Research On The Three-Dimensional Manifold Method And Its Preliminary Application To Geosciences

In the domain of earthquake research, the issues on stress and the strain fieldsplay an important role in the processes of earthquake preparation, occurrence andadjustment after the event. The space observation technologies, such as GPS, haveprovided efficient, stable and accurate results for large-scale crustal movementmonitoring, which can be used as effective data constraints for studying the aboveissues. Because both the GPS velocity field and strain field can be used to analyzecrustal deformation, and used as the valid constraint in numerical simulation, thecomparison research on the GPS strain rate computing methods is necessary. Inaddition, considering the dynamic evolution of stress and strain fields in theearthquake process, it is needed to study the three-dimensional Numeric ManifoldMethod （3d-NMM） which can realize dynamic constraint, large deformationsimulation and coupled simulation of continuous and discontinuous deformation.Based on the above analysis, there are three aspects in this thesis.1. I discuss the calculation precision and robustness of the GPS strain rate fieldsin continuous mode through simulation and actual data analysis. In this process, Idevelop the least-squares collocation method on spherical surface which can reflectthe actual deformation distribution, and then analyze its difference with othermethods.First, large scale （75°E¹35°E and20°N⁵0°N） analyses of1°grid samplingdata and decimated50%data by resampling （then erasing data in two5°×10°regions）reveal that the Delaunay method has poor performance and that the other threemethods show high accuracy. The correlation coefficients between theoretical resultsand calculated results obtained with different errors in input data show that the orderin terms of robustness, from good to bad, is least-squares collocation, sphericalharmonics, multi-surface function, and the Delaunay method. The influence of datasparseness on different methods shows that least-squares collocation is better thanspherical harmonics and multi-surface function when sample data are distributed froma2°grid to a1°grid.Second, analysis on a medium scale （90°E¹20°E,25°N⁴0°N） in1°⁰.5°gridsampling data reveals that the three gridded methods can meet the practicalrequirements in the intensive distribution of input data. Meanwhile, the sensitivities toerrors of these three methods present a slight difference. The least-squares collocationmethod is better than the other two methods to a slight degree. In view of theinfluence on the strain rate computation caused by data sparseness, the correlationbetween calculation results and theoretical results of multi-surface function andspherical harmonics attenuate faster than that of least-squares collocation with thedata sparseness increasing, which demonstrates that these two methods require a higher density of data distribution.Last, in view of analysis of strain rate results for the Chinese mainland during1999²004, the stability of least-squares collocation is highest. It can produce areasonable strain field for the decimated50%dataset without a significant increase ofstrain rate errors. The edge effect of spherical harmonics is influenced by thedistribution of input data, and the edge effect is notable when the input data are sparse.The distributions of the strain rate and errors calculated from multi-surface functionare influenced by the distribution of input data too. This method is not stable. Thespherical harmonics and multi-surface function methods are affected by the geometrydistribution of input data, such as distortion of results in eastern China.Therefore, the least-squares collocation method is best in terms of robustness,edge effect, error distribution, and method stability. The most important reason is thatthe covariance function of least-squares collocation is calculated from statistical dataand can therefore reflect the real distribution of GPS velocity fields, which differsfrom other methods that depend on repeated trial calculations such as the harmonicdegree of spherical harmonics and kernel function, smoothing factor, and adjustmentnode selection of multi-surface function methods. The parameter selection of theleast-squares collocation method is not affected by humans. For that reason, differentresearchers can obtain identical results to those obtained using this method if the inputdata meet the real requirements.2. I study the3d-NMM from multiple perspectives, such as the characteristics ofthree-dimension simplex integration, formula deduction of matrices’ elements andtheoretical tests of simulation results.First, based on the analysis of simplex integration formula, the integrationalgorithm for two-dimension and three-dimension written by C++language are given,and the calculation procedures of simplex integration are described with concreteexamples in this thesis. I analyze the accuracy of the simplex integration throughcomparing the theoretical results with calculated results, in which parameters includethe area （volume） and gravity centers, and the integral regions are regular but withhollows. Then, the influence of graphic conditions on the results of simplexintegration is discussed through comparing the theoretical results with calculatedresults in different edge length ratioes (10^-610⁶). Furthermore, the adaptivecharacteristics of simplex integration algorithm are discussed through analyzingintegral results in irregular integral regions with hollows. Lastly, the precision ofsimplex integration is discussed in which the integral term is high-order polynomial,and the integral regions are irregular with hollows. The results show that the relativeerror between calculated results and theoretical results is about10^-1510^-14, and thegraphics conditions have minimal impact on integral results and the impact exhibits no systematic characteristic. In conclusion, the difference between the calculatedresults and theoretical results is caused by computing error, and the results of simplexintegration have high stability and accuracy.Second, I present the detailed expression of all matrices of3d-NMM in an actualreference system, including stiff matrix, initial stress matrix, point load matrix,volume load matrix, inertia matrix, velocity matrix, point displacement matrix,contact matrix, friction matrix and mathematical constraint of displacement.Assembling all above matrices, I realize3d-NMM simulation in continuous anddiscontinuous deformation. Because of all formulas of NMM is deduced in an actualcoordinate system without isoparametric transformation, the constraint of point loadand point displacement can be applied on every point in the element. Meanwhile, Ihave developed the constraint algorithm of multi-displacement combinations whenrealizing the3d-NMM algorithm, which can realize constraint of relative movementand the solving strategies include the least-squares method and mandatory constraintsmethod.Last, I use the simulation examples including continuous and discontinuousdeformation to test the correctness of the above matrices’ formulas, including stiffmatrix, initial stress matrix, point load matrix, volume load matrix, inertia matrix,velocity matrix, point displacement matrix, contact matrix and friction matrix. Thecomparison between simulated and theoretical results of cantilever beam with16elements reveals that the algorithm has high efficiency, and I have tested the validityof stress accumulation module through this example. The example of multi-contactsurface simulates the elastic rebound phenomenon, and verifies the validity ofopen-close iteration algorithm, which illustrates that the program can realizestatic-dynamic transformed simulation in the limit equilibrium conditions.By studying the simplex integration algorithm, formulas deduction of theelements of all matrices, description of key algorithms and examples testing, thisthesis has discussed the algorithm realization and simulation results of3d-NMM, andverified the effectiveness of the program and high precision of the simulation results.3. The occurrence of the Wenchuan earthquake in2008is closely related to themovement and deformation of the Qinghai-Tibet Plateau. Because the eastwardmovement of the Bayan Har block was obstructed by the South China block, largeamounts of strain energy were accumulated in the Longmenshan fault over a longperiod of time. The research on deformation characteristics before the Wenchuanearthquake can help understand the earthquake preparation process more clearly. Inorder to analyze the earthquake preparation process, I have studied the directiondifference between the principal stress and principal strain rate, the distribution ofGPS strain rate fields on different spatial scales, and the simulation results of multi-step3d-NMM.First, I use the least-squares collocation method to process the stress azimuthdata of World Stress Map （WSM） plan and GPS velocity fields, and calculate theresults of stress azimuth and strain rate in the Chinese mainland, in which theprocedures include data preprocessing, covariance parameters estimating andcollocation calculating. In the data processing, I have tested the unbiasedness andnormality of fitting residual of the stress azimuth and GPS velocity fields, andanalyzed the error distributions of gridded results of stress azimuth and that ofprincipal compressive strain rate. Based on analysis to the precision of results, thesimilarities and differences between principal stress azimuths and principal strain rateazimuths are analyzed. The results illustrate that these two fields are unanimous incommon. One significant difference region is located in the eastern region of theQinghai-Tibet block, in which the azimuth of stress is NE direction and that of strainrate is EW direction. These characteristics show the eastern region of theQinghai-Tibet block has accumulated extrusion strain in EW direction for the pastfew years. Another significant difference region is located in the western region of theXiyu block, in which the azimuth of stress shows transformation from NW in west toNE in east, but the azimuth of principal compressive strain rate calculated from GPSvelocity shows NS direction.Second, I have used the GPS velocity field, consisting of approximately700points in western China, to research the characteristics of the GPS strain rate fieldbefore the Wenchuan8.0earthquake, for which the least-squares collocation straincalculation method is used. GPS principal strain rate shows that the compressiondeformation in SN-NNE direction and the tensile deformation in EW-NWW directionare the main features of the Qinghai-Tibet plateau. The direction of principalcompressive strain in the middle-eastern region of the Qinghai-Tibet plateau isgradually deflected to the east from the south to the north and from the west to theeast. The GPS strain rate in the EW direction in the Qinghai-Tibet block shows thatthe extensional deformation is dominant in the western region of the block （west to92°E）, while compressive deformation is dominant in the eastern region of the block（from92°E to100°E）. The above characteristics demonstrate that the eastwardmovement of the Qinghai-Tibet plateau is resisted by the North China block andSouth China block, so the strain energy is accumulated on the boundary, which helpscause the Wenchuan earthquake. Although the hypocentral region of the Wenchuanearthquake is not located at the high zone of shear strain rate, it lies at the highextrusion zone of surface strain rate and strain rate in east-west direction. Furthermore,the deformation environment in the seismogenic fault is extrusion with dextrorotation,and the scale of the extrusion deformation in its west region is greater than that in itseast region. On a fault scale, the strain accumulation is very slow, and the extrusion strain accumulation had reached its limit before the Wenchuan earthquake.Last, I use multi-step3d-NMM algorithm to study the characteristics ofextrusion stress and deformation fields. The results illustrate that the GPS velocityand strain rate fields derived from least-squares collocation and3d-NMM are in ahigh degree of consistency. These consistent characteristics verify the validity of the3d-NMM program. The principal strain rate field and the simulation results of3d-NMM reveal that the strain accumulation rate in the southern segment of theLongmenshan fault is faster than that in the middle-north segment during the periodof1999²007. Furthermore, the accumulation of the extrusion strain has reached limiton the middle-north segment of Longmenshan, but the strain can be accumulated onthe south segement.The deformation characteristics before the Wenchuan earthquake reflect that inthe late period of the seismogenic cycle the elastic deformation of the Longmenshanfault zone was clearly at its limit and in a weakened deformation state compared tothe surrounding region. The characteristics that the strain accumulation rate was fasteron the south segment of Longmenshan than that on the middle-north segment inrecent years reveal that the atresia extent on the middle-north segment was strongerthan that on south segment, which was beneficial to the earthquake occurrence. As aresult, when we study the strain rate field, we should analyze on different spatialscales, combined with tectonic deformation.In general, I pay more attention to method research in this thesis. The GPS strainrate can be used in both crustal deformation analysis and testing the correctness of3d-NMM program. The research of3d-NMM is focused on formula deduction andresult test, and I take much notice on guaranteeing the correctness of the algorithm.The simulation example presented at the end of this thesis is simple. In order toexpand the applications of3d-NMM in the wider field of geoscience, more in-depthresearch should be conducted.