, a consequence of the assumption that the input and output functions are
spatially in phase.
The model of the topography and deformation of a uniform viscoelastic plate in response
to eroding harmonic topography embodied by equations (20,21) is referred to as model 1.
W(k, t) can be used to calculate gravity anomalies (e.g. Banks et al. 1977) and, in turn,
isostatic response functions and flexural stresses induced by the topography and its erosion,
all as functions of time, and these are discussed elsewhere (Stephenson 1981, 1983). In
the present case, topographic heights H(k, t ) only are considered. Model parameters are the
topography and substratum densities po and p m , flexural rigidity D ,viscous relaxation and
erosion time constants 7 and u ( k ) ; model 1 is additionally dependent on initial topography
Lo&). The results are not expected to be very sensitive, within reasonable bounds, to the
choice of po and pm (cf. Banks et al. 1977) and they are assumed to be 2700 and
3300 kg m-3 respectively. For purposes of making a general assessment of the behaviour of
modelled H(k, t) as parameters are varied, changes in k and D can be combined in the single
parameter $ ( k ) (cf equation lii) which varies in the range (0,l): $ ( k ) - 0 is the case in
which there is no isostatic response to topography, at short wavelengths or high D ;and
$ ( k ) = 1 corresponds to the case of local isostatic compensation (flexural rigidity D
vanishes). Examination of the structure of the model equation (21) reveals that scaling o f t
in terms of one or other of the time constants u and 7 or that scaling of u and 7 to one
another is not profiltable.
Figs 3 and 4 show examples of the natural logarithm of topographic power predicted by
model 1 for various parameters. The topographic power is simply the square of the predicted
amplitudes as provided by equation (21). These results are presented as functions of time
normalized so that they are zero when t = 0 and easily comparable to the observations presented
later. This means they have been normalized by the square of the initial instantaneous elastic
deformation, H 2 ( k , 0) = [ 1- $ ( k ) p o / p mI2 I Lo(k) [*. These normalized model topography
power spectral predictions are written as S i ( k , t).
The natural logarithm of normalized topographic power In [S,’, ( k r ,t)] , for a given wavenumber k,, in the case of elastic plate rheology (i.e. 7 + w ) , described by the square of
equation (19i) plots as a straight line with slope -201, [i.e. exponential decay constant; cf
equation (1 Sii)] within the bounds 0 < $ ( k )Q 1 (Fig. 3). Note that changes in u effect
only a scaling change in the curves. The reason topographic decay slows as $ increases is
because for greater $, as isostatic compensation becomes increasingly local, more of the
initial load Lo(k) is ‘buried’ in the lithosphere’s flexural downwarp. With less exposed topography the erosion rate is reduced and the life span of the topography is enhanced.
The general features of the model 1 viscoelastic results, considering first Fig. 4(a) for
which u = lOOMa, are as follows. (1) For small viscous relaxation time constant, 7 < u , e.g.
7 = 1Ma, there is an early phase of rapid double exponential topography reduction,
associated more with the relaxation of the stresses incurred by the initial elastic deformation
rather than with erosion, after which time the topography becomes almost locally compen-
397
Flexural models of continental lithosphere
Figure 3. Normalized topography decay curves SH for elastic plate model 1 or 2: $ varies as shown;
slopes are determined from equation (19ii).
‘OCOl-
--
--
_____
+
1
-8
-4
I
ul
Y
c
-12
-
-6
500-
r = l O O Ma
0
500
1000
1500
5.500 Ma
2000
2500
1000
1500
2000
2500
-4
-12
U = 500 Ma
5 =looMO
-16
-8
Figure 4. Nornlalized model 1 topography decay curves SH t o r (a) u = lOOMa and (b) u = 500 Ma;
parameters $ and r vary as shown. Decay curves of instantaneously locally compensated eroding topography are plotted with dashed lines for references.
sated and decays in a like manner [subparallel to the $ = 1 or local compensation curve
which is dashed on to Fig. 4(a, b) for reference] . As $ is smaller this early reduction phase is
enhanced because the initial elastic flexure was lessened, The timing of this phase, however,
is not significantly affected by changes in $. (2) As T increases, the plate response becoming
‘more’ elastic and ‘less’ viscous, the early viscous relaxation phase is suppressed and, rather,
the decay curve at first mimics the elastic plate decay curve and then begins to diminish,
R. Stephenson
398
ultimately attaining a simple exponential decay form with a relatively decreasing decay
constant. (3) Fig. 4(b) illustrates model 1 results for the same ($, 7 ) pairs as Fig. 4(a) but
with u increased to 500 Ma meaning that the rate of erosion has been considerably reduced.
Note the change of vertical scale between (a) and (b). Examination of Fig. 4(a, b) reveals
that changes in u approximate the effect of a o-scaling factor along the time axis: as u
increases, the onset of decay rate reduction occurs after a greater length of time and is
favoured by larger values of 7 . As a result the period during which the curves resemble their
respective elastic decay curves is much longer,
For all sets of parameters the decay curves eventually become exponential with time
constants equal to twice the least negative root of sz+crs+(3=0 (cJ equations 17, 20)
which i s a' - [ a 2 -4fl]"2. The greatest negative constant achievable is that of the local
compensation case. In general, the magnitude of the time constant is reduced by greater
u and 7 and lesser $ (larger D , higher k ) . While this suggests that topography preservation
is favoured by a strong, viscous plate it does not follow that the most favourable situation
is that of a strong, infinitely viscous (elastic) plate in which case the rate of decay is
enhanced, being controlled by the most negative root of s2 + as + (3 = 0 .
3.2
MODEL^
model 1 it was assumed that topography is suddenly applied to the lithosphere at t = 0.
It is implicit in such a model that even while tectonic processes are building the topography
the underlying lithosphere has a large degree of flexural strength. This may not be realistic
if orogeny is accompanied by thermal weakening of the lithosphere and if the orogenic
processes occur over a period of time which is short compared with the cooling time of
continental lithosphere. It may be that during this time, as topography is created and modified, the lithosphere is unable to sustain flexural stresses resulting from vertical loading of
wavelengths as great as those of present interest, Thus, as the tectonic regime responsible for
the orogenic episode dissipates, the topographic load existing at the end of the orogeny may
be locally compensated by low-density crustal roots. Concurrently and subsequently, the
lithosphere cools and, ultimately, will attain flexural competence at the wavelengths of
interest. The crustal roots compensating the topography may in this way become 'frozen'
into the cooled and thickened continental lithosphere as they apparently do in oceanic
lithosphere. In oceans, surficial topography created at or near ridge crests where the
lithosphere is very thin appears always to be locally compensated by crustal thickening
regardless of its age. On the other hand, new topographic loads applied to old, cooled and
thickened, oceanic lithosphere result in a flexural isostatic response (Watts 1978; Cochran
1979; Detrick & Watts 1979). Heat flow (Pollack & Chapman 1977; Sclater et al. 1980)
and seismological data (Kono & Amano 1978) suggest that continental lithosphere thickens
with age in a fashion similar to oceanic lithosphere. Ocean crust topography does not
significantly erode. On continents, however, if a load with 'frozen in' local compensation is
partially eroded after the time at which the lithosphere acquires flexural competence, the
resulting negative load will be compensated flexurally .
This alternative model of the evolution of the isostatic character of continental
lithosphere resulting from erosion, referred to as model 2, can be quantified by making only
minor revisions to the system of equations developed for model 1. Assume that at a time
t , > 0 the lithosphere suddenly becomes competent and that at t, it supports a locally
compensated surface load Ll(k); t c is presumed to occur a sufficiently long time after
orogenesis such that most of the associated thermal anomaly has been dissipated. Ll(k) can
be partitioned, in the usual manner, into (1) measurable topography Hl(k) and ( 2 ) the
locally compensating 'root' Wl(k) = - p o / A p H l ( k ) where A p is the density contrast at the
1il
399
Flexural models of continental lithosphere
base of the root. Any erosion occurring prior to t = t , is inconsequential because it is
assumed to have been compensated locally. In terms of the subsequent dynamic evolution of
the lithosphere, Hl(k) may be thought of as being suddenly applied to a thin viscoelastic
plate at t,. Hl(k) is not a load resulting in flexural stresses, however, because of the low
density compensating root assumed to accompany it. The plate at this time is taken to be in
a n undeformed state. Subsequent erosion E(k, t ) , t > t,, producing eroded topography
H(k, t ) , is assumed to result in flexural rebound W(k, t). A mass wasting condition,
analogous to equations (8) may be adopted such that
Hi(k, t ) = H(k, t ) - W(k, t ) - E(k, t ) =
0;
(Hl(k);
t<
1,
t > fc;
it is schematically illustrated in Fig, l(b).
Model 2 can be formalized in terms of the linear system shown in Fig. 2(b) which can be
solved in a manner similar to model 1. The function H l ( k , t ) drives the system inasmuch as
it is its erosion which acts as the effective plate load resulting in isostatic plate deformation
which in turn modifies the topography being eroded. The relationship between erosion and
topography assumed before is unchanged (equations 1 1).
Laplace domain solutions for model 2 are
fi(k,s) = exp[-t,s]
[--s2 +-a-Is + p
s i- $ ( k ) / 7
H,(k)
and
Po
W(k,s) = $ ( k )
~
exp[-t,s]
a@-'
s + 1/r
Pm
where a and p are the same as for model 1 (equations 18ii, iii).
Solutions for an elastic plate with erosion are found from L-' [(23)] and L-' [(24)] with
00'7
using the convolution theorem of Laplace transforms:
where a, is defined as before (equation 19ii); a, = 0 if there is no erosion. In time the elastic
model predicts vanishing topography but with the persistence of a remnant crustal root:
Isostatic response functions for elastic model 2 are therefore not constant in time but rather
predict varying degrees of isostatic overcompensation of old topography and may help to
explain why non-erosional models of observed isostatic response functions [op. cit.] require
minimal lithospheric strength (Stephenson 1981).
Complete expressions for model 2 H(k, t ) and W(k, t ) are
R . Stephenson
400
and
ii
(1 - exp [- (t-tc) 4 2 1 [ 1 + ta/2- r p ~ l } H,(k);
ff2 =
4p
where A; = p m - p o ; -rl and -r2 are the roots of s2 + 01s + 0 = 0 as for model 1. The end
point predicted by model 2 is one of hydrostatic equilibrium; lim(t+m)H(k, t ) and W(k, t )
are zero and po/ApHl(k) respectively. Hydrostasy is achieved at the Earth's surface by
erosion and within the lithosphere by viscous flow. Model 2 gravity anomalies and state of
stress as functions of time are discussed elsewhere (Stephenson 1981, 1983a, b).
Elastic topography power decay curves, normalized by the initial topographic power
IHl(k) 1 2 , are equivalent to those of model 1 (cf equations 19, 25). Viscoelastic decay
curves, for the same sample parameters as before, are illustrated in Fig. 5(a, b). The major
difference from model 1 is that they do not display the early viscous relaxation phase of
topographic decay, characteristic of model I , because the topography is in an apriuri state
of local isostatic compensation, Otherwise, the essential pattern of the respective sets of
curves is similar, particularly with respect t o the onset and degree of decay rate reduction
as it relates to the choice of values of the model time constants u and T . This was to be
expected, of course, since the double exponential decay of each model is controlled by
equivalent parameters: the roots -rl and -r2 o f s 2 +(us + 0 = 0.
The important feature of the model predictions is that SA(k,, t ) for a given wavenumber
k , decays exponentially if the lithosphere behaves elastically (or if there is local compensation) throughout the term of the evolution of the topography but that topographic decay
may be significantly suppressed if viscous relaxation is allowed; that is, the time constant
of the exponential decay which is ultimately achieved is substantially reduced.
-16
-
inl
lU=
100 Ma
\
Lc=
500 Mo
-4
-8
-3
LU
-4
= 500 Ma
Figure 5. Normalized model 2 topography decay curves SH for (a) u
parameters j i and r vary as shown.
=
l O O M a and (b) u = 5 0 0 M a ;
Flexural models of continental lithosphere
40 1
4 Application of models to observations
Power spectra of topography of five variously aged North American geological provinces
(Fig. 6) have been computed and are presented with 95 per cent confidence bounds in
Fig. 7. Raw data were 5 ’ x 5‘ average elevations provided by the National Geophysical and
Solar-Terrestrial Data Center in Boulder, Colorado. These were averaged within 50 x 50 km
cells on a map having a Lambert conformal conic projection and the power spectrum of each
region was derived from the Fast Fourier Transform of the gridded values (e.g. Rayner
1971), tapered around the edges in order to suppress noise introduced by their finiteness
(e.g. Tukey 1967). The highest, or Nyquist, wavenumber observable is 0.01km-’ . The final
‘smoothed’ power spectral estimates & ( k ) shown in Fig. 7 are averages of ensembles of raw
transformed data falling within 0.001 km-’ wide annuli concentrically circumscribing the
wavenumber domain origin, thus obscuring directional variations in topographic power. This
was justified by prior inspection of power spectra for each region calculated separately for
four directions (N-S, E-W, NE-SW and NW-SE) which showed that nowhere among the
five sets of computed spectra did a single spectral estimate for a given direction with 95 per
cent confidence significantly differ from all three of the other spectral estimates at the same
wavenumber but in different directions. Fig. 7 shows that the power of topography of
various wavenumbers k is systematically smaller for regions of relatively greater tectonic age,
evidence of an age dependence of the degree of erosion and/or stress relaxation in the
lithosphere.
In order to compare the observed data to the topography decay models, data for each k
are separately considered as functions of time, thus requiring simplifying assumptions
regarding the age of the topography of each of the sampled tectonic provinces. While
acknowledging that the formation of the structure and topography of continental crust
Figure 6 . Generalized tectonic map of North America (after King & Edmonston 1972) showing the
main geological provinces and chosen study areas. The corner letters refer to the study areas as follows,
(W) Cordilleran, (A) Appalachian, (G) Grenville, (C) Churchill and (S) Superior. Legend: ( 1) Phanerozoic
Appalachian, Cd - Cordilleran; (2) Canadian Shield structural provinces, Ch - Churchill,
orogens, Ap
Gr - Grenville, Nn - Nain, S p - Superior, Sv - Slave; ( 3 ) cv - regions of Proterozoic and Phanerozoic
sedimentary cover. The area labellcd (T) is not referred t o in the test.
~
402
R. Stephenson
[W] Cordilleran
[ A ] Appalachian
[GI Grenville
[C] Churchill
[S] Superior
1
l5
?
k-l[km]
0b2
obs
dos
&e
oio
Figure 7. The observed topography power spectra $H as functions of 1 k 1. For clarity the 95 per cent
confidence intervals of the Grenville ( G ) and Churchill (C) spectra are omitted but are approximately
the same size as the others. Relative tectonic ages of geological provinces increase from (W) to ( S ) .
occurs during successive orogenic periods in complex processes encompassing hundreds of
millions of years, it shall be assumed here, for purposes of the simple models of postorogenic topographic evolution being considered, that the age of topography is equivalent
to the age of the last recorded tecto-thermal event, after which tectonic forces rapidly
dissipate. Further, because the topography being considered, with the exception of that of
the Cordilleran region, is geologically very old with ages probably much greater than the
thermal cooling time constant of the continental lithosphere, the same ages of topography
will be assumed for both models 1 and 2. In fact, the appraisal of the observations is
expected to be far more sensitive to the choice of model parameters (a, T , 0)than to the
choice of ages, given the very long period of time under consideration.
Thus, the topographic ages of the Grenville, Churchill and Superior structural provinces
of the Canadian Shield are assumed to be equivalent to those of the Grenvillian, Hudsonian
900--1100, - 1600-1800 and
and Kenoran orogenies dating respectively from
2400-2600 Ma (Stockwell 1964). An erosional hiatus throughout the Canadian Shield
of some 100-200Ma duration in the lower Palaeozoic, known from scattered outliers and
the preservation of strata beneath Hudson Bay and adjacent areas, is ignored because of its
relatively short duration and because geological observations indicate that most of the
present detectable erosion of the Shield occurred prior to the hiatus (Ambrose 1964). As for
the surviving Palaeozoic cover of the Churchill and Superior provinces, its extent is small
enough that the observed power spectra are dominated by the cratonic topography of each.
That the topographic power of the younger Churchill study area is generally greater than
that of the Superior is evidence of the validity of this assumption.
The most recent orogenic episodes affecting the Appalachian structural province were
the Acadian in the north during the Devonian period, 350-400Ma ago, and the Alleghanian
in the south during the late Palaeozoic, 250-300Ma ago (Stearn, Carroll & Clark 1979). The
differential timing of Appalachian deformation was a result of the progressive closure of the
Iapetus (Proto-atlantic) Ocean from north to south with consequent continental collision. A
-
-
Flexural models of continental lithosphere
403
median age of 300-350Ma is therefore assigned to the topography of the Appalachian study
area. Considerable erosion of the Appalachians had already occurred by the time of
Mesozoic continental rifting and the opening of the present Atlantic Ocean (Stearn et al.
1979) and it is assumed that uplift possibly associated with rifting was broad and uniform
enough so as not to affect greatly the spectral configuration of the pre-existing topography.
The most recent deformational event affecting the crust of the Cordilleran study area was
the Laramide orogeny which began near the beginning of the Cenozoic era, - 65 Ma ago. In
the Canadian portion of the study area it had, for the most part, ended early in the
Oligocene, - 35 Ma ago (Douglas et al. 1970), although uplift of the Coast Mountains, based
on mapping of deformed erosion surfaces, persisted until the Pliocene, 4 - 7 M a ago
(Wheeler & Gabrielse 1972). The United States portion of the study area is dominated by
the Basin and Range province, the present mountain ranges of which having formed since the
Early Miocene, -20-25Ma ago (Hamilton & Myers 1966). The relief of the Rocky
Mountains, east of the Basin and Range province, was primarily developed during an uplift
phase in the late Pliocene (Stearn et al. 1979). Essentially, the Cordilleran region of North
America is one which is currently tectonically active (e.g. Atwater 1970; Hamilton & Myers
1966) and, for the purposes of the present analysis, in which its topographic character is
compared with that of regions very much older, it shall be considered to have not yet
entered a period of post-tectonic erosional and isostatic evolution. Thus, in terms of models
1 or 2 , the observed topography power spectrum of the Cordilleran study area is assumed to
be equivalent to [ 1- $(k)po/p,]Z /L,(k) l2 or IHl(k) 1' respectively.
This is a fundamental assumption with important implications which will be discussed
later. It is a convenient assumption because it allows the observed spectral estimates to be
normalized by those of the Cordilleran region and plotted in a form easily comparable to the
model decay curves (Figs 3-5); that is, as functions of time since formation for each of the
sampled wavenumbers. These normalized data, referred to as $+
( k ) , are presented in
Fig. 8(a-i) which shows that, in general: (1) topographic power at a given wavenumber
decreases with age and that most of the reduction occurs during the first few hundred
million years after stabilization and (2) the shorter the topographic wavelength, the greater
the net power reduction at a given time, attesting to a decreasing erosion time constant
u ( k ) a s k increases.
A first-order fit to the topography power spectra decay data can be found by simple
linear regression for each k,. The regression lines are assumed to pass through the origin and
are shown as solid lines on Fig. 8. The slopes of these regression lines, which could be
interpreted in the context of either the $ ( k ) = 0 or $ ( k ) = 1 extrema of an elastic plate
model (i.e. either no isostatic compensation or complete local compensation) are plotted as
a function of wavelength in Fig. 9 where they illustrate the observation noted above that u
may be decreasing as k increases. It is emphasized, however, that the exact form of the
u ( k ) function is not central to ensuing arguments and it is noted that the results imply
that u ( k ) does not greatly vary in the range of k in which observations exist.
There is a systematic lack of fit between the observed power spectra data and the fitted
regression lines (Fig, 8). The latter are representative of the simple exponential decay of
topography predicted by the elastic rheological model whereas the data are more suggestive
of an initial relatively rapid topographic reduction followed by less rapid exponential decay.
Such a decay history conforms with those of the viscoelastic models which comprise an early
phase of double exponential decay preceding a final phase of' single exponential decay (cg
Figs 4 and 5).
There are several ways the observations can be compared to the models. For example, it
may be presumed that the Appalachian and older data points, on the basis of their perceived
collinearity on the logarithmic decay diagrams, may define the predicted mature, singly
404
t
0
500
1000
?[MaJ
[Ma]
1500
2000
2500
500
.>'.-
1000
1500
2000
2500
0
500
1000
1500
Zoo0
2500
.;.*..:....-._
s
.
...
'-I..
...s
( c ) k;l=
(a) kr-l=600km
286 krn
( f ) k r ' = 154 km
$A
Figure 8. Normalized topography decay data
with 95 per cent confidence intervals, observed at
spectral wavelengths: (a) 6 0 0 k m , (b) 4 0 0 k m , (c) 2 8 6 k m , (d) 2 2 2 k m , (d) 1 8 2 k m , (f) 154km, (g)
133 km, (h) 118 km, (i) 105 km. Choice of time axis confidence intervals is qualitative. Also shown are:
( 1 ) qualitative envelopes drawn to constrain possibly acceptable models; (2) best-fitting linear regression
lines passing through the origin (solid lines); ( 3 ) sample overall acceptable model 1 decay curves
(D = 10Z'Nm, T = 1 Ma, u = 350Ma; dashed lines); and (4) sample overall acceptable model 2 decay curves
(D= lOZ4Nm,T = LMa, u = 250 and 300 Ma; open and fiiled dots respectively).
exponential, topographic decay phase of the models. Thus, the observed slopes ( m ) of these
data would be equal to twice the smallest quadratic root, - r l , of s2 + a s + p = O .
m = -2r, = - a - [ 0 1 ~ - 4 p J ~ ' ~ ,
where 01 and /3 are functions of wavenumber and the model parameters u , 7 and D (cf.
equations 17ii, iii). This property is equivalent for both models 1 and 2. Additionally the
t = 0 intercept (b) defined by the collinearity of the Appalachian and older data can be
related to intercepts predicted by the normalized model equations. In this case models 1 and
2 differ:
b l = -2111 { [ Y ~ - Y ~ ] / [ O ~ - ~ ~ ] ~ / ~ }
and
b2 = -2 In { [$ ( k ) / r-rl]/ [a2- 4/3]1'2} .
Owing to the few data available, their variability induced by noise and/or hidden
parameters, and the difficulty of choosing reasonable estimates of their standard errors,
rigorous inversions of the data in terms of the described model functions m, bl and b2 have
not been performed. Rather, using very liberal qualitative guidelines, envelopes shown in
Fig. 8 have been drawn which are meant to constrain all possible straight lines which could
reasonably be expected to be representative of the Appalachian and older data. The
maximum and minimum slopes and intercepts of all the lines constrained by these envelopes
for each of the observed wavenumbers are listed in Table 1. Using these bounds, a threedimensional parameter search (a,r and D ) was conducted to find acceptable models for each
of m, b l and b 2 . Based on the evidence of Fig. 9, u was initially taken to be constant as a
Flexural models of continental lithosphere
b
405
-
500r
'0
V ( k ) = 0,I
100 200 300 400 500 600 700
k-' [kml
Figure 9. Values of the erosion time constant u (filled circles; left-hand scale) based on linear regression o f
observed topography decay data; bounds are based o n regression of the upper and lower 95 per cent
confidence limits of the data (Fig. 8). I n this extrema model u may be proportional to k - ' / 3if it vanishes
as k - m, an indication that the postulated erosion model (equation 10) is not inappropriate. Also shown
(squares; right-hand scale) are values of u from u-adjustments to acceptable model 2. u may be
proportional to k-'14.
function of k during each of the parameter searches. (Surface and substratum densities po
and pm were assumed to be, as before, 2700 and 3300 kg m-3 respectively.)
In the case of m , an acceptable model was one which predicted all nine slopes (one for
each wavenumber for each set of parameters) to fall withiii the established bounds. In the
models, changes due to increasing u are easily compensated by the other parameters with the
result that the observed ranges of m do not provide tight constraint. Note that in the case of
the viscoelastic models the results are solely dependent on the TOproduct. This reflects the
controlling effect of the ratio J / ( k ) / rin (Y and 0 (equations 17ii, 17iii). As D increases, for
the range of k under consideration, J/ tends to vary according to D-'.
Acceptable intercept models were those which satisfied all the observed envelopes with
the exception of the k;' = 4 0 0 k m case (Fig, 8b) which no otherwise acceptable set-of parameters could reproduce. Otherwise acceptable models always predict bl or b2 = 0 for the
400 km case; in light of the biasing effect and dubious appearance of the t = 1000Ma
(Grenville) 400 km data point this inconsistency has been overlooked. Accepted model 1
Table 1. Acceptable bounds on search functions for each observed wavelength
(cf. Fig. 8).
k ; I (km)
rn(Ma-')
-b
-SH(t = 2500Ma)
(a) 600
(b) 400
(c) 286
(d) 222
(e) 182
( f ) 154
(8) 133
(h) 118
(i) 105
0-0.00136
0-0.00128
0.00020-0.00148
0.00008-0.00120
0.00008 -0.001 28
0.00020-0.00 136
0.00004-0.00 100
0.00016-0.00108
0.000 12-0.001 16
0-2.3
0.3-3.3
0-1.6
0-2.0
0-1.4
0-1.9
0.8 -2.3
0.8-2.3
1.0-2.7
1.5-3.5
2.4-3.5
2.1-3.7
2.2-3.1
1.6-3.2
2.4 -3.5
2.4-3.5
2.7-3.9
3.O -4.3
R . Stephenson
406
Table 2. Acceptable models in u--'r - D parameter space for discrete values
( , model 1; model 2).
Search
function
m
b,
u(Ma)
400-700
800-1400
1500- 1800
1900
22000
300
350-1000
1100-1600
1700
1800
T-W
TD = (0.5-5)
T~
G
1
D(Nm)
<1020
h 102'
< 10z2
1022- 1 0 2 ~
>loz2
x
s x 102~
TD G 2.5 x
TD G
T~ G 0.25 x
100
200-300
400-600
;:700
h
T(M~)
100
200
300
400-700
800-1500
1600-2400
S~(2500)'
300
350
SH(25 001,
300
350
350
< 250
G 500
(0.5-5) x 1023
(o.s-i)x loz4
TD = 1 oz4
TD = 2.5 x I oZ4
TD = 5 X
TD =
s~ =
T~
=
TD = (1-2.5) X 10'"
= 2.5 x l o z 4
T
'
m
1 OZ0
intercepts thus constrain D = 1OZ1Nm but do not greatly limit 7 through a range of u
(Table 2); for model 2, the TDproduct is reasonly constrained as a function of u ,
Thus, the results so far are virtually insensitive to choice of u. However, if the models
are further tested so that they must reasonably predict the degree of decay of topographic
power indicated by the Sk(t = 2500Ma) data (Superior province), again set by the
qualitative bounds established by the envelopes drawn in Fig. 8, then this problem can be
overcome. The observed bounds are listed in Table 1 and the results of the parameter
searches in Table 2. Acceptable models satisfied all nine sets of observations. The net topographic decay by t = 2500 Ma tightly constrains u to be in the range 300-350 Ma.
The results of the parameter searches for model 1 are summarized in Fig. 10(a); only the
D = 10"Nm plane of a-7-D space is shown because the bl search constrained D to this
value only, Acceptable values of 7 are those < 220 Ma. The mechanism which effects the
acceptability of models in this region of parameter space is the same throughout the range
and relates not to erosion but rather to the initial phase of rapid topographic decay
resulting from viscous relaxation of instantaneous elastic flexural stresses. Once this phase
has been completed, topography is in a state of 'essentially' local isostatic equilibrium and
remains in such a state while it continues to decay in a singly exponential manner. (Topography in fact is consistently slightly overcompensated at this and subsequent times.) The
onset of the secondary phase is governed mainly by $ (k)and occurs very early in the case of
the longer-wavelength models (k;' > 286 km) because of the wavelength dependence of
stress relaxation in viscoelastic models. A sample set of decay curves, using D = 102'Nm,
Flexural models of continental lithosphere
(a) M o d e i I
h
-L,
.
ID.1O2'Nml
l
0
I
2
1
Log
r
(b) M o d e l 2
I
800
3
407
[Ttcm]
-
20
22
2'4
20
logr D
Figure 10. Results of ( a ) model 1 (D= 10"Nm plane of U-T-D space only) and ( b ) model 2 ( u - - T D
space for finite T ) parameter searches. Acceptable models based o n slopes, model 1 or 2 t = 0 intercepts,
and net decay of topographic power at t = 2500 Ma fall in regions labelled m, b , or b , and S respectively.
All three criteria are satisfied by parameters within the shaded areas.
~ = 1 M aand u = 3 5 0 M a , is plotted with dashed lines in Fig. 8, Note that the sA(3SOO)
criterion is barely satisfied at short wavelengths, k;' < 133 km.
For viscoelastic model 3 the rheological properties of the plate combine in a single
effective parameter TO; acceptable models are shown in u-TO space in Fig. 10(b). No
purely elastic models were found acceptable for all three search functions rn, b2 and
&(2500Ma). Acceptable viscoelastic models fall within the region CJ = 275-375 Ma and
TD= ( 1-2.S) x
Nm Ma.
In all the parameter search analyses so far, the u spectrum has been assumed to be
constant despite the intuitive expectation, in part supported by Fig. 9, that u should
decrease as k , increases. Therefore, great care was taken to ensure that the results presented
so far are not sensitive to a wavenumber dependency in u . Secondarily, in the case of the
acceptable models, attention was given to whether internally consistent adjustments to u
[such that no a ( k l 1< u ( k 2 )if k l < k 2 ] could actually serve to improve the fit of the acceptable models to the data. The opposite proved to be the case for model 1 but for model 3
consistent a ( k , ) adjustments yielded improved results. A a-adjusted set of topography decay
curves for a sample acceptable model 2 ( D = 1024Nm; T = 1 Ma) is plotted with open and
filled circles in Fig. 8. Values of u used in the adjusted calculations (350 and 300Ma for
filled and open circles respectively) are illustrated in Fig, 9 (right-hand scale); their distribution suggests u may be proportional to k-"4 if it vanishes as k+m.
The plotted sets of sample model topography decay curves are characterized by
moderately good fits to observations at the small wavelengths k;' < 183km. At larger
k;' the curves are effectively singly exponential, a condition dependent upon small T
compared to u such that topography essentially remains in local isostatic compensation
throughout its erosional lifetime.
5 Discussion
A feature of the theoretical normalized topography decay curves common to both models
1 and 3 is their single exponential decay for the largest of the observed wavelengths. The
models' least successful aspect in this regard is their reproduction of the Appalachian
spectral power ( t = 335 Ma) which is consistently less than that predicted.
408
R. Stephenson
This problem is particular!y acute for model 1 because much of the topographic decay
which is predicted for the first few hundred million years is the result of viscous relaxation
of elastic flexural stresses rather than of erosion. This is in contrast t o geological evidence
which suggests that erosion during a mountain range’s early post-tectonic life is likely t o be
significant (e.g. Stearn et al. 1979; Ambrose 1964). Model I also failed in respect t o adjustments of a ( k ) values during the misfit analysis described earlier. In this light the most
interesting feature of model I results is the implied low lithospheric strength ( D G 10” Nm)
being similar t o that found from continental isostatic response functioo studies (Banks et ul.
1977; McNutt & Parker 1978; Banks & Swain 1978; Stephenson 1978) which employ
different kinds of data (gravity anomalies being the discriminant in such studies) but in
which equivalent loading models were assumed. Specifically, the assumed loading model is
that of a competent lithosphere, with subsequently time-invariant physical properties, preexisting the emplacement of surface loads and that the loads are complex in geographical
distribution. The present results are taken as independent confirmation that isostatic
response function studies (ibid.) and the present model 1 topography decay study are
measuring properties characteristic of the lithosphere at the time of loading rather than
those more characteristic of its subsequent state. The term ‘more’ is used because it is almost
certain that the physical properties of the lithosphere are not time-invariant and, therefore,
that modelled parameters, being time-integrated, are ‘effective’ parameters only. (In the case
of large distributed loads they may also be space-integrated.)
The model 2 loading scheme represents an attempt to overcome problems of syn-tectonic
lithospheric rheology in order to judge the post-tectonic case better. The parameter searches
show that acceptable models are viscoelastic rather than elastic; this is because viscous
relaxation in the lithosphere has the tendency to reduce the erosion-induced decay rate of
spectral topography to less than the exponential decay of the purely elastic case. The
analysis can only discriminate among products of the rheological parameter pairs (D, T) and
acceptable values of these parameters fall in the respective ranges 1 023-1025Nm and
50-0.5 Ma. The effective thickness o f the continental lithosphere indicated by flexural
rigidities such as these is comparable t o that suggested by analyses of individual continental
loads (cf. 1024-1025N m ; Walcott 1 9 7 0 ~Forsyth
;
1979; Cochran 1980). More important is
that topographic heights observed in tectonically very old regions are apparently sustainable
by a lithosphere with a relatively small viscosity (lOz4-1Oz6Pa s) such that the topography
does not simply flow away through geological time spans. The indicated parameter pairs
cited above, moreover, compare not unfavourably with those found necessary b y Beaumont
(1981) to model the structure and stratigraphy of the Alberta Foreland Basin (1OXNm;
27.5Ma) and by Lambeck (1983) t o chart the evolution of the Amadeus and associated
basins in central Australia (- 102’Nm; 10-50Ma with regional compression).
The linear models are first-order approximations and their lack of success t o satisfy the
observations completely, given that the misfits are strongly dependent upon load wavelength and, conjunctively, load amplitude (power increases markedly at longer wavelengths;
cf. Fig. 7), is probably due to physical non-linearity in the real Earth. How the non-linearity
may be manifest in the Earth is not obvious. I t is probably not correct to presume that an
effective rheological non-linearity in models preserving a uniform thin plate geometry might
be expected to return smaller effective parameters D and T for large loads. The effect of this,
as is illustrated in Figs 3-5, would be t o reduce the early ( < 3 2 5 M a ) rate of topographic
decay rather than to enhance it as is required b y the data. This may not be applicable, however, because of the relative magnitudes of the topography and erosional components of the
total load, the former being much greater during the early evolution o f the models.
Less equivocal is the possibility of an effective erosion time constant which increases
through time; this may be equivalent to a non-linear erosional process o r one in which
Flexural models 0.f continental lithosphere
409
erosion rate would be proportional to some power of the topographic height greater than
unity. This, in combination with an elastic plate rheology, could lead to acceptable models
of the observed decay data. In such a case the ultimate decay constant of topography
( t > 300Ma) of given wavelength would depend solely on D and u. The observed m parameter search results tabulated earlier (Table 2) for this model require either a very low
flexural rigidity (D<
Nm) for continental lithosphere, apparently unlikely in the light
of other studies as noted elsewhere, or a very long erosion time constant (a z 1900Ma).
Approximate erosion rates implied by the latter constraint appear to be too small compared
to those observed in regions of Palaeozoic topography such as the Appalachians (Menard
1961) but are no more than one order of magnitude smaller than those suggested for
cratonic surfaces (Yilgarn Block, Western Australia) by Finkl (1 982). Finkl does point out,
however, that recent erosion rates of cratonic regions based on sediment yield rates appear
to be considerably higher. Indeed, a mature erosion time constant of the order delimited by
the viscoelastic model results reproduces quite well the average cratonic denudation rates
known from available observations (cf. Finkl 1982). Thus, it appears to remain necessary to
invoke viscoelastic models as opposed to purely elastic ones to reduce long-term topography
decay rates.
A procedure fundamental to the present work was the normalization of the observed
topography power spectra. Thus the directionally averaged initial spectral configuration of
the tectonically older regions was assumed to be similar to that existing at the present for
the Cordilleran study area. It is emphasized that this does not imply that the spatial
configurations of the presumed set of young mountain ranges are necessarily comparable. It
perhaps does follow, however, that similar mountain buiIding processes, presumably in a plate
tectonic framework, are implicitly assumed to have occurred since the Archean. Davies
(1979) discusses evidence in support of this assumption that a stable continental lithosphere
comparable in thickness to that existing today existed as early as the Archean. In the
context of the Palaeozoic Appalachians it is noted that the Cordilleran and Appalachian
mountain-belts consist of similar gross structural elements (Dewey & Bird 1970) and that
accordingly there is no reason to believe that the initial topographic power of each would be
greatly different.
An error of *100-700Ma in the choice of the present as the initialization time in the
models, given the very long-term evolution being considered, is unlikely to be significant in
terms of the implications of the models. It is possible that the Cordilleran region may yet
undergo hundreds of millions of years of orogenesis before commencing a model ?-type
post-tectonic evolution and it cannot be speculated whether during such time the present
topographic power would be reduced by massive synarogenic erosion or whether it would
be continually rejuvenated or even supplemented. In either case, should the correct initialization time be several hundred million years hence and should the present analysis be
repeated then, the requirements of magnified early decay of topography as facilitated by the
viscoelastic models is potentially negated. For example, consider the mature exponential
decay lines, characterized by slopes m in Fig. 8, to be extrapolated linearly into the future,
leftwards on the diagrams, to intercept with some future Cordilleran spectral power. In
elastic models such as these would be, however, the arguments presented earlier with respect
to required small flexural rigidity or very long erosion time constants prevail as before.
6 ConcIusions
Continental topographic power at given wavelengths has been found from North American
geological provinces to decrease systematically as the tectonic age of the underlying crust
410
R. Stephenson
increases. This long-term decay of topography has been quantiatively modelled in terms of
erosion and the incumbent flexural isostatic response of the continental lithosphere. The
lithosphere has been treated as a thin (Maxwell) linear viscoelastic plate in the models and
erosion was assumed to occur at a rate linearly proportional to the height of topography at
any given time.
Two kinds of loading models were considered. Model 1 assumed that a flexurally competent lithosphere is suddenly loaded by topography which subsequently erodes whereas
model 2 ignores the tectonic emplacement and erosion of topography during orogenesis and
considers only the load effected by erosion beginning some time later when the lithosphere
has cooled and strengthened .
The main aim of the study was to evaluate the mechanical behaviour of continental
lithosphere in response to large-scale distributed loads applied through geological time spans.
Previous efforts in this regard have relied on the calculation of admittance functions between
Bouguer gravity anomalies and continental topography and have faltered because of poor
coherence (low signal to noise ratio) and because of the ambiguities inherent in the interpretation of gravity anomalies.
The main conclusions are as follows:
( I ) The form of the data and the models’ general success in reproducing them support the
notion that long-wavelength ( 102-103 km) topography on continents, once it has been
constructed by successive orogenic upheavels, evolves in the normal course of events mainly
by erosion through hundreds of millions of years without significant tectonic disturbance.
Thus, the results imply that the topography and crust tectonic provinces within cratons are
the remnants of palaeo-mountain ranges which initially had topographic relief comparable to
that observed in present-day young mountain ranges.
(2) The important theoretical result is that both models predict that the decay rate of
very old continental topography is reduced, that is, the survival of this topography is
favoured, by a lithosphere in which viscous stress relaxation occurs. This is in contrast to the
view that the existence of topography and gravity anomalies in very old regions such as the
Canadian Shield is evidence of a finite yield strength in the Earth’s crust in the geological
time frame (e.g. Jeffreys 1976). In the present case stress relaxation was modelled in terms
of Newtonian viscosity as small as 10w-102s Pa s.
( 3 ) Both models, because of viscous flow, predict topography decay rates which decrease
through time, especially rapidly during the first few hundred million years, in general
accordance with geological observations and with topographic power calculated from North
American geological provinces.
(4) The North American spectral data interpreted through model 1 returned low values of
the flexural rigidity D of continental lithosphere, - 1021-1022Nm, comparable to those
found through isostatic response function studies of continents in which a similar loading
model was employed. This is taken to be independent evidence that the response function
results are measuring not so much present-day D but that which was effective when topography was being formed.
( 5 ) Model 2 constrains the product of the rheological parameters rD to a value,
1024-102sNm Ma, which compares favourably to that found from studies of individual
loads on continental lithosphere (cf. Sleep & Snell 1976; Beaumont 1981; Lambeck 1983).
While this cannot be considered proof of the validity of the effective viscoelastic model at
the wavelengths and time-scales of interest, it strengthens the appropriateness of such a
model and supports the ability of the long-term decay data to be modelled in the fashion
they have been here. These results show that model 2, which incorporates a thermal origin
for continental topography, is preferable to the commonly invoked model 1 which does not.
Rexural models of continerltal lithosphere
41 1
(6) A purely elastic uniform lithosphere model cannot be ruled out by the analysis but
would require either a small effective flexural rigidity (< 1022Nm) or very large effective
erosion time constants (> 1900Ma). The former must be judged in the light of studies of
individual continental loads (op. cit.) while the latter appears unlikely.
(7) The third model parameter resolved by the data, the erosion time constant u, has a
value which lies in the range 200-400 Ma for harmonic topography in the wavelength range
102-103km. There is some evidence in the data for the intuitively reasonable result that
u is inversely proportional t o the wavenumber k . This is equivalent t o a topographic gradient
dependence in erosion rates. A relationship of the form u = wk-' with w = (50-100)Ma
km-' and E < 0.4 is suggested. This value of E is less than what might have been expected on
the basis of analytical erosion models from geomorphology literature (e.g. - 2 ; Culling
1960), a discrepancy probably explicable by the greater length and time-scales considered
in the present work.
(8) In terms of the employed linear erosion model, u should be viewed as not only a
space-integrated but also as a time-integrated effective parameter with smaller values for
younger, shorter periods of erosion. This is suggested by a systematic wavelength-dependent
misfit between models and observations.
(9) The wavelength-dependent misfit of the models may also be evidence of non-linear
rheological effects in the continental lithosphere but does not unduly jeopardize the general
conclusions noted above and, specifically, that viscous relaxation of stresses in the lithosphere is compatible with and, in fact, favours the retention of very old continental
topography such as that found in the Canadian Shield.
-
Acknowledgments
Much of rhe work presented is derived from the author's PhD thesis at Dalhousie University,
Halifax, for which the supervision of Chris Beaumont, Department of Oceanography;
facilities and financial support from the Department of Geology; and financial support from
the Department of Energy, Mines and Resources (Ottawa) through grants to Dr Beaumont,
are gratefully acknowledged.
The present work was completed while the author held a post-doctoral fellowship at the
Australian National University. Professor Kurt Lambeck of ANU is thanked for critically
reading an early manuscript.
References
Adams, J., 1980. Contemporary uplift and erosion of the Southern Alps, New Zealand: summary, Bull.
geol. Soc. Am., 9 1 , 2 - 4 .
Ainbrose, J. W., 1964. Exhumed paleoplains of the Precambrian Shield of North America, Am. J. Sci.,
262,817-857.
Atwater, T., 1970. Implications of plate tectonics for the Cenozoic tectonic evolution of western North
America, Bull. geol. SOC.A m . , 81, 35 13-3536.
Banks, R. J., Parker, R . L. & Huestis, S. P., 1977. Isostatic compensation on a continental scale: local
versus regional mechanisms, Geophys. J. R. astr. SOC., 51, 431 -452.
Banks, R. J. & Swain, C. J., 1978. The isostatic compensation of East Africa, Proc. R. SOC. A, 364,
33 1-35 2.
Beaumont, C., 1978. The evolution of sedimentary basins on a viscoelastic lithosphere: theory and
examples, Geophys. J. R. astr. SOC., 55,471-497.
Beaumont, C., 1979. On rheological zonation of the litho4phere during flexure, Tecronophys., 5 9 ,
347-365.
Beaumont,C., 1981. Foreland basins, Geophys. J. R. astr. Soc., 65, 291-329.
Cazenave, A., Lago, B., Dominh, K . & Lambeck, K., 1980. On the response of the ocean lithosphere to
seamount loads from GEOS 3 satellite radar altimetry observations, Geophys. J. R. 4str. SOC., 63,
233-252.
412
R. Stephenson
Cochran, J . R., 1979. An analysis of isostasy in the world’s oceans 2. Midocean ridge crests, J. geophys.
Res., 84,4713-4729.
Cochran, J . R., 1980. Some remarks o n isostasy and the long-term behavior of the continental lithosphere, Earth planet. Sci. Lett., 46, 266--274.
Culling, W. E. H., 1960. Analytical theory of erosion, J. Geol., 68, 336-344.
Davies, G. F., 1979. Thickness and thermal history of continental crust and root zones, Earth planet.
Sci. Lett., 44, 231-238.
Davies, P. A. & Runcorn, S. K. (eds), 1980. Mechanisms of Continental Drift and Plate Tectonics,
Academic Press, London, 362 pp.
Detrick, R. S. & Watts, A. B., 1979. An analysis of isostasy in the world’s oceans 3. Aseismic ridges,
J. geophys. Res., 84, 3637-3653.
Dewey, J. F. & Bird, J. M., 1970. Mountain belts and the new global tectonics, J. geophys. Res., 75,
2625 -2647.
Doetsch, G., 1974. Introduction to the Theory and Application of the Laplace Transformation, SpringerVerlag, New York, 326 pp.
Dorman, L. M . & Lewis, B. T. R., 1970. Experimental isostasy. 1. Theory of the determination o f the
Earth’s isostatic response to a concentrated load, J. geophys. Res., 75, 3357-3365.
Dorman, L. M. & Lewis, B. T. R., 1972. Experimental isostasy. 3. Inversion of the isostatic Green
function and lateral density changes, J. geophys. Res., 77, 3068-3077.
Douglas, R. J. W., Gabrielse, H., Wheeler, J. O., Stott, D. F. & Belyea, H. R., 1970. Geology of western
Canada, in Geology and Economic Minerals of Canada, pp. 365-488, ed. Douglas, R. J. W.,
Geological Survey of Canada, Ottawa.
England, P. C. & Richardson, S . W., 1980. Erosion and the age dependence of continental heat flow,
Geophjx J. R. astr. Soc., 62,421-437.
Finkl, C. W., 1982. On the geomorphic stability of cratonic planation surfaces, 2. Geomorph., 26,
137-150.
Forsyth, D. W., 1979. Lithospheric flexure, Rev. Geophys. Space. Phys., 17, 1109-1 114.
Hamilton, W. & Myers, W. B., 1966. Cenozoic tectonics of the western United States, Rev. Geophys.,
4,509-549.
Hanks, T. C., 1971. The Kuril Trench-Hokkaido Rise system: large shallow earthquakes and simple
models of deformation, Geophys. J. R. astr. Soc,, 23, 173-189.
Haxby, W. F., Turcotte, D. L. & Bird, J. M., 1976. Thermal and mechanical evolution of the Michigan
Basin, Tectonophys., 36,57-75.
Jeffreys, H., 1976. The Earth, 6th edn, Cambridge University Press, 574 pp.
King, P. B. & Edmonston, G. J., 1972. Generalized Tectonic Map of North America, United States
Geological Survey, Washington DC.
Kono, Y . & Amano, M., 1978. Thickening model of the continental lithosphere, Geophys. J. R. astr. SOC.,
54,405-416.
Lambeck, K . , 1981. Flexure of the ocean lithosphere from island uplift, bathymetry and geoid height
observations: the Society Isltinds, Geophys. J. R. astr. Soc., 67, 91-1 14.
Lambeck, K . , 1983. Structure and evolution of the intracratonic basins of central Australia, Geophys. J.
R. astr. Soc., 74, 843-886.
Lambeck, K . & Nakiboglu, S. M . , 1981. Seamount loading and stress in the ocean lithosphere 2. Viscoelastic and elastic-viscoelastic models, J. geophys. Res., 86, 6961 -6984.
Lambeck, K . , Penney, C. L., Nakiboglu, S . M. & Coleman, R., 1984. Subsidence and flexure along the
Pratt-Walker seamount chain, J. geophys. Res., in press.
Lewis, B. T. R. & Dorman, L. M., 1970. Experimental isostasy. 2 . An isostatic model for the USA derived
from gravity and topographic data, J. geophys. Res., 75, 3367-3386.
Louden, K . E., 1981. A comparison of the isostatic response of bathymetric features in the north Pacific
Ocean and Phillipine Sea, Geophys. J. R. astr. SOC.,64, 393-424.
Luke, J. C., 1972. Mathematical models for landform evolution, J. geophys. Res., 77, 2460-2464.
McKenzie, D., 1977. Surface deformation, gravity anomalies and convection, Geophys. J. R. astr. SOC.,
48, 21 1-238.
McKcnzie, D. & Bowin, C., 1976. The relationship between bathymetry and gravity in the Atlantic
Ocean,J. geophys. Res., 81,1903-1915.
McNutt, M., 1979. Compensation of oceanic topography, an application of the response function
technique to the Surveyor area, J. geophys. Res., 84, 7589-7598.
McNutt, M., 1980. Implications of regional gravity for state of stress in the Earth’s crust and upper
mantle, J. geophys. Res., 85,6377-6396.
Flexural models of continental lithosphere
413
McNutt, M. & Menard, H. W., 1978. Lithospheric flexure and uplifted atolls, J. geophys. Res., 83,
1206- I21 2.
McNutt, M. & Parker, R. L., 1978. Isostasy in Australia and the evolution of the compensation
mechanism, Science, 199, 773-775.
hlenard,H. W., 1961. Someratesofregional erosion,J. Geol., 69, 154-161.
Nadai, A., 1963. Theory of Flow and Fraciure ofSolids, Vol. 2, McGraw-Hill, New York.
Penck, W., 1923. Die Morphologisclze Analyse, trans. by H. Czech and K . Boswell as Morphological
Analysis o f Land Forms, MacMillan, London, 1953.
Pollack, H. N. & Chapman, D. S., 1977. On the regional variation of heat flow, geotherms and lithospheric thickness, Tectonophys., 38, 279-296.
Pugh, J. C., 1955. Isostatic readjustment in the theory of pediplanation, Q. Jl geol. SOC. Lond., 111,
36 1-369.
Rayner, J . N., 1971. A n Introduction to SpectralAnalysis, Pion, London, 174 pp.
Roberts, G. E. & Kaufman, H., 1966. Table of Laplace Transforms, W. B. Saunders, Philadelphia,
367 pp.
Sandwell, D. T. & Pochls, K. A., 1980. A compensation mechanism for the Central Pacific, J. geophys.
Res., 85, 3751-3758.
Sclater, J. G., Jaupart, C. & Galson, D., 1980. The heat flow through oceanic and continental crust and
the heat loss of the Earth, Rev. Geophys. Space Phys., 18, 269-3 11.
Sleep, N . 11. & Snell, N. S., 1976. Thermal attraction and flexure of mid-contincnt and Atlantic marginal
basins, Geophys. J. R. astr. Soc., 45, 125-154.
Stearn, C. W., Carroll, R. L. & Clark, T. H., 1979. Geological Evolution of North America, 3rd edn,
Wiley, New York, 566 pp.
Stephenson, R., 1978. Isostatic response of lithosphere in Canada, Ahstr. Progr. geol. Ass. Can./Min.
Ass. Can., 3, 498.
Stephenson, R., 1981. Continental topography and gravity, PhD rhesis, Dalhousie University, Halifax,
Canada, 34 1 pp.
Stephenson, R., 1983a. Erosion-induced uplift and stresses in south-eastern Australia. Ahstr. geol. Soc.
Aust., 9 , 6-7.
Stephenson, R., 1983b. Stresses resulting from erosion of old topography in Canada, Abstr. Progr. geol.
Ass. Can./Min. Ass. Can./Can. geophys. Un., 8, A65.
Stockwell, C . H., 1964. Fourth report on structural provinces, orogenics, and time-classification of rocks
o f the Canadian Precambrian shield, Pap. geol. Surv. Can., 64-17 (11), 1-21.
Tukey, J . W., 1967. Spectrum calculations in the new world of the Fast Fourier Transforms, in Advanced
Seminar on Spectral Analysis of Time Series, pp. 25-46, ed. Harris, B., Wiley, New York.
Turcotfe, D. L., McAduo, 1).C. & Cdldwcll, J . C., 1978. An elastic-pcrfcctly plastic analysis of the
bending of the lithosphere at ;I trench, Tectonophys., 47, 193-205.
Van Houten, F. B., 1981. The odyssey of niollase, in Sedimentation and Tectonics in Alluvial Basins,
ed. Miall, A. D.,Spec. Pap. geol. Ass. Can., 23, 35-48.
Walcott, R. I., 1970a. Flexure of the lithosphere at Hawaii, Tectonophys., 9 , 4 3 5 - 4 4 6 .
Walcolt, R. I., 1970b. An isostatic origin for basement uplifts, Can. J. Earth Sci., 7, 931-937.
Walcott, R. I . , 197Oc. Flexural rigidity, thickness and viscosity of the lithosphere, J. geophys. Res., 75,
394 1-3 954,
Walcott, R . I . , 1976. Lithosphere flexure, analysis of gravity anomalies, and the propagation of seamount
chains, in The Geophysics of'~ h cPacific Oceanic Basin and its Margin, eds Sulton, ti. H.,
Manghnani, M. H. & Moberly, R., Geophys. Monogr. A m . geophys. Un., 1 9 , 4 3 1 -438.
Watts, A . B., 1978. An analysis of isostasy in the world's oceans I . Hawaiian-Emperor seamount chain,
J. geophys. Res., 83,5989-6004.
Watts, A. B. & Cochran, J . R., 1974. Gravity anomalies and flexure of the lithosphere along the
Hawaiian-Emperior seamount chain, Geophys. J . R. astr. SOC., 38, 119-141.
Watts, A. B., Cochran, J . R. & Selzer, G., 1975. Gravity anomalies and flexure of the lithosphere: a threedimensional study of the Great Meteor Seamount, north-east Atlantic,J. geophys. Res., 80, 13911398.
Watts, A. 8 . CG Ribe, N. M., 1982. Thc distribution o f intraplate volcanism in the Pacific Ocean basin: a
spectral approach, Geophys. J . R. astr. Soc.. 71, 333-362.
Wheeler, J . 0. & Gabriclsc, H. (eds), 1972. The Cordillcran structural province, in Variafions in Tectonic
Styles in Canada, eds Price, R. A. & Douglas, R. J . W., Spec. Pap. geol. Ass. Can., 11, 1-81.