JOURNAL
OF GEOPHYSICAL
RESEARCH,
VOL. 94, NO. B2, PAGES 1655-1664, FEBRUARY
10, 1989
Displacementof SurfaceMonuments'Vertical Motion
FRANK K. WYATT
Instituteof Geophysics
and PlanetaryPhysics,Universityof California, SanDiego, La Jolla
Severalmeasurements
of verticalgroundmotionat PitionFlat Observatory,
California,indicatethe overall
weaknessand instability of the Earth's weatheredsurfacewith respectto the underlying rock. Cumulative
long-periodmotionsof order 0.5 mm per year dominatetheserecords,thoughsmallerelasticdeformations
causedby precipitationloading,atmosphericloading,and tidal strainsare evidentat higherfrequencies;all of
thesehelp to characterize
the near-surface
material. The long-periodrecordssuggestthat near-surface
weathering is the dominantinfluenceon monumentmotion, at least at this site on crystallinerock in a semi-arid
environment.Rainfall loadinggives an averageverticalmodulusof 2.6 GPa for the materialin the uppermost
26 m of the ground,comparedwith 88 GPa for graniteundermoderateconfiningstress;atmosphericloading
givessimilarresultsbut indicatesthe groundis permeableto airflow at periodslongerthana few hours. Earth
tide recordsshowPoisson'sratio to be 0.09 in contrastto the normativerangeof 0.2-0.25, establishingthat horizontal strainscoupleonly weakly into vertical ones,so that vertical strainsnear the surfaceare a poor measure
of arealstrain. The form of the ground-surface
displacementpowerspectrumindicatesthat analysesof geodetic
surveyswould be improvedwith the inclusionof a monument-positioning
errorbudgetthat increaseswith time.
Becauseof groundinstability,andthe generallysmallrateof crustaltectonicmotions,deeplyemplacedmonumentswill be neededfor observationalprogramsdesignedto detectshort-termchangesin crustaldeformation
over baselengths
of order 10 km andless.
INTRODUCTION
Direct
measurements
of crustal deformation
are limited
to
observationson, or very near, the ground surface--most often
betweenmonumentsspecificallydesignedfor repeatedsurveying.
Near the ground surface,weatheringalters the once competent
rock massto a weakenedstate,makingit an especiallynoisyplace
for such measurements.The size of the displacementnoise is
small enough,and scale of long-term deformationssufficiently
broad,that averagesof the deformationover large areasand long
timesgenerallygive usefulresults[e.g.,Savage,1983]. However,
over distancesof 10 km and less(oftencomparableto the depthof
seismicity),and for time scalesof years and shorter,monument
instabilitycan be the limiting noise factor. Understandingand
quantifyingmonumentstability is thus essentialfor interpreting
results from the more detailed studies of crustal deformation.
displacement
measurements
reportedon hereextendto depthsof
only26.5 m, roughlythesamedepthasthewatertable.
The most important set of measurementsaffected by
millimeter-size vertical motions is precise geodetic leveling
[Vani[eket al., 1980]. For overa century,spiritlevelingbetween
widelyspacedbenchmarkshasprovidedmostof theevidencefor
what is knownaboutverticalgrounddeformations--fromregions
of active faultingand volcanismto areasof groundwater
withdrawal. As to sourcesof noise,levelingstudiesby Savageet al.
[ 1979] andReilingeret al. [ 1984] foundthatthe observedground
tilt spectrumhasmostof its energyat shortspatialwavelengths,
andsuggest
elasticwarpingof the groundas the probablecause.
Such signalsarise from the couplingof thermal stresses,or
regionalstresses,
intogroundtilt, throughtopography
or material
inhomogeneities.
The resultsfrom this study,however,suggest
thatinelasticprocesses
are moreimportant.
Continuously
recordingstrainmeters
and tiltmetersare another
class of measurementsrequiring stable bench marks [Agnew,
1986]. Theseare observatory-based
instruments
meantto record
This paperon verticalmonumentmotionis intendedto complement an earlier one on horizontal motions [Wyatt, 1982], and
reportson measurements
not availablethen. All of the observations consideredhere are from Pition Flat Observatory(PFO), deformationsover distancesless than 1 km, with precisions
located in southern California
between
the San Jacinto and San
Andreas fault zones. Details of the geological setting are
describedin the earlier paper. In brief, the site is an uplanderosional surface inclined to the southwestwith a 3% slope. Its
approaching
1 partin 10•ø.Heretheneedforexceptionally
stable
monumentsis obvious:motionsof only 0.1 mm/yr correspond
to
strainor tilt ratesof 10-6 yr-• on 100-m-long
instruments.
Thisis
muchlargerthanmosttectonicsignalsand,because
onlya pairof
geomorphology
suggests
thatthe12-km
2flatis a pediment,
whose monuments are involved, there is no network averaging to
suppressthe monument-displacement
noise. Savage [1983]
base level is defined by a line of rock exposuresalong its
southwestedge. A layer of sandabout0.3 m thick coverscrystalline rock composedof biotite-richhornblendegranodiorite.Wyatt
[1982] and recent vertical seismicprofiles (J. Fletcher, personal
communication,1987) show a linear increaseof P wave velocity
from 0.5 km/s at the surfaceto 2 km/s at a depthof 15-20 m, indicative of weakenedrock in this range. Below this depththe velocities tend toward values more characteristicof solid granite at
low confiningpressures(3.5 to 5.5 km/s). The deepestvertical
shows that strain rates in the western United States are of order
1-3 x 10-7yr-•; instrument
stability,includingattachment
to the
ground,mustbe substantially
betterthanthis to recordcrustal
deformationsfaithfully.
The recordspresentedin this paperare from long baselength
tiltmetersand from other instrumentsspecificallydesignedto
measurebenchmark stability. Long baselengthtiltmeters[Wyatt
et al., 1984] provide a precisemeasureof elevationchanges
betweenmonumentsseparatedby as much as 1 km. (Greater
separation
is possible,but not generallypractical.)To identify
Copyright1989 by the AmericanGeophysicalUnion.
those monument motions distinct from tectonic ones, it is assumed
Papernumber88JB03600.
thattruetectonicsignalsare gradual,independent
of environmen-
0148-0227/89/88JB-3600505.00
tal conditions,and commonto colocatedinstruments,so that any
1655
1656
WYATt:
DISPLACEMENT OF SURFACE MONUMENTS: VERTICAL MOTION
unreasonablyhigh tilt rates, or correlationsbetweenweatherand
changesin tilt rate, or tilts appearingonly on individual instruments running in parallel with others,are taken as evidenceof
bench mark instability. The best argumentfor this reasoningis
that correcting the long baselengthtilt records for their endmonumentmotions (measuredindependently,to shallow depth)
removes most occurrences of anomalous tilt.
MEASUREMENTSAND TECHNIQUES
Wyatt [1982] found persistenthorizontalmotionsof order 0.1
mm per year for large gabbromonumentsinstalledat a depthof
about 2 m. The largestmotionswere associatedwith periodsof
unusually heavy precipitation, which seem to accelerate the
weatheringof the underlyingrock. Wyatt [1982] speculatedthat
if indeed weathering were the controlling factor in monument
instabilitythen verticalmovementsmight be even largerthan horizontal ones,throughongoingdissolutionof the underlyingrock
column.
LFT/LOA
As an initial test of this, and as the starting point for the
developmentof an instrumentcapableof recordingtectonicdeformations,a 50-m-long tiltmeter was built at PFO in 1979. This
usedan uninterruptedfluid surface(ideally an equipotentialsurface) as the referencelevel, forming a Michelson-Galetiltmeter
[Beavan and Bilham, 1977]. The end-monumentswere two gab-
bro columns(1.8 m3) attached
to the Earthat depthsof 3.7 m
(Figure 1). Differencingthe fluid heightmeasurements
from each
end gives the differentialelevationchange,or tilt, over the 50-m
Fig. 2. Sideview of oneof theend-vaults
for thecurrent535-m-longfluid
baseline. The results of this experiment(discussedin the next tiltmeter(LFT), includingits 25.3-m verticallaseropticalanchor(LEA).
Any spuriousmotionsof the gabbrotable are registeredequallyas an
apparentheightchangeof theequipotential
fluidsurfaceandasa displacement relativeto the bottomof the LeA, allowingfor their elimination
from the tilt record.
section)verified the inadequacyof such surfacemonumentsfor
the recordingof tectonicsignals.
The next stepwas to build a long baselengthtiltmeter(535 m)
incorporating
opticalanchorsat eachend (Figure2). An optical
anchor [Wyatt et al., 1982] is simply an optical interferometer
designedto detectany changesin distancebetweenthe ground
surfaceand anchoringpointsat depth. In this application,for the
measurement
of verticaldisplacements,
the opticalanchorsextend
vertically from the tiltmeter's end-monumentsto the bottom of
26.5-m-deepboreholes.Any localizedmovementof eitherof the
m
SFT
monumentsthen appearsboth as a changein tilt (a changein
monumentheightrelativeto the fluid surface)and as a changein
vertical displacement,as recordedby the optical anchor. Subtractingthe two opticalanchormeasurements
from the tilt record,
itself formedby differencingthe fluid level observations
from the
two ends, correctsfor monumentinstability,at least down to
depthsof 26.5 m. The monuments
for thisinstrument
are gabbro
slabs
sitting
upon1-m3concrete
pierswhichwereformed
directly
on thefloorsof subsurface
vaults.Figure2 presentsthedetails.
PitionFlat Observatory
is alsothe sitefor a largecollection
of
geodeticbench marks. Most of these (about 70) are classB rod
marksinstalledby researchers
from the Universityof California,
SantaBarbara(UCSB) in 1979-1980 and surveyedat leastannually since then [Sylvester, 1984]. Class B marks are small-
diameterstainlesssteelrods driven into the grounda moderate
Fig. 1. Side view of one of the end-monumentsusedin the initial 50-mof the groundat
long shortfluid tiltmeter(SFT). The gabbrocolumnswere cementedinto distance[Floyd, 1974]. Owing to the toughness
mostof therodmarksthereextendonlyto depths
the bottomof large-diameterboreholesandthen insulated.The enclosures the observatory,
were temperaturecontrolled.
of 1 m, muchlessthanthe 4-m depthusuallyachievedfor classB
WYATt: DISPLACEMENT
OFSURFACEMONUMENTS:VERTICALMOTION
resultof the intenserains which precededthe earthquake,causing
water to percolate into the ground around the monuments.
Somehowground shakingseemsto have promptedthe reversal
and eventualdissipationof this rainfall-induceddifferentialdisplacement,perhapsby promotingdrainagein the critical volume
just below the monuments.
Vertical displacementrecordsfrom the West monument(at site
Rho) and the East monument (Tau) of the 535-m tiltmeter are
presentedin Figure 4. Becausethe final tilt signal from this
instrument(when correctedfor the end-monumentdisplacements)
indicates no more than 0.06mm/yr of differential monument
motion betweenthe endpoints,and this may well be true crustal
tilt, it is not displayedhere. The most conspicuous
event in this
figure occurs following the rainstormsof early 1983. These
stormsled to the rapid uplift of both monuments:at Rho 1 mm,
and at Tau 0.4 mm. Actually this event had severalphases,not
evident on this scale, with the monumentsreacting distinctly to
2.0
Short
Fluid
Tiltmeter
95.65
1657
(50rn)
ø
_
1.o
I
0.0
o
successive storms. Because the vault floors are not sealed, once
the loose soil immediately surroundingthe vault became water
saturated,the vaults flooded,but only by a few centimeters.To
preventsimilar flooding,A-frame roofswere constructedover the
vaultslate in 1983 (indicatedby the asterisksin Figure 4). However, the vault floors remained wet until near the start of 1985
Precipitation
•
-2.0
,
,
,
•
6.25cm/div
....
1980
I
....
1980.5
I
1981
Time (yeer)
Fig. 3. The completerecordfrom the SFT, includingthe tiltmeter'sfluid
temperature.Vertical units are given both in radiansof tilt (over 50 m)
and as the relative changeof heightbetweenthe two end-monuments.A
positivetilt trendindicatesgroundtippingdownat an azimuthof 95.65ø,
azimuthmeasuredeastfrom north. The broadeningof the tilt recordis due
when dehumidifiers were added (cross-hatchsymbols). This
becamenecessary
becausepersistentlyhigh vault humidityruined
several electronic components. The effect of adding the
dehumidifierswas unexpected:both monumentsbegan to settle
and have continuedto subsidesincethen. An explanationfor this
is givenin the next section.
Further evidenceof groundinstabilitycomesfrom small tilt-
2.0
to the Earth tides.
marks.The UCSB levelinglinesalsoincludesevenclassA bench
marks(rodsin boreholes,anchoredat a depthof 9 m) installedby
the National GeodeticSurvey (NGS), as well as ties to both the
original50-m tiltmeterand the current535-m instrument.Five of
the class A marks, installed in 1986, are clustered around the west
end of the long tiltmeterfor direct comparisonwith its optically
anchored monument; a sixth one is at the east end.
OBSERVATIONS
Figure 3 presentsthe data from the shortfluid tiltmeter (SFT)
from August 1979 throughJanuary1981, at which time its sensors
were removed for use in the long baselengthinstrument. The
records show a cumulative tilt of 1.5prad, down to the west,
much too large to be reflectingbroad crustal deformation. The
importanceof this figure is that it shows an apparentelevation
changebetweenthe instrument'send-mounts,amountingto 0.08
mm. Even theseclosely spacedmonuments,which must be sharing nearly identical geological and environmental conditions,
show substantial differential
motion.
The small bump at 1980.15 in Figure 3 (marked by a vertical
line) deservesspecialmention. Its peakcorresponds
to the time of
a moderate earthquakeonly 13 km away (Mc 5.5 [Frankel,
1984]). The bump is not the signatureof a regional deformation
becausecolocatedstrainmetersdid not show any unusualsignals
around this time, at deformation-levels orders of magnitude
smaller[Wyatt, 1988]. Most likely the onsetof this featureis the
._
]I iPreciplt(:]tlO
[ Im/div::)
I
o 1985.0
II ,.,.I [,19•4
,,[,0 ,a[J19•5
Ifi,,.._
JI 19•6
][[I0•, ,,J,,
0
Time
(years)
Fig. 4. Fouryearsof verticaldisplacement
recordsfromtheopticalanchor
gaugesat thetwo endsof theLFT. Thesemeasurements
givethedisplacement of monumentsrestingat 2.5-m depth, relative to 26.5 m. Also
shownis thegroundtemperature
at a depthof 0.5 m andthe dailyprecipitation. The asterisksshowwhen A-frame weatherprotectionhousingwas
addedover the vaults,while the cross-hatch
symbolsindicatewhen vault
dehumidifiersbeganoperation.
1658
WYATT' DISPLACEMENTOF SURFACEMONUMENTS:VERTICALMOTION
2.0
NW Monument (NW motion positive)
1.0
SE Monument (SE motion positive)
O.
•
198,3.0
19d40
Time
19•50
(years)
19•6.0
erally important:soil moisture sensorsat 1 m show negligible
changeto all but the mostprolongedstorms,while the waterwell
recordsshow little clear responseto even the biggestrainfalls.
Most rainfall is either absorbedby the very near-surfacematerial,
whereuponit eventuallyevaporates,
or runsoff laterally.
Figure 6 presentsan expandedview of the vertical displacementsfor two representative
rainfalls in 1983. This plot includes
records from the optical anchors and from two mechanical
(invar-rod) displacementgaugesdevelopedby researchersfrom
Lamont-Doherty Geological Observatory (LDGO), Columbia
University (R. Bilham and J. Beavan, personalcommunication,
1988; Wyatt et al. [1984]); the LDGO W records are from a
monumentonly 11 m distantfrom Rho, with LDGO E a similar
distancefrom Tau. The responseof the adjacentmonumentsis
similar for the two rainstorms,the westernmonumentsshowing
generallylargerdisplacements
(1.1/•m per cm of rainfall) thanthe
easternones(0.75/•m per cm). Apparentlythe groundis stronger
in the vicinity of the easternmonuments.
Cross-spectral
analysiselucidatestwo otherfactorscontrolling
vertical displacements:thermoelastic stressesand barometric
loading. Figure 7 showsa heavily averagedpower spectrumof
the monumentmotionat Rho alongwith scaledspectraof the air
temperatureand atmosphericpressure.Analysis showsthat the
admittance function is not a constant for either of these factors;
rather,the effect they have on vertical displacementis dependent
Fig. 5. Horizontaldisplacementrecordsfrom horizontalopticalanchors
for the sameperiodpresented
in Figure4. The displacement
gaugesspan on frequency.For air temperaturethe influencebecomeslessat
roughlythe samedepthintervalasthe verticalonesof Figure4.
higherfrequencies,while for atmosphericloadingit is the opposite. At periods of a week the temperature coefficient is
140nm/øC and at a day only 60 nm/øC;the atmosphericeffect is
1.6nm/Pa at periodsof a few days, increasingto 2.8 nm/Pa at 12
meters attached to the end-monuments
at Rho and Tau. These
sensorsshowtilts of order2 x 10-4 rad, with recordswhichlook hours (S2), and 3.5 nm/Pa at 6 hours. In Figure 7 the pressure
much like the displacementrecordspresentedin Figure 4. This spectrumis plottedusingthe admittanceat S2,whereit is the conalsocontributes
heavilyto the S•
tilt correspondsto differentialmotionsof 0.2 mm acrossthe 0.9- trollingfactor,thoughpressure
m2baseof themonuments.
Suchlargedisplacements
aresurpris- and S3 spectralpeaks. (The pressureamplitudes,for typical
conditions,
are40 Pa for S•, 60 Pa for S2,and 10 Pa
ing consideringthat the concretemonumentswere formeddirectly atmospheric
of assuminga constantadmittance
on a preparedsurfaceof intactgranitewhich, from the excavation for S3.)The inappropriateness
viewpointat least,was competent.The monumentsthusmustbe
averagingthe displacements
occurringacrossthe surfaceof this
seeminglycompetentmaterial.
For comparisonwith the verticalmovementsof Figure4, Figure 5 showsthe horizontaldisplacementrecordsfor monuments
MONUMENT
DISPLACEMENT
NW
and SE of the NW-SE
laser strainmeter for the same time
•
period (this plot is a continuationof Figure 8 in Wyatt [1982]).
MonumentNW undergoesa maximum displacementof 0.5 mm
relative to 24-m depth, while SE moves about0.2 mm. Both of
Response
to Precipitation
these monuments are in vaults of similar construction to those at
the endsof long fluid tiltmeter(Figure2), but withoutthe addition
of A-frame roofs or dehumidifiers.
The character of the horizon-
tal motions is akin to the vertical ones, but with somewhat less
amplitude.
Episodesof rainfall loading are easily identifiedin the vertical
•
'
LDCO
E
.
displacementrecords,providingthat the precipitationis large
enoughand occursquickly enoughthat the associated
displacementsare not lost in the backgroundnoise. A closeinspectionof
Figure4 showsnumeroussmalldownwardjumpsassociated
with
rainfall. These jumps are assumedto reflect the vertical strain
Precipitation
inducedby an increasein the amount(weight) of water held by
the near-surfacematerial.To quantifythisresponse,
the rain must
not occurso quickly as to causelateralrunoff, nor so slowlythat
2 !0
220
230
240
250
260
270
280
eitherevaporationallowsthe water to dissipatebetweenshowers
time (Day # - 1983)
or that the surfacematerialbecomessaturated,allowingdown- Fig. 6. Vertical groundmotionat a depthof 2.5 rn in responseto surface
ward flow. At this site the last mechanismdoesnot seemgen- loadingby rainfall. Precipitationis givenasdaily total.
• (10
½rn/dtv)
WYA'I'T: DISPLACEMENTOF SURFACEMONUMENTS:VERTICALMOTION
Vertical Displacement Power Spectrum
"Grav"); the other end of the original 50-m tiltmeter was not
includedin the field surveys.
Observationof the newer classA monumentsbegansoonafter
their installation,in 1986, and they have been resurveyedonly a
few times since then by each of two groups. Results from this
sparsedata set show much greaterstability,generallywith about
-40
Rho
-60
1659
0.3-mm
S2
standard deviation relative to Rho.
DISCUSSION
-80
This section presentsfirst a descriptionof the mechanisms
responsiblefor long-periodvertical groundmotionsin weathered
crystallinerock. These motionsare then quantifiedby meansof
their power spectrum,allowing for a discussionof the consequencesof theseobservations
on programsof crustaldeformation
measurement.Finally, the recordsare usedto place constraintson
variousmeasuresof rock androck-jointmoduli.
-lOO
Response to
Air Pressure(@ S2)
.....
-120
A•r Temperature(@ S•)
Weatheringand Other Causesof Long-TermDisplacement
i
& i
i iii
I
10-6
i
i
i
i
i i i i
i
i
10-5
i
i
i
I III
I
I
10-4
Frequency (Hz)
Fig. 7. Verticaldisplacement
powerspectraldensityaround1 cpd,showing contributions
from air pressureandair temperature,
andassumingconstantadmittance.In this figure,the air pressure
responseis takento be the
valuedeterminedfor a periodof 12 hours;the air temperatureresponseis
that for 24 hours.
throughout
ismostevident
atfrequencies
of3-5 x 10-6Hz,where
the scaled,and nominally causativepressurespectrumactually
exceedsthe observeddisplacement
spectrumslightly.The air temperatureis mostimportantat 1 and3 cyclesper day (cpd),butthe
physicalmechanism
for this is lesscertain.The groundtemperature is generallytaken to attenuateexponentiallywith depth,
dependingon the ratio of the depthto a thermallengthconstant,
which itself dependson frequency.At theseperiodsthe ground
temperature's
thermallengthconstant
is about0.2 m, makingsimple half-spacethermoelastic
effectsat the vault bottom(2.5 m)
vanishinglysmall, but this ignoresthose perturbingeffects
broughtaboutby the vault's presence[e.g., Agnew, 1979], and
possible topographic/inhomogeneity
factors [Harrison and
Herbst, 1977]. A direct instrumentaleffect is anothercandidate
mechanism
to explainthe observed
powerat 1 cpdandin its harmonics. The instrument temperature coefficient is about
At this site, the dominant influence on vertical motion of near-
surface bench marks is precipitation, through weathering
processesactivatedby changesin soil moisture. The correlation
of displacements
with periodsof heavy precipitationis mostclear
in the horizontalmovementrecordsof Wyatt [1982], but it is also
quite evidenthere. The displacement
recordsshow long-period
excursionsonly after thoroughwetting of the ground, when the
top 1 m of the ground becomessaturated,allowing moistureto
propagatedownward, or during periodsof desiccation.Weathering causesthe rock to decomposevia granulardisintegration[Isherwood and Street, 1976]; the fully loosenedsurfacematerial is
then erodeddownslope,leaving only a thin veneerof soil over the
decaying rock. The depth of extensive weathering, or
grussification,can be quite deep. Stierman and Healy [1985]
report evidenceof weatheringto a depth of 70 m at a settingin
California similar to PFO. There they find the changefrom weakened to competentrock to be quite abruptand most likely related
to water-rockchemistryin the vadosezone--the regionabovethe
water table. (Severalboreholesat that site gavethe water table to
be anywhere between 65 and 110 m, thereby showing the
difficulty with the very conceptof a water table in certaintypesof
fracturedcrystallineterrain.)They attributethe low strengthof the
graniteto alterationof biotite and chloritemontmorillonite.
Isherwood and Street [1976] ascribethe rapid weatheringof
biotite-rich rock to the mineral's expansionwhen subjectedto
moisture. The expansionof the biotite and the subsequent
formation of microfracturesin the quartz and feldsparscan have the
2.5/•m/øC(or 15/•m/øCif we includethethermalresponse
of the
concretemonument),drivenby daily variationsof about0.02øC
insidethe temperature
regulatedinstrumentenclosure.Together, effectof reducing
thebulkdensityof therockfrom2.7 x 103to
air pressureand air temperature
causethe daily verticaldisplace- 2.0x 103kg/m
3. Figure4 presents
strongevidencefor this
mentsignalsat Rho andTau (= 1.5/•m p-p) to be highlycoherent. mechanism:first the end-monuments
of the tiltmeter are uplifted
The otherclassof verticaldisplacement
observations
at the site by as much as 1 mm following the heavy rainfall, only to show
arethoseof the geodeticbenchmarks.In his 1984reporton bench subsidence once the vault and the ground beneath it are
mark motionsat PFO, Sylvester[1984] reportedapparentlyran- dehumidified. Presumably,in the natural course, the ground
dom motions for class B rod marks installed at shallow depths swells up when wetted, eventuallyto sink below its initial level
(includingmeasurements
to a specialNGS referencepad, 5 m on (becauseof partial rock dissolution)as the material returnsto its
a side and 0.3 m thick), with standarddeviationsabouttheir mean normal moisturecontent. In this particular instance,the process
heightrangingfrom 0.5 to 3.2 mm. In sevensurveysspanninga was apparentlyfrozen in two states:after the rainfalls of early
periodof 3« yearsthe meanstandarddeviationwas 1.0 + 0.5 1983, rain shieldspreventedsubsequent
rainsfrom permeatingthe
mm. The moststablesurveyingresults(standarddeviationof 0.2 nearby ground, but (1) the rain shield and the vault itself
mm) came from elevation differencesbetween Rho and Tau suppressed the normal evaporative processes, then (2)
(UCSB: 7337 and 7336), betweenRho and a classA monument dehumidifierswere added,causingthe underlyingmaterialto lose
installedby UCSB (UCSB: 05N), and betweenRho and the east its moisture.
monumentof the original 50-m tiltmeter (UCSB: 7313, or
The conditionsin the groundimmediatelyunderthe vault (the
1660
WYATT: DISPLACEMENTOF SURFACEMONUMENTS:VERTICALMOTION
first 1-3 m) appearto control the vertical motions;it is this same
material that showsthe highestdegreeof weathering. Motions of
Power Spectral Density
-
Vertical Displacement
1 mmoverdistances
of 1 m implystrains
of 10-3. Whilethisis a
•'//.
-2O
substantialdeformation,it is small comparedwith the responseof
the expansivesoils discussedin soil mechanicsliterature [Chen,
1975; Yong and Warkentin, 1975]. These soils show strainsof
Near-Surface
Monuments
10-2 in response
to 5% changesin soil moisture.By most
accounts,the groundat PFO wouldbe consideredquitestable.
Soil creep is another candidatefor explaining the long-term
motionspresentedin this study,but Wyatt [1982] did not find any
correlationbetweenlateral groundmotionsat a few metersdepth
and groundslope,which rules out this as the controllingmechanism. Since,on the average,the groundis not creepingdownslope
and yet is weathering,it would seem likely that the long-period
vertical motions should exceed the horizontal ones. This is true,
but onlyjust so, as shownby comparisonof Figures4 and 5.
All the observationsof groundmotion at PFO are from depths
of 1-3 m, but what might the rate of subsidence
be at the ground
surface?Averagingover large areas,Schumm[1963] found that
surfacedenudationratesare an exponentialfunctionof drainage
basin relief, the averagemaximumrate being 1 mm/yr for areas
similar to PFO. Saundersand Young [1983] discussdownslope
soil creep,surfacewash,and chemicalsolution(alsocalled weathering in that paper),alongwith othermechanisms
of grounddenudation. The highestrates,again of order 1 mm/yr, are given for
semi-arid regions and caused mostly by surface wash, or by
intenserainstormsfalling on rock poorly protectedby vegetation.
Indeed they suggestthat the rate of 1 mm/yr may representthe
thresholdwhich preventsestablishmentof substantialvegetation
cover. Both the thinnessof the vegetationand the displacement
measurementsof the underlying rock at PFO are in accord with
this high rate.
As an example of ground motion under other circumstances,
Riley [ 1970] reportedon long-baselinetiltmeterrecordingsof land
surfacedeformationfrom hydrocompactionof moisture-deficient
fan deposits.Hydrocompactionoccurswhen highly poroussediments, above the water table, are rewetted for the first time since
deposition,resultingin weakenedintergranularbondsand settling
under the overburdenload--a geologicalsettingmuch different
from PFO. Recognizingthe need for vertical anchoring,46-mdeep invar-type compactiongauges were built at the ends of
fluid-level tiltmeters[Riley, 1970, 1986]. He found initial ground
swelling with increasedsoil moisture (0.3 mm when the end
vaultsflooded),a frequency-dependent
barometricresponse(indicating a vertical modulusof 0.25 GPa at periodsof a few hours),
m
-1 O0
-120
-140
I
IllIll
I
I
I
10 -7
IIIIII
i
10 -6
i
i
i iiiii
i
10 -5
i
i
i iiiii
i
10 -4
i
i
10 -3
Frequency (Hz)
Fig. 8. Severaldecadesof the verticaldisplacement
powerspectrum
for
near-surfacemonuments,showingthe generaldependence
on frequency
(f-2) characteristic
ofBrownian
motion.
Theshaded
region
indicates
the
typicalrangeof resultsfor the lowerfrequencies;
at higherfrequencies
the
spectrashowlittle scatter.
powerlaw, powerinverselyproportional
to thefrequencysquared,
which is descriptiveof one-dimensionalBrownian motion, or a
random-walkprocess.Wyatt [1988] givesthe standarddeviation
for suchprocesses
as a functionof the intervalT betweenobservations. Over the frequencyband wherethe displacement
power
spectrum
maybewrittenasP(f)= P0(2/rf) -2,thestandard
deviationof displacement
is givenby cr= (1/SPoT)l/2.
Herethisis
validabove10-6Hz (about12 days),withP0= 1.8x 10-17m2/s
givingcr= 3.0x 10-9(ms-1/2)T 1/2,or cr= 3.3/•m in 2 week's
time. The equivalentstandarddeviationfor horizontalmotionat 2
weeksis just one half of this, or 1.7/•m. This fo,rmulation
progressivelyunderestimates
the standarddeviationat timeslonger
thanthis,asthespectrum
diverges
fromthesimple
f-2 powerlaw,
indicatingeven greater energy at the longer periods. Empirical
studiesshowthatfor P (f) = P 0(2zrf)-a, crmaybe approximated
by (1APoTa-1)l/2,
foro•in therange1.5-2.6. Overintervals
of 5
months,which coincideswith the lowest frequencyresultsgiven
in Figure 8, displacementsmay thus amount to something
between0.01 and 0.24 mm, corresponding
to the lower and upper
boundsof that plot. Many more data are needed to define the
spectrallevelsat periodsof a year andmore.
Strictly Brownianmotion (a = 2) is governedby the diffusion
and thermaelastic effects, all in common to what is found at PFO.
equation[Lavenda,1985], providinganotherusefulway to underEven the hydrocompactioneffect, 0.4-0.8 mm of displacement
stand monumentmotion. The root-mean-squareof a particle's
over the 46-m lengthof the verticalextensometers,
was commendisplacement
is givenby (2DT)1/2,whereD is the diffusion
surate.In sharpcontrast,recordsfrom the 30.5-m-long tiltmeters
coefficient(D = lAP0). A monumentmay thereforebe considered
showed differential subsidence of 2.5 mm, even after correction
as an individual particle, albeit a macroscopicone, diffusing
for end-monumentmotion, indicating substantialdisplacement
below the bottom of the extensometers.
through
theground.Thediffusion
coefficient
forthis,10-17m2/s,
is roughlya factorof 10lø smallerthanthecoefficient
for most
Power SpectralLevelsand their Consequences
for
Measurementsof Crustal Deformation
mentionedin the next section).
Figure 8 presentsthe verticaldisplacementobservations
of Figure 4 in anotherform, as their power spectraldensity. This figure
is comparableto Figure 9 of Wyatt [ 1982] and showsthat the vertical displacementpower levelsare about6 dB higher(a factorof 2
in amplitude)than the power levels forming the horizontalspectrum, except at the lowest frequencieswhere both data setsshow
considerable scatter. Generally, the spectra display the same
other individual "particles"in nature(e.g., thermaldiffusivity, as
Despite the generalrecognitionof the effects of benchmark
instabilityon inferred vertical crustalmotions[e.g., Karcz et al.,
1976], mostreviewsof geodeticlevelingmake little mentionof it.
This is true because,in somesense,adequatelong-termcoupling
to the earth is outsidethe domain of leveling (most statistical
evaluationsare basedon discrepancies
betweenforward andbackward runs, or loop misclosure),and becausebenchmark stability
is generally adequate for surveying purposes. Also, for
WYATT: DISPLACEMENTOF SURFACEMONUMENTS:VERTICALMOTION
1661
baselengths
longerthan 10 km, monumentmotionsare usually far exceedsthe theoreticalrandomsurveyerror (the first term in
smallcomparedwith long-periodtectonicsignals.
(1)), we may use these recordsto estimateP0. For the better
For measurements
of crustaldeformationover shorterdis- UCSB classB monumentsat PFO /•t is about 0.65 mm in a
tances,
whether
byleveling
orfromtiltmeters,
monument
stability year'stime,givingP0= 2.7x 10-•4m2/s(assuming
o•= 2), and
is crucial.Savageet al. [1979]givethevariancein tilt for level- for themoststablemonuments,
emplaced
at severalmetersdepth,
ingbetween
twobenchmarksa distance
L apartas
/•t is about0.18mmin a year,orP0 = 2.1 x 10-•5m2/s.
cYtiqt
= 2ot2/L+ 2fi2/L2
(1)
Estimatesof Rock Properties
where(207L)•/2isthestandard
deviation
in therelative
change
of
Seismicvelocitiesin the upper20 m at PFO are well below
elevation
dueto measurement
error,and(2fi2)1/2is dueto bench those for solid rock. This could be due to the results of weather-
mark instability from all sources (e.g., simple decoupling,
ing, to inherentrock porosity(microcracks,
scale 1-100/•m
thermoelastic/inhomogeneity
effects). The first term comesfrom
[Birch,1961]), or because
of numerous
weakjoints. In theirstudy
the statistically independentrandom errors which accumulate
of jointing and seismicvelocitiesin fracturedgranite,Moos and
according to the square root of the survey distance,
Zoback[1983] foundthatwhilelarge-scale
jointsareimportantin
o•= 1.6x 10-Sm
m (or 0.5 mm/km
m) for the highestorder
explainingthe increaseof velocitywith depth(perhapsby promodemleveling[Vanidek
et al., 1980]. The second
termclearly
moting chemicaland mechanicalalterationnear their surfaces),
dependson the type of monumentsused and the environmental
compositionand microscopicpropertiesof the rock itself are the
and geologicalsetting. Basedon the assumptionof uniform tiltmainfactorsgoverning
velocity.Thissuggests
thatweathering
or
ing, Savageet al. [ 1979] find/• = 0.25 mm for classB rod marks
naturally occurring microcracksare the primary mechanisms
drivento a depthof 5-6 m at sitesin centralCalifornia;Sylvester
affectingthe near-surfacematerial.
[ 1984] givesa somewhatlargervalueof 1.0 mm for the generally
The nonlinear stress-strain behavior of intact rock at low
shallowermarksat PFO. Using/• = 0.5 mm andtakingL to be 1
confining stressis generally ascribedto microcracks[Walsh,
km (such that the two terms of (1) contributeroughly equally)
1965a;WalshandGrosenbaugh,
1979]. As confiningstresses
are
givescr = 1.0/•rad, a largeuncertaintyin the measurement
of tec-
increased,the cracksclose and the rock eventuallybecomes
nearlylinearly elastic. But for crystallinerock at pressures
less
In searchingfor an explanationfor the high degreeof nonunithan100MPa (i.e., abovedepthsof 3.8 km in theEarth,assuming
form movementexhibitedby adjacentbenchmarks,Savageet al.
lithostaticloadonly)theeffectof microcrack
openingis dramatic:
[1979] review severalearlier studiesof small-apertureleveling
Young'smoduluschangesfrom nearly70 GPa underpressureto
arrays.They note that large-aperturearraystend to averageout
20 GPa [Brace, 1965], and Poisson's ratio falls from 0.2-0.25 to
this short-wavelength
noise,presumablybecausethe tilt episodes
tonic deformation.
seenon the smaller arraysare not coherentacrossthe entirety of
the largerones. The preferredmechanismfor this is elasticvertical displacementsof order 1 mm causedby the couplingof thermoelastic,tectonic,or other strainsinto vertical motionsthrough
topographicor local materialinhomogeneities[e.g., Harrison and
Herbst, 1977].
Weatheringof the groundsurfaceis an alternativeexplanation,
consistentwith the resultsfrom PFO. In this context,weathering
may be consideredsimplyas causingpoor couplingof the monumentsto the Earth's crust,thoughat the groundsurfaceitself this
might not be evident. The presumptionof weatheringas the cause
of long-period, short-wavelengthtilt noise has a most important
implication. The form of the displacementpower spectrumindicatesthat for any finite interval T, the processwill be dominated
by energywith frequencynear 1/T and that all partsof the series
will be correlated (not statistically independent); as Figure 4
makesclear, averagesover longer and longer time periodsdo not
necessarilyconvergeto fixed values [Mandelbrot and McCamy,
1970]. The form of (1) is thusincompleteand shouldbe modified
to include a time dependenceof the standarddeviation due to
cumulativemonumentinstability:
cYti21t
= 2a2/L + 2[•o2/L
2+ 2[•t2/L
2
(2)
near zero [Walsh, 1965b;Jaeger and Cook, 1976]. Complicating
mattersis whetherdynamictests(monitoringseismicvelocitiesof
mega-Hertz pulses) or static tests are used to measure these
parameters;the effects of open crackscausedynamic techniques
to be in error by as much as several hundredpercent at atmospheric pressure[Simmonsand Brace, 1965]. Thus the relationship between seismic velocities and the true elastic moduli of
near-surfacerocksis poorly determined,exceptthat sinceseismic
waves are less likely to causeactualdisplacementsat crack faces
than do static loads, dynamic tests give results biased toward
greaterstiffness;they tendto providean upperlimit on the rock's
moduli. Poisson'sratio v, rigidity/•, and Young's modulusE
may be written in terms of compressionand shearvelocities Ve
and Vs, and densityp:
v=
1 (Ve/Vs)2-2
2 (v• / rs)2 _ •
la =pVs2
E = 2/•(1 +v)
For the values of seismicvelocitiesat PFO (J. Fletcher,personal
communication,
1987) this givesE- 1 GPa and v = 0.10 in the
depthrange0-7.5 m, with E-7 GPa andv = 0.11 between7.5
and 15 m, jmnpingto E -50 GPa,v = 0.22, between15 and35 m.
The changein magnitudeof Young'smodulusprovidesgoodevidence that the rock below about 20 m is sound(not friable), while
where/•0 is meantto includeonly thosemonument-positioningthe rapidincreasein Poisson'sratio arguesfor confiningstresses
nearatmospheric,
v
errors not increasingwith time (e.g., reoccupationerrors, true higherthanlithostatic.(Forconfiningstresses
decoupling,
cyclicelasticdisplacements),
and]3t = («P oTa-1)
1/2. is expectedto be low, and seismicvelocitiesactually tend to
underestimate
it [Jaegerand Cook, 1976].) It is alsopossiblethat
In fact, at PFO, elasticeffectsseemto be insignificant.
of moisture
Sylvester[1984] presentsthe time history of vertical motions the jump in v and E are, in part, a consequence
for representativemonumentsat PFO. Excluding the first of the saturationconditionslikely prevailingbelow 15 m.
Rainfall loading gives a better determinationof the average
two 1982 surveys,most all the plots in this article show a correlafor deformation of sequentialobservations,as would be expectedfor samples rock rigidity. Farrell [1972] reviewsthe equations
of Brownian motion. Since the variation in these displacements tionsof an elastichalf-spacecausedby surfaceloads.For an arbi-
1662
WYAI'T:DISPLACEMENT
OFSURFACE
MONUMENTS:
VERTICAL
MOTION
Vertical and Areal Strain Spectra
trary distributionof surfacepressureP acting on a homogeneous
elastic half-space, the vertical strain at the surface is given
by ez =-P / 2(;I,+/•),
where ;I, is Lam6,'s constant
(;L= 2/•v/(1- 2v)). As discussed
earlier,rainfall is assumedto
Vertical Strain •v
be absorbedin the upper 2.5 m of the ground,above the floor of
the vaults. Using an averageof the loading displacements
given
previously,dividing by the lengthof the opticalanchorbelow the
vault floor (24 m), and convertingthe rainfall into a load, gives a
vertical modulus(-P/ez) of 2.6 GPa. Assumingfor the moment
v=0.1
(so ;I, =0.25/•),
(Interval12 - 265 m)
-8O
S1
M2
Tau
-lOO
we have E =2.3 GPa for Young's
ß
modulusover the depth range 2.5-26.5 m, a value in reasonable
v
-120
accordwith the seismicallydeterminedvalues.
Atmosphericloading should cause the same signalsas does
precipitation,but cross-spectralanalysisbetweenthe air pressure
and vertical displacementsshowsa diminishingeffect towardlow
-140
frequencies.Again takingv = 0.1, and dividingby the instrument
Areal Strain •A
length, we may convertthe observeddisplacementvaluesto estimatesof Young's modulus. Here, the groundapparentlybecomes
-160
much stiffer at low frequencies,being 6.4 GPa for 6-hour periods
10-5
10-4
and 13.9 GPa at a few days. These values, but especiallythe
Frequency (Hz)
long-periodone, seem to contradictthe rainfall loading results,
but may be explained as the consequenceof airflow into the Fig.9. Comparison
of verticalandarealstrainspectra
fortherecords
from
permeable
ground.An infinitelypermeable
rockmasswould PFO.ForPoisson's
ratioof 0.25thearealstrain
spectrum
should
unidisplay infinite strength with respect to air pressureloading,
becauseno net downward pressurewould be createdon its surface. The permeabilityof the groundsurfacemeanslong-period
air pressuresignalswill not deform it as much as short-period
pressurefluctuations.
The annualtemperatureswingis anothercauseof cyclicalvertical motions. Harrison and Herbst [ 1977] showthatthe (upward)
vertical displacementat depthz, due to an imposedtemperature
variation T = To cosrot on the surfaceof a homogeneous
halfspace,is
formly exceedthe vertical strainby 9.6 dB, which it doesnot, indicating
the generallyhigh noiselevel of near-surfacevertical strain.
data seriesanalyzedsimilarly. The areal strainis from the unanchoredNS and EW 732-m-long laser strainmetersat the site (discussedby Agnew [1986]) and so gives the areal strain at 3-m
depth,averaged
overan arearoughly1km2 whichencompasses
Tau. Taking Poisson'sratio to be 0.25, and in the absenceof any
othervertical strain-generating
mechanism,we shouldexpectthe
vertical strainto be everywhere0.33 of the areal strainspectrum,
or, in decibels,9.6 dB less than the areal strain. Figure 9 shows
U(z)=1,•1-v
l+v fl Tok_
1e_kZ
{cos(ot-kz)+sin(ot-kz)}
the vertical strainis insteadgenerallygreater,indicatingthat the
vertical strain continuum must be noise of nontectonicorigin.
wheret is time,cofrequency(in rad/s),fl the coefficientof linear
This also follows from the long-term vertical displacement
expansion,
andk = (rO/2tC)
1/2,tCthethermaldiffusivity.Thetherrecordsof Figure 4, which, when convertedto strain,give rates
mal wavenumberk for tc= 7.5x 10-7m2/s(typicalfor surface
fully 2 ordersof magnitudelarger than other measuresof Earth
rocks)is [2.7m]-1 at periodsof a year. Basedonburiedground
strainat the site. In Figure 9 the power spectrallevels are nearly
temperaturesensorsat PFO, the effective amplitudeof the annual
equal at lines S• and S3, but this is an artifact of the vertical
surfacetemperaturecycle is estimatedto be 9ø C. (Becausethe
displacement'ssensitivity to air temperature(discussedearlier).
uppermostlayer of the groundat PFO is a particularlygood insuThe spectrallevels alsooverlapat S2,but here the apparentagreelator the true surfacetemperaturecycle is somewhatbigger than
ment is causedby responseto air pressure. Only at M2 (12.42this.)With• = 8 x 10-6øC-1andv = 0.1, thisgivesthe annual
hour period) is the vertical strainexceededby the areal strainby
cycle at a depthof 3 m (just below the vault floor) as 0.11 mm (pmore than 10 dB and showing a clear responseabove the
p), delayed about 110 days from the temperaturecycle. A close vertical-strain noise continuum. For Rho and Tau this difference
inspectionof Figure 4 showsthat this model fits the observations
in M2 spectralevels is 21.5 and 18.5 dB, respectively.Taking the
fairly well. For monumentsinstalledat depthsof only 1 m (e.g.,
meanvalue of 20 dB, Orez = -0.1 eA, impliesV = 0.09.
the classB rod marksat PFO) the theoreticalsignalwould be 0.23
Theoreticalargumentsbasedon the openingof microcracksat
mm, roughlytwice as large.
low pressures[Walsh, 1965b; Kemenyand Cook, 1986] predict
The assumptionof Poisson'sratio has little impacton the forethat a linearrelationshipshouldexist betweenPoisson'sratio and
going results,neverthelesswe can derive an independentestimate
Young's modulus,which is given by v =(vo/Eo)E, where the
for it too. At the traction-free surface of homogeneouselastic
subscripted
terms are the limiting values obtainedat very high
half-spacethe areal strain (t•A = t•1q-t•2, whereel ande2 are any
confiningstresses.Jaegerand Cook [1976] plot observations
of
two orthogonalmeasuresof strainon the free surface)is relatedto
this which show considerablescatter.The resultshere, of quite
the vertical strainby
low valuesfor v and E near the surface,are generallyconsistent
--V
with this relationship.
E z -EA
1-V
Movements on fractures and joints are. another mechanism
Althougheven the meaningof
For v = 0.25, its nominal value, ez =--0.33EA. Figure 9 shows affectingvertical displacements.
the power spectraof both the vertical strainat Tau (where the joints in the uppermostlayersis ambiguous,below somedepth
groundis somewhatstifferthanat Rho) andthearealstrainfrom a they are likely the controllingfactor. Engelder [1987] defines
WYATI':DISPLACEMENT
OFSURFACE
MONUMENTS:
VERTICAL
MOTION
1663
joints as being "a breakor crackin rock displayingopeningdis- rockis strongexceptfor jointing,thegeneralform of (3) oughtto
placementsand no appreciableshear displacement."Twidale apply.
CONCLUSIONS
[1982] notes that most all granitesnear the Earth's surfaceare
fractured,often in joint systemsthat parallel the groundsurface,
Weatheringof the groundseemsto be the primary factor conthe result of erosionalunloading. In their studiesof borehole
trolling
the long-perioddisplacementof surfacemonuments,at
seismic velocities, Stierman and Kovach [1979] and Moos and
Zoback [1983] found a systematicdiscrepancybetweenin situ
seismicvelocities(spanningintervalsof 1-2 m) and laboratory
estimatesof pressurizedcore samplesfrom wells with numerous
fractures,indicatingthe influenceof fractureson intactproperties.
In anotherstudyof six wells drilled in graniteto depthsof 25 m
and in three wells drilled to about 1 km, Seeburgerand Zoback
[ 1982] foundonly a slightdecreasein the numberof fracturesper
unit lengthwith increasingdepth. At PFO, variousboreholelogs
showfracturespacingS averagingfrom 1.5 to 5 m, similarto the
resultsreportedby Seeburgerand Zoback [1982]. Fracturesare
thusimportantandpervasive.
Bandis et al.'s [1983] laboratorystudy providesan excellent
empiricaldescription
of rock-jointdeformation.Jointing'scontributionto the overallmoduliof a rock massmay be computedby
knowledgeof the rock-joint'sstiffness(bothnormaland in shear),
togetherwith only a few otherobservations
of joint closureand
joint spacing[e.g., Fossum,1985]. The normal stiffnesskn, at a
givennormalstressCrn,is definedasdcr,/ da, wherea is thejoint
aperture. Walsh and Grosenbaugh[1979] show that the normal
joint stiffnessshouldbe proportional
to normalstress,kn = • Crn,
andreport• = 1.2x 104m-• forjointedgranite.We maytherefore use an estimateof normaljoint stiffnessat somedepthand
knowledgeof averagejoint spacingto determinethe effective
modulusfor any intervalin the verticalcolumn.
In a study of PFO grounddeformation,causedby transient
lowering of the water table, Evans and Wyatt [1984] found
kn = 20 GPa/m for a joint at 100-m depth,under 36 m of lithospheric load (26 kPa/m, above the water table) and 64 m of
reducedload (16 kPa/m, loadingreducedby hydrostaticpressure
in the affectedjoint). This gives a total effective load of 1.92
least at this one site on crystalline rock in a semi-arid environment. The general consistencyof survey leveling results [e.g.,
Savage et al., 1979; Sylvester,1984; and Reilinger et al., 1984]
and near-surfacedeformationalmeasurements[Wyatt et al., 1988]
from a range of locations would suggestthis observation is
broadlyapplicable. Once the surfacematerialis sufficientlywetted to allow the moisture to migrate downward, accelerated
weatheringof the rock begins,often lastingfor months,with large
displacements
persistingthroughoutthe desiccationperiod.
For measurementsof tectonic deformation, the apparently
chaotic motions of very near-surfacemonumentsmay be generally so large as to becomethe limiting factorin detectingshortterm signalsoccurringover distancesof 10 km and less. Averaging over many monuments,while certainto improvematters,ultimately will be limited by any systematic component in the
ground's responseto environmentalchanges. The form of the
vertical displacement power spectrum suggeststhat, in the
analysisof field observations,including an error term dependent
on time (its square root or a slightly higher power) should
improve estimates of true ground deformation. Most all the
recordsdiscussedhere indicate anchoringto depth as the solution
to the problem of monument instability. The superior performance of class A geodeticmarks at PFO testifiesto this. Their
design,rod marksisolatedfrom the surfaceandextendingto 10-m
depths [Floyd, 1974], demonstratesthe geodetic community's
long-standingrecognitionof the needfor deepanchoring.As the
weatheringof surfacerocks may extendbeyondeven this depth,
still deeperanchoringmay be neededfor observationalprograms
seekingvery high accuracyover short periods and short spatial
distances.
MPa,andthisimplieskn(z) = 1.0x 1040'n(2), in goodagreement Finally, vertical strain measurementsnear the ground surface
are not likely to yield useful estimatesof tectonic strain. The
recordsfrom the near surfaceare quite noisy, being at least an
order of magnitudenoisier than the equivalenthorizontalstrain,
(assuming
thejointsto be nearlyhorizontalandwith evenspacing
while below the water table anothermechanismcomesinto play:
S ), we needonly sum the contributions
of the individualjoints.
horizontaljointing permitschangesin the water pressureto induce
Equation(3), basedon the definitionsfor the vertical modulusand
large vertical strains [Evans and Wyatt, 1984]. Also, the
normal joint stiffness,gives the general expressionfor this
inherently low value of Poisson'sratio at low confining stress
betweendepthsof z • and z 2:
meansthat horizontalstrainscouple only weakly into vertical
strains.
_p
z2/s
1
]-•
z2-z• > S
(3)
Ez
I=z•/S
+I
withthevalueof • above.
To determine
the effective
vertical
modulus
of a column
L
-- =(z2 z•)[ • •n(lS)
This is most easily applied for jointing above the water table
whereit is assumedthatCrn(Z)=pgz. From the surfaceto some
depthz the vertical modulusis then
z/S 1
-e/Sz=•Spgz
/=1
'
with the upperfew cracksdominatingthe result. (Note that the
infiniteharmonic
series•/o__•
1/l is divergent.)
For • = 104m-1,
S = 2 m, and z = 26 m, this equationgives-P/ez = 4.2 GPa and
E = 3.7 GPa (whenv = 0.1). Thisresultis essentiallythe sameas
the value of E determinedby rainfall loading, lending some
Acknowledgments.
This researchwas supportedby grantsfrom the
NationalScienceFoundationand the U.S. GeologicalSurvey. I wish to
thankArthurSylvesterfor the manyyearsof preciselevelingdata,Roger
BilhamandJohnBeavanfor the useof observations
from theirlongbase
tiltmeter at the site, and William Kaula, Ross Stein, and the National Geo-
deticSurveyfor the installationandmonitoringof the classA benchmark
cluster.Leo Weuve sawto the integrityof all the field records,for whichI
am indebted.Keith Evans,in additionto his mostthoroughreviewof the
initial manuscript,
introducedme to the subjectof joint stiffnessin rock,
andsodeservescreditfor severalof the ideaspresented
here. DuncanCarr
Agnew,JohnLangbein,and RossSteineachprovidedparticularlyvaluable comments on earlier versions of this article.
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1664
WYA'IT: DISPLACEMENT OF SURFACEMONUMENTS: VERTICAL MOTION
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