Estimation for partially observed Markov processes

Many stochastic process models for environmental data sets assume a process of relatively simple structure which is in some sense partially observed. That is, there is an underlying process (X_n, n 0) or (X_t, t 0) for which the parameters are of interest and physically meaningful, and an observable process (Y_n, n 0) or (Y_t, t 0) which depends on the X process but not otherwise on those parameters. Examples are wide ranging: the Y process may be the X process with missing observations; the Y process may be the X process observed with a noise component; the X process might constitute a random environment for the Y process, as with hidden Markov models; the Y process might be a lower dimensional function or reduction of the X process. In principle, maximum likelihood estimation for the X process parameters can be carried out by some form of the EM algorithm applied to the Y process data. In the paper we review some current methods for exact and approximate maximum likelihood estimation. We illustrate some of the issues by considering how to estimate the parameters of a stochastic Nash cascade model for runoff. In the case of k reservoirs, the outputs of these reservoirs form a k dimensional vector Markov process, of which only the kth coordinate process is observed, usually at a discrete sample of time points.     

Geographical and vertical variation of the earth's seismicity

Information concerning a total number of 13700 instrumentally recorded earthquakes is used to study the geographical and the vertical distribution of the Earth's seismicity. From these earthquakes, which form four complete samples of data (M 7.0, 1894–1992; M 6.5, 1930–1992; M 6.0, 1953–1992; M 5.5, 1966–1992), 11511 are shallow (h 60 km), 2085 are of intermediate focal depth (61 h 300 km) and 564 are deep focus earthquakes (301 h 720 km). The parameters a and b of the frequency-magnitude relationship were calculated in a grid of equally spaced points at 1° by using the data of earthquakes located inside circles centered at each point. The radius of the circles increased from 30 km with a step of 10 km until the information for the earthquakes located inside the circle fulfil three criteria which concern the size of the sample used to compute these parameters at each point of the grid. The results are given in a qualitative way (epicenter maps) as well as in a quantitative way (mean return periods).     

The conditional probability of earthquake occurrence and the next large earthquake in Tokyo, Japan

The use of probability distribution ofrecurrence times as described by theexponential, Weibull and Rayleihgprobability densities form the core of theprobabilistic seismic prediction analysispresented in this paper. Using these threeprobabilistic models we derive threeformulas to calculate the conditionalprobability P(t|t) than an earthquakeevent will occur in the time interval (t, t+ t), provide that it has not occurredin the elapsed time t since the last largeearthquake (M 6.4) in the Tokyo area.This paper proposes a new method toestimate the time interval t foroccurrence of a new large earthquake inTokyo area. This time interval is measuredafter the elapsed time (t) since the lastlarge earthquake. To do this we use thethree formulas for the conditionalprobability P(t|t) and the criterionof the maximum conditional probability ofearthquake occurrence.Using a list of historical earthquakeswhich have occurred in the Tokyo area asgiven by Usami (1976, pp. 235–243), wefound that: (1) Using the exponentialmodel, it is estimated that a highlydamaging earthquake magnitude M 6.4, mayoccur before the year 2009.50, orequivalently before June 2009; (2) Usingthe Weibull model, it is estimated that thedamaging earthquake (M 6.4) may occurbefore the year 2129.80, or equivalentlybefore October 2129.     

Experimental investigation of turbulent processes in a complex terrain

In the present paper the first results of the international KOPEX-86 experiment are presented. The experiment took place at the Kopisty Atmospheric Observatory of the Institute of Physics of the Atmosphere in Prague as part of a special project of the Commission of the Academy of Sciences in Planetary Geophysics (KAPG) in June and July 1986. Using 4 ultrasonic anemometers at 4 levels up to 80 m, Doppler-SODAR and wind-, temperature- and radiation balance gradients from 2 to 80 m, a complex investigation of the atmospheric boundary layer in an industrial area was made. The authors present the first results of

Secular variations in the curvature of equipotential surfaces and directions of verticals due to geodynamic phenomena

Summary Formulae and numerical estimates are given for the non-periodical variations in the curvature of equipotential surfaces, horizontal forces and directions of the vertical, caused by the decrease in the second zonal geopotential harmonic, by the decrease in the angular velocity of the Earth's rotation and by secular polar motion.

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am ¶rt; u ua a nu¶rt;uu uu uu nm, maua u u u ma, a u m aauunmuaa, u mu au u u ¶rt;uu n.

     

Anomaly of geomagnetic variations in the southeastern margin of the Bohemian Massif

Summary Magnetic variations were recorded along three profiles crossing the southeastern margin of the Bohemian Massif. The data were processed in order to get induction vectors (Wiesevectors) and in-phase and out-of-phase induction vectors (Schmucker-vectors). Several events of field variations were separated into external and internal parts. The same events were also treated by a physical-statistical approach. Taking into account these results, we were able to delineate a zone of electrical inhomogeneity. It is in close relation to the Moravo-Silesian lineament. The depth of the internal anomalous field source was estimated at 20 to 25 km.

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¶rt;a u uuau n uu n nu, nu — u. u¶rt;uu ( u), n u u n¶rt; u n u¶rt;uu ( ), n ¶rt;u n uau u u u n au ¶rt; uu-uuu ¶rt; u ¶rt;¶rt;u na¶rt; nu -uu . ¶rt; au uu a n a u n¶rt; 20–25 .

     

A note on secular variation of gravity due to non-tidal geodynamic phenomena

Summary Secular non-tidal variations of geopotential and gravity are estimated due to secular decrease of the second zonal geopotential harmonic, secular polar motion and deceleration of the Earth's rotation.

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am a nuu uunmuaa u u u mmu, a u m aauunmuaa, ¶rt;uu n u u mu au u.

     

Mechanics of seismic instabilities induced by the recovery of hydrocarbons

We review earthquake distributions associated with hydrocarbon fields in the context of pore pressure diffusion models, poroelastic stress transfer and isostasy theory. These three mechanisms trigger or induce seismic instabilities at both local scale (D5 km) and at regional scale (D20 km). The modeled changes in stress are small (1 MPa), whatever the tectonic setting. Each mechanism corresponds to different production processes. (1) Local hydraulic fracturing due to fluid injection induces seismic-slip on cracks (M _L3) within the injected reservoir through decreasing the effective stress. (2) Pure fluid withdrawal causes pore pressure to decrease within the reservoir. It triggers adjustments of the geological structure to perturbations related to the reservoir response to depletion. Poroelastic mechanisms transfer this stress change from the reservoir to the surrounding levels whereM _L5 seismic instabilities occur either above or below the reservoir. (3) Massive hydrocarbon recovery induces crustal readjustments due to the removal of load from the upper crust. It can induce larger earthquakes (M _L6) at greater distance from the hydrocarbon fields than the two other mechanisms.Due to the mechanical properties of the shallow rock matrices involved, seismic slip triggered either by mechanism (1) or (2), is a second-order process of the main elastoplastic deformation. for a minimum of 80% of commercially productive basins, most of the local deformation is reported as aseismic, i.e., there is no evidence forM _L3 earthquakes. Nevertheless, the induced stresses vary as a function of time in a manner that depends on the hydraulic diffusivity (i.e., permeability) of the reservoir and surrounding rocks. Because small earthquakes (M _L3) indicate changes in stress and pore pressure, monitoring of seismicity is a means of assessingin situ reservoir behavior.The less constrained seismic response to hydrocarbon recovery is the possible connection between local fluid manipulations, triggered earthquakes and major regional earthquakes. Positive feedback mechanisms suggest that the region of seismic hazard changes is much larger than the area where hydrocarbons are extracted. These observations and models testify that fluid movement and pore pressure changes (increase or decrease) play important roles in the mechanics of earthquakes and in the triggering of natural earthquakes.     

Source spectra and RMS acceleration of Mexican subduction zone earthquakes

Strong motion (SM) data of six Mexican subduction zone earthquakes (6.4M _S8.1) recorded near the epicentral zone are analyzed to estimate their far-field source acceleration spectra at higher frequencies (f0.3 Hz). Apart from the usual corrections such as geometrical spreading (1/R), average radiation pattern (0.6), free surface amplification (a factor of 2), and equal partitioning of the energy into two orthogonal horizontal components (a factor of 1/ ), the observed spectra are corrected for a frequency dependentQ(Q=100f), a site dependent filter (e ^–kf ), and amplification ofS waves near the surface (a factor of about 2 atf2Hz). We takeR as the average distance from the rupture area to the site. If we model the high frequency plateau (f1 Hz) of the source spectra, by a point source ^–2-model, and interpret them in terms of Brune's model we obtain between 50 and 100 bars for all earthquakes. The low-frequency broadband teleseismicP wave spectra, corrected witht ^*=1.0 s, agrees within a factor of two with SM source spectra near 1 Hz. The ^–2-model is inadequate to explain the observed source spectra in a broad frequency range; these resemble spectra given byGusev (1983) with some differences.SM source acceleration spectra require significant corrections to explain observed spectra and RMS acceleration (a_rms) (a) at farther coastal sites for extended sources due to directivity effect and (b) at inland sites (100R200 km) because of unaccounted path and site amplification and/or invalidity of body-wave approximation. The observed spectra and a_rms at these sites are significantly greater than the predicted values from the estimated source spectra.     

Seismic velocity-ratio in the crust and uppermost mantle in southeast Wellington province,New Zealand

The regional variation of the seismic velocity-ratio () over a 200 km long traverse has been studied by means of microearthquake surveys. The Wadati-plot method is used with a minimum of four P and S arrivals for each of 49 earthquakes. The area as a whole is found to be characterized by a value of 1.74–1.76 for earthquakes of depth 12–40 km, except in a 50 km long section near Wellington, where is low at 1.60. This low has been attributed to the fault zones in the region. A small change of is observed between the upper crust (5 km) and lower crust (12 km), but there is no change of between the lower crust and uppermost mantle.     

On the frequency distribution of interoccurrence times of earthquakes

The distributions of discrete frequency, N, versus interoccurrence time, t (in days), of M 7 earthquakes in the Taiwan region during the 1900–1994 period, M 6 earthquakes in the north-south seismic belt of China during the 1900–1990 period, and M 5.5 earthquakes in Southern California, U.S.A., during the 1914–1995 period are studied through two statistical models (gamma function and exponential function). Results show that both the exponential function and gamma function can describe the distributions. However, the former is more appropriate than the latter. This indicates that the three time series of earthquakes have a significant component of Poisson processes, even though the tectonic conditions, the fault distributions and the size of the three seismic regions are different.     

Seismicity and seismic hazard mapping of northern Algeria: Map of Maximum Calculated Intensities (MCI)

An earthquake catalogue covering the period1716–2000, comprising 2430 events, has beencompiled for the region lying between3°W-9°E and 31°N-38°N. It results fromraw data of IGN, ISC, USGS and Algeriansources, enabling an input consisting oforigin time H, geographical coordinates(longitude and latitude) and at least one of thefollowing parameters: surface wavemagnitude Ms, body wave magnitude Mb,epicentral intensities Io. Empiricalrelations permit transformations of Mb andMs into Io. The output consists in H,, , Ms, Mb, Io, and focal depth h whichis fixed to 10 km. The number ofevents falls to 1458 characterised by Ms 3.3 and Mb 3.6, or Io III. The fixed depth is suggested by thebest documented Algerian macroseismic mapsthat also lead to an empirical intensityattenuation law. A first application ofthis catalogue allows the drawing up of anupdated Seismicity and a MaximalCalculated Intensities (MCI) Maps ofAlgeria. The MCI map is obtained by usingthe empirical attenuation law: theintensities inferred by the whole eventsconstituting the catalogue are computed atnodes of a 5×5-km grid covering the area ofstudy. The corresponding maximum value isassigned to each node. The MCI map producedthat way gives precise spatial informationin comparison with Maximum ObservedIntensities (MOI) maps obtained in previousmacroseismic studies. This document may beuseful in mapping the seismic hazard inNorthern Algeria, without attachingprobabilities to ground-motionparameters.     

Effects of collision-induced breakup on drop size distributions in steady state rainshafts

New equations and techniques for dealing with drop breakups are developed and applied to the modelling of the evolution of raindrop spectra in rainshafts. Breakup experiments byMcTaggart-Cowan andList (1975) served as data base.No matter what the original size distribution, the spectrum evolution will always lead to a Marshall-Palmer type equilibrium di tributionN=N ₀e^–D, with =constant andN ₀ proportional to the rainfall rateR. (D stands for raindrop diameter.) ForR29 mm h^–1 and an original Marshall-Palmer distribution, the required fall height to reach equilibrium is 2 km.The equilibrium distributions are characterized by linear relationships betweenR, the radar reflectivity factorZ, the liquid water content LWC and theN ₀ of the Marshall-Palmer distribution. Possible explanations for the discrepancy with observations are given.The fact that the all-water processes cannot produce drops withD2.5 mm (as confirmed by observations) leads to the conclusion that observed large raindrops withD5 mm represent melted hailstones and have not yet reached an equilibrium distribution. These latter conclusions were reached within the original assumption of videspread, steady state precipitation.     

Propagation of surface waves on the surface of a non-homogeneous aeolotropic cylindrical shell

Summary The object of the present paper is to investigate the propagation of surface waves on a non-homogeneous aeolotropic cylindrical shell surrounded by vacuum. The elastic constantsc _ij (i, j=1,2...) and density of the material of the shell are assumed to be of the form and respectively, where _ij, ₀ are constants andk ₁,k ₂ are any integers.     

Determination of the geopotential scale factor from satellite altimetry

Summary The geopotential scale factor R ₀ =GM/W ₀ has been determined on the basis of satellite altimetry as R _{0=(6 363 672·5±0·3) m} and/or the geopotential value on the geoid W ₀ =(62 636 256·5±3) m ² s ^–2 . It has been stated that R ₀ and/or W ₀ is independent of the tidal distortion of surface W=W ₀ due to the zero frequency tide.

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¶rt;a nmu amumuu u ama amnmuaa R ₀ =GM/W ₀ =(6 363 672,5±0,3) m u/uu aunmuaa a nmuu¶rt;a W ₀ =(62 636 256,5±3) m² s^–2. m, m R ₀ u/uu W ₀ auum m nm amu a a nuu ¶rt;au nmu W=W ₀ .

     

Molecular refraction in the Earth's mantle

The Drude law (molecular refraction) for the temperature radiation in a monoatomic model of the Earth's mantle is derived. The considerations are based on the Lorentz electron theory of solids. The characteristic frequency (or eigenfrequency) of independent electron oscillators (in energy units, ) is identified with the band gapE _G of a solid. The only assumption is that solid material related to the Earth's mantle has the mean atomic weight A21 g/mole, and its energy gap (E _G) is about 9 eV. In this case the value of molecular refraction (in cm³/g) is (n ²–1)/=0.5160.52, where andn are the density and the refractive index at wavelength _D=0.5893 m (sodium light), respectively. The average molecular refraction of important silicate and oxide minerals with A21, obtained byAnderson andSchreiber (1965) from laboratory data, is , where denotes the mean arithmetic value calculated from three principal refractive indices of crystal. For the rock-forming minerals with 19A<24 g/mole the new relation was found byAnderson (1975).     

Radioactivity and geochemistry of granulites and some related rocks from the saxonian granulite complex

Summary The distribution of radioactive(Th, U, K), major and selected trace(Rb, Sr, Ba, Y, Zr, V, Cr, Ni) elements of granulites from the Saxonian Granulite Complex was studied. Similarly to the South Bohemian granulites, the Saxonian granulites can be divided according to the contents of their major and trace elements into two main groups, groupA containing mostly acid and subacid granulites (K ₂ O>2.5%, SiO ₂ >68%), and groupB containing mostly intermediate and basic granulites (K ₂ O<2.5%, SiO ₂ <68%). Statistically significant differences between groupsA andB were found for all major oxides and several trace elements(Rb, V, Cr, Ni). The Saxonian granulites follow the same calc-alkaline trend as the South Bohemian, granulitesA being placed mostly in the rhyolite field and granulitesB mostly in the dacite, andesite and basalt fields of this trend. The investigated granulites are characterized by a considerable scatter ofTh andU contents accompanied by very variableTh/U ratios; theTh andU concentrations of granulitesA are substantially lower than is usual for rocks of corresponding acidity.

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¶rt;a an¶rt;u a¶rt;uamu(Th, U, K) u ua ¶rt;u(Rb, Sr, Ba, Y, Zr, V, Cr, Ni) m aum n¶rt;a aaum na. naa, m u¶rt;aum n uu aam n aaum u ¶rt;u am aua, u u uu. aum n u uu ma a¶rt;um ¶rt; ¶rt;nn; nnA nua¶rt;ama a au¶rt; u au¶rt;aum (K ₂ O>2,5%, Si O ₂ >68%), nnB ¶rt;u u aum (K ₂ O<2,5%, SiO ₂ <68%). ¶rt; muunnau mm mamumuu m au ¶rt; a u u ¶rt; m ¶rt;u m(Rb, V, Cr, Ni). auaum n¶rt;¶rt;m um- m¶rt; a u -uaum;aumA a¶rt;ma a uum n, uaumB a a ¶rt;aum, a¶rt;um u aam n m m¶rt;a. ¶rt;aum — u unnA — aamum uu ¶rt;au da¶rt;uamu mTh uU.

     

Triggered earthquakes as stress gauge: Implication for the uppercrust behavior in the Grenoble area,France

A sequence of moderate shallow earthquakes (3.5M _L5.3) was located within the Vercors massif (France) in the period 1961–1984. This subalpine massif has been a low seismic area for at least 5 centuries. During the period 1962–1963, 12 shallow earthquakes occurred in the neighborhood (10 km) of the Monteynard reservoir, 30 km south of the city of Grenoble. The latest fourM _L4.0 earthquakes occurred in 1979–1984 either at larger distance (35 km) or greater depth (10 km) from the reservoir. Two triggering mechanisms are suggested for this sequence: (i) the direct effect of elastic loading through either increased shear stress or strength reducing by increased pore pressure at depth; (ii) the pore pressure diffusion induced by poroelastic stress change due to the reservoir filling.The weekly water levels, local balanced geological cross sections, and focal mechanisms argue for two types of mechanical connection between the earthquake sequence and the filling cycles of the Monteynard reservoir. The seismic sequence started with the 1962–1963 shallow earthquakes that occurred during the first filling of the reservoir and are typical of the direct effect of elastic loading. The 1979 deeper earthquake is located at a 10 km depth below the reservoir. This event occurred 16 years after the initial reservoir impoundment, but one month after the previous 1963 maximum water level was exceeded. Moreover the yearly reservoir level increased gradually in the period 1962–1979 and has decreased since 1980. Accordingly we suggest that the gradual diffusion of water from reservoir to hypocentral depths decreases the strength of the rock matrices through increased pore pressure. The transition between the two types of seismic response is supported by the analysis ofM _L3.5 earthquakes which all occurred in the period 1964–1971, ranging between 10 and 30 km distance from the reservoir. The three other delayed earthquakes of the 1961–1984 seismic sequence (M _L4 during the 1979–1984 period) are all located 35 km away from the reservoir. Based on the seismic activity, the estimates for the hydraulic diffusivities range between 0.2–10 m²/s, except for the first event that occurred 30 km north of the reservoir, the filling just started. The lack ofin situ measurements of crustal hydrological properties in the area, shared by most of the Reservoir-Induced-Seismicity cases, prevents us from obtaining absolute evidence for the triggering processes. These observations and conceptual models attest that previous recurrence times for moderate natural shocks (4.5M _L5.5) estimated within this area using historical data, could be modified by 0.1–1 MPa stress changes. These small changes in deviatoric stress suggest that the upper crust is in this area nearly everywhere at a state of stress near failure. Although the paucity of both number and size of earthquakes in the French subalpine massif shows that aseismic displacements prevail, our study demonstrates that triggered earthquakes are important tools for assessing local seismic risk through mapping fault zones and identifying their possible seismic behavior.     

The four primary geodetic parameters

Summary The four primary geodetic parameters defining the geodetic reference system are discussed from the point of view of their physical meaning and current estimation of their actual accuracy. The geopotential scale factor has been treated as the primary geodetic parameter defining the Earth's dimensions.

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¶rt;am m nu¶rt;uu naama, n¶rt;u¶rt;u um mumu, mu u uu a u mmu. ama amnmuaa ¶rt;am am nu¶rt;u naama, n¶rt; a u.

     

Characteristics of the mid-latitude ionosphere suitable for modelling ionospheric filtration of ULF waves

Summary An approximate method of one-dimensional modelling of the plasma of the Earth's ionosphere is demonstrated for purposes of studying the ionospheric filtration of ULF waves (micropulsations). Apart from the basic local parameters, characterizing the plasma, also derived local characteristics have been defined, i.e. the mass of the so-called effective ion and its effective collision frequency . Drawing on existing empirical models of the mid-latitude ionosphere, vertical profiles (50 km h 1000 km) were determined of the characteristics N_e N_i, v_e, and for the daytime and nighttime mid-latitude ionosphere under low and enhanced solar activity, which can be used to study the ionospheric ULF filter.

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aa nu uum m¶rt; ¶rt;a ¶rt;uau na ¶rt; u ¶rt; nmm uu u umauu (unau). ¶rt; u mu naamau na n¶rt; mu m aamumuu — aa m. a. mu ua, , u mua amma mu, . a mu nuuu ¶rt; ¶rt;um u u n¶rt; mua nuu (50 h 1000 ) aamumu N_e N_i, v_e, u ¶rt; u u ¶rt;u um nu u u amumu. ¶rt;u nam nu uuu u uma.