Determination of Rate Constants of Catalytic Reaction by Spectroelectrochemistry with Microelectrodes

The at constants of catalytic reaction of ferrocyanide ascorbic acid and ferro cyanide histidine system were determined by transmitted spectroelectrochemistry using a group of cyclindrical microelectrodes, It is the first time to find that the reaction can still be considered as the pseudo first order reaction when tilt concentration of ascorbic acid or histidine is close to and even slightly lower than the concentration of ferrocyanide. The determined rate constants are in agreement with the reported values, A reasonable explanation was given,

SPECTROELECTROCHEMISTRY WITH CYLINDRICAL CARBON-FIBER MICROELECTRODES

Transmittance spectroelectrochemistry can be performed using a group of cylindrical microelectrodes. A dependence of absorbance on electrolytic charge during the potential step was derived. The rate constant of catalytic reaction of the ferrocyanide-ascorbic acid system was determined using single potential step-open circuit relaxation chronoabsorptometry. This is the first report that the reaction can still be considered as a pseudo-first-order reaction when the concentration of ascorbic acid is close to and even slightly lower than the concentration of ferrocyanide. The determined rate constant is in agreement with the reported value. The reason is that the diffusion of ascorbic acid toward electrode surface is contractive and the diffusion of the electrogenerated ferricyanide from the electrode surface to the bulk of solution is expansive.

A SINGULAR EXACT SOLUTION FOR NONLINEAR SHALLOW-WATER ON A ROTATING EARTH

In this paper, we present an exact solution for nonlinear shallow water on a rotating planet. It is a kind of solitary waves with always negative wave height and a celerity smaller than linear shallow water propagation speed square-root gh. In fact, it propagates with a speed equal to (1 + a/h) square-root gh(1 + a/h) where a is the negative wave height. The lowest point of the water surface is a singular point where the first order derivative has a discontinuity of the first kind. The horizontal scale of the wave has actually no connection with the water depth.

Highly nonlinear internal solitary waves over the continental shelf of the northwestern South China Sea

Large amplitude internal solitary waves (ISWs) often exhibit highly nonlinear effects and may contribute significantly to mixing and energy transporting in the ocean. We observed highly nonlinear ISWs over the continental shelf of the northwestern South China Sea (19A degrees 35'N, 112A degrees E) in May 2005 during the Wenchang Internal Wave Experiment using in-situ time series data from an array of temperature and salinity sensors, and an acoustic Doppler current profiler (ADCP). We summarized the characteristics of the ISWs and compared them with those of existing internal wave theories. Particular attention has been paid to characterizing solitons in terms of the relationship between shape and amplitude-width. Comparison between theoretical prediction and observation results shows that the high nonlinearity of these waves is better represented by the second-order extended Korteweg-de Vries (KdV) theory than the first-order KdV model. These results indicate that the northwestern South China Sea (SCS) is rich in highly nonlinear ISWs that are an indispensable part of the energy budget of the internal waves in the northern South China Sea.

Photocatalytic degradation of CI Acid Blue 80 in aqueous suspensions of titanium dioxide under sunlight

The dye C.I. Acid Blue 80 (AB80) was easily degraded by TiO2-P25 assisted photocatalysis in aqueous dispersion under irradiation of sunlight. The optimal reaction conditions were [TiO2] = 2.0 g/L, pH = 10, [H2O2] = 5 mmol/L. The photocatalytic reaction followed pseudo-first order kinetics. The adsorption of AB80 onto TiO2 was in accord with Langmuir equation.

二维流形三角网格模型顶点法矢量估计

提出一种以顶点的一邻域中三角形在该顶点处的顶角与对应三角形的面积比值加权三角面法矢量估计二维流形三角网格模型顶点法矢量的方法.回顾了现有的五种顶点法矢量估计方法,然后给出了新的方法.设计了利用理论法矢量与估计法矢量的夹角作为误差评价标准的实验,应用球体和椭球体模型分析了所涉及的6种估计方法的性能。

地震波模拟：非规则网格离散的数值解法、吸收边界及其实际应用

This dissertation presents a series of irregular-grid based numerical technique for modeling seismic wave propagation in heterogeneous media. The study involves the generation of the irregular numerical mesh corresponding to the irregular grid scheme, the discretized version of motion equations under the unstructured mesh, and irregular-grid absorbing boundary conditions. The resulting numerical technique has been used in generating the synthetic data sets on the realistic complex geologic models that can examine the migration schemes. The motion equation discretization and modeling are based on Grid Method. The key idea is to use the integral equilibrium principle to replace the operator at each grid in Finite Difference scheme and variational formulation in Finite Element Method. The irregular grids of complex geologic model is generated by the Paving Method, which allow varying grid spacing according to meshing constraints. The grids have great quality at domain boundaries and contain equal quantities of nodes at interfaces, which avoids the interpolation of parameters and variables. The irregular grid absorbing boundary conditions is developed by extending the Perfectly Matched Layer method to the rotated local coordinates. The splitted PML equations of the first-order system is derived by using integral equilibrium principle. The proposed scheme can build PML boundary of arbitrary geometry in the computational domain, avoiding the special treatment at corners in a standard PML method and saving considerable memory and computation cost. The numerical implementation demonstrates the desired qualities of irregular grid based modeling technique. In particular, (1) smaller memory requirements and computational time are needed by changing the grid spacing according to local velocity; (2) Arbitrary surfaces and interface topographies are described accurately, thus removing the artificial reflection resulting from the stair approximation of the curved or dipping interfaces; (3) computational domain is significantly reduced by flexibly building the curved artificial boundaries using the irregular-grid absorbing boundary conditions. The proposed irregular grid approach is apply to reverse time migration as the extrapolation algorithm. It can discretize the smoothed velocity model by irregular grid of variable scale, which contributes to reduce the computation cost. The topography. It can also handle data set of arbitrary topography and no field correction is needed.

复杂介质中地震波场数值模拟研究——基于广义正交多项式迭积微分算子法

In this paper, we propose a new numerical modeling method – Convolutional Forsyte Polynomial Differentiator (CFPD), aimed at simulating seismic wave propagation in complex media with high efficiency and accuracy individually owned by short-scheme finite differentiator and general convolutional polynomial method. By adjusting the operator length and optimizing the operator coefficient, both global and local informations can be easily incorporated into the wavefield which is important to invert the undersurface geological structure. The key issue in this paper is to introduce the convolutional differentiator based on Forsyte generalized orthogonal polynomial in mathematics into the spatial differentiation of the first velocity-stress equation. To match the high accuracy of the spatial differentiator, this method in the time coordinate adopts staggered grid finite difference instead of conventional finite difference to model seismic wave propagation in heterogeneous media. To attenuate the reflection artifacts caused by artificial boundary, Perfectly Matched Layer (PML) absorbing boundary is also being considered in the method to deal with boundary problem due to its advantage of automatically handling large-angle emission. The PML formula for acoustic equation and first-order velocity-stress equation are also derived in this paper. There is little difference to implement the PML boundary condition in all kind of wave equations, but in Biot media, special attenuation factors should be taken. Numerical results demonstrate that the PML boundary condition is better than Cerjan absorbing boundary condition which makes it more suitable to hand the artificial boundary reflection. Based on the theories of anisotropy, Biot two-phase media and viscous-elasticity, this paper constructs the constitutive relationship for viscous-elastic and two-phase media, and further derives the first-order velocity-stress equation for 3D viscous-elastic and two-phase media. Numerical modeling using CFPD method is carried out in the above-mentioned media. The results modeled in the viscous-elastic media and the anisotropic pore elastic media can better explain wave phenomena of the true earth media, and can also prove that CFPD is a useful numerical tool to study the wave propagation in complex media.

旋转轴对称介质中的qP波一阶射线追踪

Elastic anisotropy is a very common phenomenon in the Earth’s interior, especial for sedimentary rock as important gas and oil reservoirs. But in the processing and interpretation of seismic data, it is assumption that the media in the Earth’s interior is completely elastic and isotropic, and then the methods based on isotropy are used to deal with anisotropic seismic data, so it makes the seismic resolution lower and the error on images is caused. The research on seismic wave simulation technology can improve our understanding on the rules of seismic wave propagation in anisotropic media, and it can help us to resolve problems caused by anisotropy of media in the processing and interpretation of seismic data. So researching on weakly anisotropic media with rotated axis of symmetry, we study systematically the rules of seismic wave propagation in this kind of media, simulate the process with numerical calculation, and get the better research results. The first-order ray tracing (FORT) formulas of qP wave derived can adapt to every anisotropic media with arbitrary symmetry. The equations are considerably simpler than the exact ray tracing equations. The equations allow qP waves to be treated independently from qS waves, just as in isotropic media. They simplify considerably in media with higher symmetry anisotropy. In isotropic media, they reduce to the exact ray tracing equations. In contrast to other perturbation techniques used to trace rays in weakly anisotropic media, our approach does not require calculation of reference rays in a reference isotropic medium. The FORT-method rays are obtained directly. They are computationally more effective than standard ray tracing equations. Moreover the second-order travel time corrections formula derived can be used to reduce effectively the travel time error, and improve the accuracy of travel time calculation. The tensor transformation equations of weak-anisotropy parameters in media with rotated axis of symmetry derived from the Bond transformation equations resolve effectively the problems of coordinate transformation caused by the difference between global system of coordinate and local system of coordinate. The calculated weak-anisotropy parameters are completely suitable to the first-order ray tracing used in this paper, and their forms are simpler than those from the Bond transformation. In the numerical simulation on ray tracing, we use the travel time table calculation method that the locations of the grids in the ray beam are determined, then the travel times of the grids are obtained by the reversed distance interpolation. We get better calculation efficiency and accuracy by this method. Finally we verify the validity and adaptability of this method used in this paper with numerical simulations for the rotated TI model with anisotropy of about 8% and the rotated ORTHO model with anisotropy of about 20%. The results indicate that this method has better accuracy for both media with different types and different anisotropic strength. Keywords: weak-anisotropy, numerical simulation, ray tracing equation, travel time, inhomogeneity

地震波传播模拟的边界元―体积元积分方程法及其近似求解

To improve the efficiency of boundary-volume integral equation technique, this paper is involved in the approximate solutions of boundary-volume integral equation technique. Firstly, based on different interpretations of the self-interaction and extrapolation operators of the resulting boundary integral equation matrix, two different hybrid BEM+Born series modeling schemes are formulated and validated through comparisons with the full-waveform BE numerical solutions for wave propagation simulation in a semicircular alluvial valley and a complex fault model respectively. Numerical experiments indicate that both the BEM+Born series modeling schemes are suitable for complex geological structures and significantly improve computational efficiency especially for the cases of high frequencies and multisource seismic survey. Then boundary-volume integral equation technique is illuminated in detail and verified by modeling wave propagation in complex media. Furthermore, the first-order and second-order Born approximate solutions for the volume-scattering waves are studied and quantified by numerical simulation in different random medium models. Finally, preconditioning generalized minimal residual method is applied to solve boundary-volume integral equation and compared with Gaussian elimination method. Numerical experiments indicate this method makes the calculations more efficient.

喜马拉雅西构造结区岩石圈演化三维有限元动力学数值模拟

The past three decades have seen numerous attempts to numerically model stress and strain patterns in the lithosphere of the Earth on both global and regional scales. This efforts have been indispensable in identifying the features we need to include in our endeavour to develop better models of our planet’s lithosphere and they have also raised our awareness for the many unresolved issue in the deep geodynamical issues that need to be addressed in the future. Nonetheless, in most models, the lithosphere is treated as a single layer with depth-averaged properties, and as the same distribution in the stress and strain fields, and as deforming under plane strain. All these above make a great hander for its reality and degree of recognition. As the beginning in this paper, some principal numerical models and results on the evolution of Tibetan plateau are reviewed and analyzed. Then, the geological and geophysical expedition on the Western Himalayan Syntaxis is briefly reviewed. Furthermore, we analysis the feature in deep geophysical field studies in this area and adjacent regions. Because, for most continents, stress models driven by plate boundary forces have successfully reproduced the main characteristics of the stress and strain field, we present a set of three-dimensional models of lithosphere system for a simplified geometry of the Western Himalayan Syntaxis area and its adjacent regions, where we try to match the first-order characteristics of the stress and strain fields of lithosphere since 10 Ma, and deformation and geodynamical evolution process in former 2Ma. Of course, the kinematic boundary conditions of the stress models driven by plate boundary forces were applied. The rheology plays a significant role in the lithospheric tectonics, which lead to different rheological parameters were used in different works although the have the same constitutive equations in models. So, in this paper we do not aim to produce all characteristics of the Western Himalayan Syntaxis areas’ stress and strain fields by the choices of various parameters, but rather the dynamic response between various rheological parameters and stress and strain fields. We have chosen to concentrate on the importance of rheology and lateral strength variations for lithospheric stress and strain patterns and use our findings to build a model of the Western Himalayan Syntaxis areas. In doing so, we want to go beyond purely elastic models or purely viscoelastic models. Compared the results of the crust viscosity in the Western Himalayan Syntaxis areas, we believed that, when various viscoelastic models are adopted, the selection of the coefficient of viscosity in the Western Syntaxis area has important influence on the its uplifts and evolutions. A wider uplift ranges and gently elevation was observed at the same time when a lower viscosity was used in our models, and vice versa. Data of stress magnitudes are not available, but it is clear that the stress levels must be at or below the failure threshold of rock under compression. Under these criteria, the calculation results show that the viscosity in the Western Syntaxis area should be smaller than 1023Pa.s When elastic model is adopted in relatively rigid Tarim basin, obvious changes are induced to the stress and strain fields of the whole Western Syntaxis area. We found that rigid block of lithosphere reduced stress levels within its interior and that, at the edges of such regions, stress orientation can change. Furthermore there is no evidence that such rigid regions act as stress barriers in that they shield areas in opposite sides of the structure from the influence of one another. In our models, the upper crustal material of the Western Syntaxis area does not turns to move westward. Whereas, because of the stress and strain fields have been decoupling at the interior of the lithosphere, we can get the results that the deep material must not move westward.

断层封闭性及其与油气成藏的关系研究-以沾化凹陷孤岛地区为例

Godao area, located in the east of the Zhanhua depression of Jirang sag in Bohai Bay Basin, is the studying area in my dissertation. It is first time that fault sealing properties and the related relationship with the pool forming are studied in Gudao area. On the base of the analysis of the regional tectonics, the author has studied the tertiary structural evolution of the Gudao area and distinguished the fault's level and put forward the distinguishing principle. The geometrical feature, mechanical characters, developmental mechanism and history of the boundary faults in the tectonic unit of this area are all studied and emphasized especially. The buried history of oil-generating depression (that is Gudao depression) and the history of oil and gas migration simultaneously are discussed, the juxtaposition relationship between boundary fault evolutionary history and oil and gas migrated history are expatiated. To the geological condition of the Gudao area, three level faults sealing properties of this area were discussed in detail. Their characteristics of behavior and the intrinsic relationship between their sealing and oil and gas migrated reservoir are elucidated. The pool-forming models related to fault seal are exposed. The author has studied the lithologies of different order of faults, the relationship of occurrence assemblage analysis, normal stress of fault plane in different depth and shale smear factor faults. Then analysis their role in the various faults sealing and confirms fault sealing marks of three different orders faults and exposes the mechanism of fault sealing. Shale smear zone formed by first order fault in lasting activities is one different type of fault breccia and mainly controlled factor to its entrapment of petroleum. Effective sealing threshold value and fault displacement is ascertained. Mainly controlled factor of second fault sealing is bigger compressive stress loaded on fault plane. According to this, quantitatively evaluated index is given. Shale smear zone is necessary condition for second fault stress entrapment. The juxtaposed relationship between the different lithologies within third order faults is most important controlled factor for its sealing. Based on various order of fault sealing features and mechanism in Gudao area, the author proposed three orders of fault sealing models. Shale smear zone sealing model, normal stress sealing model and lithologies juxtaposed sealing model are suggested to first, second and third order fault respectively. The conclusion of this studying has not only the very important theoretical significance and practical value in Gudo area but also the very important guiding role for other areas of related aspects.

接收函数方法与中国及邻区上地幔间断面的研究

The receiver function method applied in researching the discontinuities in upper mantle was systematically studied in this paper. Using the theoretical receiver functions, the characteristics of P410S and P660S phases were analyzed, and the influencing factors for detection of these phases were discussed. The stability of receiver function was studied, and a new computational method of receiver function, RFSSMS (Receiver Function of Stack and Smooth of Multi seismic-records at a Single station), was put forward. We built initial reference velocity model for the media beneath each of 18 seismic stations respectively; then estimated the buried depths of 410-km and 660-km discontinuities(simply marked as '410' and '660') under the stations by using the arrive time differences of P410S and P660S with P. We developed a new receiver function inversion method -PGARFI (Peeling-Genetic Algorithm of Receiver Function Inversion), to obtain the whole crust and upper mantle velocity structure and the depths of discontinuities beneath a station. The major works and results could be summarized as follows: (1) By analysis of the theoretical receiver functions with different velocity models and different ray parameters, we obtain the knowledge: The amplitudes of P410S and P660S phases are decreasing with the increasing of epicentral distance A , and the arrival time differences of these phases with P are shorter as A is longer. The multiple refracted and/or reflected waves yielded on Moho and the discontinuities in the crust interfere the identification of P410S. If existing LVZ under the lithosphere, some multiple waves caused by LVZ will interfere the identification of P410S. The multiple waves produced by discontinuity lied near 120km depth will mix with P410s phase in some range of epicentral distance; and the multiple waves concerned with the discontinuity lied near 210km depth will interfere the identification of P660S. The epicentral distance for P4i0s identification is limited, the upper limit is 80° . The identification of P660S is not restricted by the epicenter distance obviously. The identification of P410S and P6gos in the theoretical receiver functions is interfered weakly from the seismic wave attenuation caused by the media absorption if the Q value in a reasonable range. (2) The stability of receiver function was studied by using synthetic seismograms with different kind of noise. The results show that on the condition of high signal-noise-ratio of seismic records, the high frequency background noise and the low frequency microseism noise do not influence the calculating result of receiver function. But the media "scattering noise" influence the stability of receiver function. When the scattering effect reach some level, the identification of P4iOs and P66os is difficult in single receiver function which is yielded from only one seismic record. We provided a new method to calculate receiver function, that is, with a group of earthquake records, stacking the R and Z components respectively in the frequency domain, and weighted smooth the stacked Z component, then compute the complex spectrum ratio of R to Z. This method can improve the stability of receiver function and protrude the P4i0s and P66os in the receiver function curves. (3) 263 receiver functions were provided from 1364 three component broadband seismograms recorded at 18 stations in China and adjacent areas for the tele-earthquakes. The observed arrival time differences of P410S and P660S with P were obtained in these receiver functions. The initial velocity model for every station was built according to the prior research results. The buried depths of '410' and '660' under a station were acquired by the way of adjusting the depths of these two discontinuities in the initial velocity model until the theoretical arrival time differences of P410S and P660S with P well conformed to the observed. The results show an obvious lateral heterogeneity of buried depths of ' 410' and (660' . The depth of '410' is shallower beneath BJI, XAN, LZH and ENH, but deeper under QIZ and CHTO, and the average is 403km . The average depth of '660' is 663km, deeper under MDJ and MAJO, but shallower under QIZ and HYB. (4) For inversing the whole crust and upper mantle velocity structure, a new inversion method -PGARFI (Peeling-Genetic Algorithm of Receiver Function Inversion) has- been developed here. The media beneath a station is divided into segments, then the velocity structure is inversed from receiver function from surface to deep successively. Using PGARFI, the multi reflection / refraction phases of shallower discontinuities are isolated from the first order refraction transform phase of deep discontinuity. The genetic algorithm with floating-point coding was used hi the inversion of every segment, and arithmetical crossover and non-uniform mutation technologies were employed in the genetic optimization. 10 independent inversions are completed for every segment, and 50 most excellent velocity models are selected according to the priority of fitness from all models produced in the inversion process. The final velocity structure of every segment is obtained from the weighted average of these 50 models. Before inversion, a wide range of velocity variation with depth and depth range of the main discontinuities are given according to priori knowledge. PGARFI was verified with numerical test and applied in the inversion of the velocity structure beneath HIA station down to 700km depth.

济阳地区深部结构与含油气性讨论

Oil and gas exploration of marine strata in China's Pre-Cenozoic residual basins is regarded as a worldwide puzzle because of existent problems and cruxes. Objectively speaking, the subsurface geologic structure is complicated, and the surface conditions of some areas are tough. On the other hand, there are still many problems to be solved in oil and gas exploration technologies of Pre-Cenozoic marine fades, and theoretic cognition about petroleum geology is not profound yet. Therefore, it is principal to explore integrated geophysical research ways of Pre-Cenozoic residual basins. Seismic prospecting and geophysical integrated interpretation technologies aimed at middle Paleozoic marine facies with deeper burial and complicated geologic conditions have not formed due to bad quality of deep strata data. Pre-Cenozoic strata, and especially extension, thickness and internal structure of Paleozoic strata can not be recognized from seismic profiles, so it is hard to systematically cognize structural features and oil-gas resources prospect of Pre-Cenozoic basins. To further investigation of fabric and structural features, basin prototype, formation and evolution pattern of Pre-Cenozoic basins, and also their control over formation, migration and aggregation of oil and gas, will play a guiding and promotive role in developing new surveying areas, selecting advantageous zones and predicting oil-gas resources.This paper follows the modem macrocontrol theory of "Region constrains local, deep strata controls shallow ones", and uses the integrated geophysical method of "One guide, two hinges, three combinations and multi feedbacks'*. Based on several years of geological and geophysical results of the Shengli Oilfield, and 14 newly-joint regional seismic profiles, deep structure and oil-gas bearing capacity of the Jiyang area are discussed and new cognitions are drawn as below.Seismic identification marks Tr, Tg, Tgl and Tg2 are established for importantPre-Cenozoic geological interfaces, and promoted to the whole Jiyang area.Through area-wide tracking and clogging of important seismic reflection marker,the isochronic framework of pre-Tertiary basin is set up in the Jiyang area for the firsttime, which is vital for basin research.Integrated with geological and geophysical research results, the Jiyang area isdivided into four first-order tectonic sequences- basement, lower tectonic layer,upper tectonic layer, and top tectonic layer. The basement and lower tectonic sequence which are related to Pre-Cenozoic are studied with emphasis.Through the research of regional seismic profiles, the point of view is given thatthe Kongdian Formation of Jiyang is structural transition period. The top-bottomunconformable interface of the Kongdian Formation is found out for the first time,and the basin model is determined primarily, which lay a basis for prototype basinresearch of the Jiyang Kongdian Formation.The distribution status of Middle-Paleozoic is delineated in the Jiyang area.The maximum thickness of Paleozoic lies in the top of the south declivity of half-graben. The thickness gets thinner towards the center of Mesozoic and Cenozoic half-graben basin, and even disappears. Structural action in the west-north affects the distribution of Paleozoic residual strata.6. The features of second-order tectonic sequence of the Jiyang depression isstudied and its evolution history of is rebuilt.Combined with the 5-stage evolution history of the China continent and structure evolution features of the Jiyang area, the structure sedimentary process since Paleozoic is divided into 5 periods - basement forming , Indosinian orogenic, Yanshan negative reversal, Himalayan extension and Neogene subsidence period.Combined with the research results of gravity, magnetic surveying and regionalprofiles, this paper brings forward the idea for the first time that the western boundaryof the Jiyang depression is the Ningjin-Yangpan fracture zone, and forms aside-column assemblage with the Wudi fracture zone.The opinion that under Middle-Cenozoic basins in the middle Jiyang area theremight superimpose an old residual basin is given for the first time. And if it is provedto be true, a new exploration space will be pioneered for Jiyang and even north China.There exists many types of tectonic-stratigraphic traps formed under piezotropy,extension and compound action in Pre-Cenozoic Jiyang. Therein all kinds of burialhills are the most important oil-gas trap type of Pre-Cenozoic, which should besurveyed layeredly according to the layout of oil sources.As such a new challenging project and field, the paper systematically analyses different geophysical responses of the Jiyang area, frames the deep structure of the area, and preliminarily recognizes the Pre-Cenozoic residual basins. It breaks through to a certain extent in both theory and practice, and is expected to provide new geophysical and geotectonic clues for deep exploration in Shengli.

The Image Irradiance Equation: Its Solution and Application

How much information about the shape of an object can be inferred from its image? In particular, can the shape of an object be reconstructed by measuring the light it reflects from points on its surface? These questions were raised by Horn [HO70] who formulated a set of conditions such that the image formation can be described in terms of a first order partial differential equation, the image irradiance equation. In general, an image irradiance equation has infinitely many solutions. Thus constraints necessary to find a unique solution need to be identified. First we study the continuous image irradiance equation. It is demonstrated when and how the knowledge of the position of edges on a surface can be used to reconstruct the surface. Furthermore we show how much about the shape of a surface can be deduced from so called singular points. At these points the surface orientation is uniquely determined by the measured brightness. Then we investigate images in which certain types of silhouettes, which we call b-silhouettes, can be detected. In particular we answer the following question in the affirmative: Is there a set of constraints which assure that if an image irradiance equation has a solution, it is unique? To this end we postulate three constraints upon the image irradiance equation and prove that they are sufficient to uniquely reconstruct the surface from its image. Furthermore it is shown that any two of these constraints are insufficient to assure a unique solution to an image irradiance equation. Examples are given which illustrate the different issues. Finally, an overview of known numerical methods for computing solutions to an image irradiance equation are presented.