Secular Tilt Research Articles

Orbital dynamics of two circumbinary planets around misaligned eccentric binaries

ABSTRACT We investigate the orbital dynamics of circumbinary planetary systems with two planets around a circular or eccentric orbit binary. The orbits of the two planets are initially circular and coplanar to each other, but misaligned with respect to the binary orbital plane. The binary–planet and planet–planet interactions result in complex planet tilt oscillations. We use analytical models and numerical simulations to explore the effects of various values of the planet semimajor axes, binary eccentricity, and initial inclination. Around a circular orbit binary, secular tilt oscillations are driven by planet–planet interactions and are periodic. In that case, planets undergo mutual libration if close together and circulation if far apart with an abrupt transition at a critical separation. Around an eccentric orbit binary, secular tilt oscillations are driven by both planet–planet interactions and binary–planet interactions. These oscillations generally display more than one frequency and are generally not periodic. The transition from mutual planet libration to circulation is not sharp and there is a range of separations for which the planets are on orbits that are sometimes mutually librating and sometimes circulating. In addition, at certain separations, there are resonances for which tilt oscillations are complicated but periodic. For planets that are highly misaligned with respect to an eccentric orbit binary, there are stationary (non-oscillating) tilt configurations that are generalizations of polar configurations for the single planet case. Tilt oscillations of highly inclined planets occur for initial tilts that depart from the stationary configuration.

Tilt monitoring at wairakei geothermal field

The withdrawal of fluids from the Wairakei geothermal field has induced substantial deformation of the ground surface, with subsidence rates as great as 400 mm/yr. Substantial changes to pressures within the upper steam zone of the reservoir would be expected to be recognised in corresponding changes in the deformation rate and in particular in the tilting. Two types of tiltmeter have been tested for this purpose at Wairakei geothermal field. A 5 month test of borehole tiltmeters showed short duration tilt differences as great as 35 μrad (microradians) between meters 5 m apart. This effect is believed to be caused by differences in local response to regional subsidence. Over 5 months of testing the tilt rates from two borehole meters were in good agreement with surface tilt rates derived from a local levelling network (520 μrad/yr). The short-term test of a two-component bubble tiltmeter mounted on a horizontal concrete block gave results that were dominated by a large amplitude (150 μrad) diurnal tilt variation attributed to the response of the block to air temperature variations. However the repeatability of the diurnal response suggests that the tilt rate could be determined from as little as 3 days' observations provided temperatures of the site and instrument were also monitored. In the absence of site temperature data, daily temperature extremes (measured at about 1 km from the site) were used to eliminate the temperature-dependent response. The resulting secular tilt rate (measured over 36 days) of 950 ± 130 μrad/yr, is larger than the long-term estimate determined from precise levelling surveys.

Tilt observations using borehole tiltmeters: 1. Analysis of tidal and secular tilt

We have designed a borehole tiltmeter using two horizontal pendulums which have periods of 1 s. We have installed the instruments at seven sites in Colorado and Wyoming to evaluate the secular tilt, the tides, and the coherence between nearby instruments. Using 28 days of data from Boulder, Colorado, the standard deviations of the tidal estimates are 5% for the semidiurnal component M2 and 9% for the diurnal component O1. The estimates agree with models that include the body tide, the ocean load, and the topographic correction to better than the estimated uncertainty. Tidal measurements at Erie, Colorado, have larger, possibly nonrandom variability that may be caused by a coupling between the tides and long‐period tilts. The coherence between nearby instruments is ≳0.5 for frequencies ranging from 0.5 to 2 cycles per day, but drops to a much smaller value outside of this range. The secular tilt rate is quite variable and ranges from 0.1 to 1 μrad/yr. Measurements at Erie, Colorado, and in Yellowstone National Park, Wyoming, exhibit an annual or biannual periodicity with an amplitude of about 1 μrad, which is consistent with a 1/f2 extrapolation of the diurnal noise power to longer periods.

Twelve years of geophysical research in the Charlevoix seismic zone

Various and extensive geophysical monitoring began in 1974 in the Charlevoix Seismic Zone (CSZ). The historical record and a major earthquake in 1925 characterize this zone as the likely site of a future damaging earthquake (i.e. M = 5.5 or larger) and the aim of the monitoring was to develop the capability of predicting such earthquakes. The parameters studied included: microseismicity, seismic travel times, electrical impedance, vertical movement, horizontal movement, tilt, gravity change, and strain through water well level changes. Although no clear precursor was detected in the months before the largest seismic event that occurred during the observation period ( M = 5.0), the multiparameter experiment provided new insights into the structure and the mechanics of this active region. Observations of several hundred well-located microearthquakes have permitted the delineation of ancient rift faults that are now being activated by high horizontal compression. The geographical extent of the zone is sharply defined by the microseismicity which occurs in an area of 35 km × 80 km straddling the St. Lawrence River. Observation of P-waves from accurately-timed explosions and S-waves from earthquakes support the hypothesis of Extensive Dilatancy Anisotropy (EDA) involving saturated parallel cracks in the crust. The data have permitted an estimate of both crack density and the direction for the principal compressive stress in the region (N 30° E). Changes in electrical parameters with periods of months to years were detected at five of the six magnetotelluric stations installed in the region. Changes in the polarization angle of the earth's electrical (telluric) field correlated well (for seven years) with calculated groundwater accumulation levels at one site. Electric impedance changes showed no clear correlation from station to station, but changes at one station appear to correlate with a seismic P-wave travel time change anomaly and with the disappearance of a negative gravity anomaly at the time of an unusual number of earthquakes in the range M = 2 to 3 from mid 1976 to mid 1977. It is suggested that these correlated changes could be related to aseismic stress redistribution and fluid flow along one of the NE striking rift faults about two years before the M = 5.0 earthquake in 1979. Continuous monitoring of tilt (surface and borehole) and well-water levels was carried out at a single site (Charlevoix Geophysical Observatory). Although there were unexplained near-surface tilts of the order of the earth tide amplitude within two or three days before some earthquakes, they were either not spatially coherent or were related to small unexplained temperature changes in the tiltmeter vault. No intermediate-term (one week to one year) precursors were seen against a noise level of 1–3 μrad (RMS) for near-surface tilt observations and a noise level of 0.2–0.3 μrad (RMS) for borehole tilt measurements at a depth of 50–100 m. The threshold of detectable tidal tilt admittance changes for the borehole tiltmeters was lowered to 2% but no significant changes were seen. Four coseismic changes in the levels of water in boreholes were observed (all were drops in water level) which were much larger than expected on the basis of elastic dislocation theory. It is suggested that these could be explained either by the collapse of a dilatant zone around the respective hypocentres or by triggered aseisnuc movement on a nearby active fault. Borehole water level tides observed from 1980 to 1986 exhibit significant variations in amplitude and phase which are thought to be the result of time-varying elastic properties in a nearby structure. Some constraints can now be placed on the long-term deformation of the CSZ as a whole from tiltmeter, levelling and horizontal surveys. Data from a levelling array observed from 1976 to 1986 puts a limit of 0.54 μ rad/yr on the secular tilt rate at the observatory: a further limit of 0.1 μ rad/yr is provided by borehole tilt measurements over the period 1983–1986. A limit of 2 cm is placed on vertical movements northwest of the St. Lawrence River by levelling surveys in 1965, 1978, 1980 and 1982. No significant (90% confidence) deviatoric strain has been detected across the CSZ from horizontal control surveys carried out in 1912–1919, 1965, 1978 and 1983: the maximum allowable long-term strain rate across the CSZ from 1919 to 1983 is 0.16 μ strain/yr.

Tidal and secular tilt from an earthquake zone: thresholds for detection of regional anomalies

Tiltmeter data from an array of three boreholes at the Charlevoix observatory in the Charlevoix seismic region of Que´bec have been analysed for evidence of tectonically related signals. The secular tilt is dominated by water table induced effects which can be substantially removed by linear regression of the water level on the tilt. Short-term (days to months) anomalies are shown to be detectable at the 0.3 μrad to 1 μrad level depending on the depth of the measurement. Long-term changes in the linear drift as small as 0.1 μrad/yr would be detectable in all of the boreholes. Large spatial anomalies in the mean tidal admittance among boreholes preclude its use in refining either models of the regional crustal structure or the adjacent marine tide distribution. Strongly coherent time variations in the tidal admittance among the observations of all the major tidal constituents are shown to be generated by corresponding variations in marine tidal loading in the St. Lawrence estuary. Diurnal band variations are closely correlated with the tide gauge data. The semi-diurnal constituents show a weaker correlation because of the complex spatial pattern within the estuary of time variations in this band. Inspection of the residual admittance variations for the M 2 and O 1 constituents demonstrates that the thresholds for detecting tectonic tidal tilt anomalies are ±2% and ±5–8%, respectively. The level of earthquake activity in the Charlevoix seismic zone throughout the period of the borehole tiltmeter experiment was sufficiently low that no significant tilt anomalies were expected or were undeniably detected.

Topographic effect on tilt, strain, and displacement measurements

Deformation and stress fields are locally modified by the presense of topography. An approximate analytical method is used to give simple expressions for the effect of arbitrarily shaped topography on vertical, horizontal, and surface tilt measurements. This provides a means of making corrections to observations without the use of finite element techniques. Corrections to tilt fields are found to be directly proportional to the local slope of the topography. Comparison is made between expressions for tilt and those for strain and displacement. Although the topographic effects on secular tilt, strain, and vertical displacement are not yet measurable, we emphasize that the topographic effect on precision tidal tilt and strain measurements can be significant.

Borehole tilt measurements from Charlevoix, Québec

An array of three borehole tiltmeters near Québec City in eastern Canada is designed to study the tidal and secular response of the crust in the Charlevoix seismic zone. The objectives of this study of the first year of data from two boreholes of the array are to investigate the spatial coherency of the tidal observations and determine whether there are time variations in the tidal amplitudes and phases and to describe the main features of the secular tilt signal. The tidal analysis was done using a modified version of the HYCON harmonic analysis program with which the time‐varying tidal amplitudes and phases were determined by the sequential analysis of overlapping 2‐monthly subsets of the data. The admittance observed for the major semidiurnal (M2) and diurnal (O1) constituents varies by up to 10 and 30%, respectively, and is strongly correlated between boreholes. Comparison with admittance variations determined from two nearby tide gauges indicates a strong correlation in the amplitude fluctuations, pointing to a predominantly marine loading source for the time‐varying tilt admittance. Differences of up to 20% in amplitude and 5° in phase were found between the mean M2 results determined from boreholes 1 and 2, located only 80 m apart, indicating small‐scale distortion of the local tilt field by lateral inhomogeneities. The secular tilt from both boreholes correlates strongly with transient and seasonal water table fluctuations, suggesting the dominant influence of pore pressure effects on the nontidal tilt. A preliminary estimate of the detectability of long‐term regional trends in tilt is 0.4 μrad/yr.

Investigations of tilt measurements using shallow borehole tiltmeters

An array of shallow borehole tiltmeters has been operated at Piñon Flat Observatory since early in 1977. The data from this array are examined for coherence between the individual instruments and compared with the corresponding data from three 732‐m laser strainmeters. In general, there is no significant coherence between the signals from the tiltmeters outside the microseismic and tidal bands even though they are spaced as closely as 10 m apart. Comparisons with the strain records show that the observed tilt noise power exceeds the strain noise power by 25–40 dB over the band from 10−6 to 1 Hz. Analysis of the coherence estimates establishes the necessity for an unacceptably large (≃1000) array of instruments in order to determine the common tectonic signals. The observed secular tilt rates are from 40 to 450 times the observed secular strain rates for the same period. The theoretical noise power limit of the tilt transducer renders the instrument incapable of recording background noise at a quiet site over the band from 10−4 to 10−1 Hz.

Interrelations between secular sea level variations and subsidence of coastal areas in the North sea

The effect of secular sea level variations on coastal levelling nets was considered for some parts of the North West European Lowlands levelling. On neglecting local geological influences a simplified global model was used for treating the following problems: (1) Magnitude of secular tilt effects in NWELL due to possible sea level variations; (2) Levelling errors due to sea tide loadings.

On tilting motion of ground observed before and after the occurrence of an earthquake

A large secular tilting motion of the ground was observed with a tiltmeter at a station in the epicentral region three months before the occurrence of a destructive earthquake. After examination of the reliability of observation,the secular ground tilt of this kind is considered to be intimately related to the earthquake occurrence and may be useful for prediction of a destructive earthquake and mitigation of disaster.