models for disc infiltrometers disc infiltrometers are popular devices for determining in situ hydraulic properties of unsaturated soils this paper compares alternative solutions of richards equation for both steady state and time dependent cases the steady state solutions were in general agreement using alternative hydraulic conductivity functions of the same capillary length scale small time solutions for the nonlinear cases were consistent with linear diffusion from a disc source using an average diffusivity value which is simply related to the capillary length this offers a refinement over the one dimensional solution for short times in that the geometric effect of the circular source is included simulations for two examples indicate that the approach to the steady state solution may take considerably longer than what is commonly reported in the literature for field applications

steady infiltration from large shallow ponds exact quasi linear solutions are obtained for steady infiltration from a shallow half plane pond on the surface of an isotropic semi infinite porous medium three particular boundary value problems are considered namely constant dirichlet boundary conditions in relative permeability and vertical flux and a mixed boundary value problem the boundary conditions on the surface of the porous media for these problems correspond physically first to fully saturated conditions under the pond and fully unsaturated conditions elsewhere second to a constant vertical flux under the pond and zero vertical flux elsewhere and third to fully saturated conditions under the pond and zero vertical flux elsewhere each problem has essentially the same diffusive boundary layer or capillary fringe connecting the fully saturated and unsaturated regions the total flux per unit length of boundary due to capillarity in the first problem is infinite and for the latter two problems finite constants strong analogies between our results for a surface source and the corresponding results of waechter and philip for buried spherical and cylindrical sources are established finally the exact results for the half plane pond are used in an approximate description of steady infiltration from a large shallow pond of arbitrary shape subject to the minimum radius of curvature of the pond s perimeter being large relative to 2 where is the inverse length scale introduced in quasi linear models a boundary layer develops from capillarity at the pond s edge away from which the solution is adequately described by a diffusion equation representing a balance between the vertical flow from gravity and the horizontal flow from capillarity an approximate expression is obtained analytically for the total flow from large shallow ponds

decoding the golay code with venn diagrams a decoding algorithm based on venn diagrams for decoding the 23 12 7 golay code is presented the decoding algorithm is based on the design properties of the parity sets of the code as for other decoding algorithms for the golay code decoding can be easily done by hand

scattering functions and infiltration in the theory of the scattering of plane harmonic waves by an obstacle van de hulst theorems give simple connections between the forward scattering function and the total extinction cross section in this paper analogous theorems are proved for two and three dimensional quasi linear steady flows in saturated soils from finite cavities of arbitrary size and shape with arbitrary boundary conditions on the cavity surface these theorems connect downward wetting functions and total cavity flow rates simple relations are established between scattering functions in the wave context and wetting functions in the soil water context the important consequence is that flow rates for a large range of cavity shapes can be inferred simply and directly from extant results on forward scattering functions it is proved that at large r dimensionless radius flow is concentrated in a small angular region vertically beneath the cavity and has a gaussian distribution with angular standard deviation sr s l sorptive number l characteristic cavity length

stochastic modeling of unsaturated flow in heterogeneous media with water uptake by plant roots tests of the parallel columns model under two dimensional flow conditions in the first paper of this series rubin and or 1993 we developed a stochastic model predicting spatial moments of soil water matric potential and saturation s under conditions of unsaturated steady state flow as function of the spatial moments of log saturated hydraulic conductivity y capillarity index and plant rooting depth in this study we investigate the ability of the proposed parallel columns model to reconstruct two dimensional flow solutions and apply the model to spatial estimation using the geostatistical formalism a finite difference numerical scheme was developed to solve the two dimensional richards equation in heterogeneous flow domain and obtain detailed images of the spatial distribution of and s and their moments two dimensionally based images representing true distributions of and s in a vertical cross section x z plane of laterally heterogeneous soil were used in a synthetic spatial estimation problem for comparison with the one dimensionally based solution and with kriging estimates the results show that 1 the means of and s were practically identical for the one and two dimensional solutions 2 variance obtained from the two dimensional solution was smaller than the one dimensional whereas s variance was larger in the two dimensional case 3 a smoothing of under two dimensional flow regime resulted in spatial covariance function of a bessel type whereas two dimensional spatial covariance of s exhibited a steep decay near its origin denoting spatial discontinuity 4 the least smoothing of was observed near the surface where nearly one dimensional conditions exist 5 during transient flow spatial covariances assume their steady state form quite rapidly 6 spatial covariances predicted by the parallel columns model provided good estimates at unmeasured locations with conservative confidence regions for estimation and less conservative for s estimation and 7 analytically predicted spatial covariances provide a means for network design

the seepage exclusion problem for parabolic and paraboloidal cavities the problems of water exclusion from or entry into parabolic cylindrical and paraboloidal cavities during steady downward unsaturated seepage are solved both the quasi linear and the full nonlinear solutions are found exactly and simply the identical general mathematical character of the two types of solution and the good agreement of numerical details attest to the reliability and utility of the very much simpler quasi linear analysis a remarkable result is that for both the parabolic cylinder and the paraboloid the potential at the cavity wall is spatially uniform these geometries thus constitute separatices between apically blunter ones circular cylinder sphere where the potential at the wall decreases downstream and sharper ones wedge cone hyperbolic cylinder hyperboloid where it increases the parabolic geometries are optimal in leaking nowhere or everywhere blunter ones leak first at the apex and and sharper ones at a low point of the wall implications for the design of seepage excluding tunnels and underground repositories are indicated comparison of the present results with those for the circular cylinder and the sphere reveals the significance of two dimensionless parameters of cavity geometry 1 with sorptive number and apical total curvature and apical sharpness

eigenvalue boundary problems for the dirac operator on a compact riemannian spin manifold with mean convex boundary we analyse the ellipticity and the symmetry of four boundary conditions for the fundamental dirac operator including the global aps condition and a riemannian version of the local mit bag condition we show that friedrich s inequality for the eigenvalues of the dirac operator on closed spin manifolds holds for the corresponding four eigenvalue boundary problems more precisely we prove that for both the aps and the mit conditions the equality cannot be achieved and for the other two conditions the equality characterizes respectively half spheres and domains bounded by minimal hypersurfaces in manifolds carrying non trivial real killing spinors

multidimensional quasi linear steady infiltration toward a shallow water table the present work explores further a class of analytical solutions of moisture movement in unsaturated porous media characterized by an exponential dependence of the hydraulic conductivity on water pressure the green s function method is used to derive a general analytical model pertaining to multidimensional steady infiltration to a shallow water table with arbitrary boundary conditions and arbitrary root uptake forcing functions and for various simple source geometry the general solution is expressed in integral form from which particular analytical solutions pertaining to cases of surface and subsurface irrigation evaporation and root uptake in the presence of a shallow water table can be easily deduced from the general analytical model new solutions include the one dimensional infiltration subject to discrete extraction functions the two dimensional infiltration from a strip source with a uniform exponential and a periodic flux distribution and the three dimensional infiltration from a rectangular source at the land surface particular solutions have also been obtained for infiltration from a periodic series of point line and strip sources algebraic equations for the parameter estimation of the hydraulic conductivity function were also derived from the simple one dimensional solution the proposed model offers the analyst significant flexibility in deriving results and analyzing infiltration phenomena of practical interest and provides a basis upon which parameter estimation can be effected

burgers equation a general nonlinear solution of infiltration and redistribution 1 the general one dimensional solution of burgers equation is developed using a series of transformations and the green s function method the solution gives the distribution of the reduced water content in a semi infinite and in a finite domain for arbitrary initial moisture distributions the boundary condition at the soil surface can be either a time dependent flux or constant water content while the bottom boundary condition for the finite case is a constant water content value explicit results for a uniform discrete hydrostatic and steady state initial moisture profiles are derived an analytical solution that spans the preponding to the postponding period of rainfall is also obtained expressions for the time to ponding and infiltration equations are presented for the various cases of initial moisture profiles and a shallow water table these results allowed the quantification of the error in the various approximations to the time to ponding and the cumulative infiltration in the postponding period

fuzzy rule based models for infiltration modeling water movement in the unsaturated zone requires a great number of parameters such as the initial water content the saturated moisture content and the saturated hydraulic conductivity which can relatively easily be assessed other functions such as the suction head and the relative hydraulic conductivity which are highly nonlinear functions of the actual water content are more difficult to find furthermore infiltration experiments carried out at a few locations can only define these parameters at selected locations and a reasonable interpolation of these parameters is extremely difficult in this paper two fuzzy logic based models are presented for the infiltration process basic definitions of fuzzy sets are given and basic operations are explained principles of the rule based infiltration models are presented the rules for the first model are based on the moisture content of the adjacent layers and the depth of the wetting front using these rules the flows can be calculated as exact quantities the rules are derived from a training set which was obtained from several test runs of a green and ampt ga infiltration model separate rules are derived for the flow dynamics in the upper layer runoff and infiltration and for water flow between the layers examples of the application of the model are given the model performs with similar accuracy as the ga model using only 40 of its parameters and running 2 orders of magnitude faster a training set obtained by numerical solution of the richards equation was used to create a second fuzzy rule based model here the rules are based on the moisture content of the adjacent layers results obtained by the use of the models are compared with measured values showing a very accurate reproduction of the values

soft decoding techniques for codes and lattices including the golay code and the leech lattice two kinds of algorithms are considered 1 if is a binary code of length n a soft decision decoding algorithm for changes an arbitrary point of r n into a nearest codeword nearest in euclidean distance 2 similarly a decoding algorithm for a lattice lambda in r n changes an arbitrary point of r n into a closest lattice point some general methods are given for constructing such algorithms ami are used to obtain new and faster decoding algorithms for the gosset lattice e 8 the golay code the leech lattice

on the complexity of minimum distance decoding of long linear codes we suggest a decoding algorithm of q ary linear codes which we call supercode decoding it ensures the error probability that approaches the error probability of minimum distance decoding as the length of the code grows for n spl rarr spl infin the algorithm has the maximum likelihood performance the asymptotic complexity of supercode decoding is exponentially smaller than the complexity of all other methods known the algorithm develops the ideas of covering set decoding and split syndrome decoding

second order accurate monotone finite volume scheme for richards equation in this work we perform a theoretical and numerical analysis of richards equation for certain types of nonlinearities we provide explicit analytical solutions these solutions are used to show that conventional unconditionally monotone finite volume schemes have only first order accuracy we derive necessary and sufficient conditions for the monotonicity of finite volume discretizations and use these conditions to construct a monotone finite volume discretization accurate to second order

correspondence of the gardner and van genuchten mualem relative permeability function parameters the gardner and van genuchten models of relativepermeability are widely used in analytical and numerical solutions toflow problems however the applicab ility of the gardner model to realproblems is usually limited because empirical relative permeability datato calibrate the model are not routinely available in contrast vangenuchten parameters can be estimated using more routinely availablematric potential and saturation data however the van genuchten model isnot amenable to analytical solutions in this paper we introducegeneralized conversion formulae that reconcile these two models ingeneral we find that the gardner parameter alpha g is related to the vangenuchten parameters alpha vg and n by alpha g alpha vg 1 3 n thisconversion rule will allow direct recasting of gardner based analyticalsolutions in the van genuchten parameter space the validity of theproposed formulae was tested by comparing the predicted relativepermeability of various porous media with measured values

decoding the golay codes we introduce exceptionally simple decoding algorithms for the two extended golay codes the algorithms are based on recent methods of conway and curtis of finding the unique blocks containing five points in either the 5 8 24 steiner system or the 5 6 12 steiner system these decoding methods are simple enough to enable decoding extended golay codes by hand both of the methods involve relations between the extended golay codes and other self dual codes proofs are given explaining these relationships and why the decoding methods work the decoding algorithms are explained and illustrated with many examples 3 chap 12 has facts about the mathieu group and some details about decoding the golay codes

stochastic modeling of large scale transient unsaturated flow systems a new framework for modeling large scale transient unsaturated flow systems in spatially variable soils is proposed in order to overcome the problem of limited information about the local details of spatial soil variability a stochastic approach which assumes that local soil properties are realizations of three dimensional random fields is followed for derivation of a large scale model representation structure the three dimensionality of the local flow and the nonlinear dependence of the local flow output on the local soil properties are considered the large scale model structure is derived by averaging the local governing flow equation over the ensemble of realizations of the underlying soil property random fields the resulting mean model representation is in the form of a partial differential equation in which averaged or effective model parameters occur these effective model parameters i e effective hydraulic conductivity and effective specific moisture capacity are evaluated using a quasi linearized fluctuation equation and a spectral representation of stationary processes the large scale model representation considers the large scale effects of soil property variability and has relatively few parameters the general conclusions of this study are that soil property variability produces large scale hysteresis and anisotropy of the effective parameters the potential theoretical and practical ramifications of these results in the area of unsaturated flow modeling need to be investigated the general stochastic modeling framework developed here is applicable not only to unsaturated flow but also to other distributed parameter systems e g saturated flow and transport geothermal and oil reservoir modeling

algebraic models for disc tension permeameters four infiltration models for disc tension permeameters were compared to field and simulated data comparisons were made in order to assess the alternative values of sorptivity and steady state flow rates model 1 was the conventional analysis with assumptions of one dimensional flow at small times and wooding s steady state flow rate at large times model 2 was a modified horton equation defined to fit one dimensional infiltration at small times with an exponential decay to a steady state value for large times model 3 assumed linear diffusion for a soil with constant diffusivity ignoring gravity model 4 was based on linearized flow with gravity the number of unknown parameters to be estimated ranged from one to four sorptivity data using models 2 and 3 generally agreed with the conventional method model 1 both for the field and simulated data for the steady state infiltration rate all of the models gave generally consistent results model 2 with four parameters was the best fitting model to field and simulated data the simulated data allowed comparisons of results based on the total time over which the parameters were evaluated results were sensitive to the time of evaluation up to 0 3 1 3 hours for the sandy loam soil

algebraic decoding of 103 52 19 and 113 57 15 quadratic residue codes in this paper two algebraic decoders for the 103 52 19 and 113 57 15 quadratic residue codes which have lengths greater than 100 are presented the results have been verified by software simulation that programs in c language have been executed to check possible error patterns of both quadratic residue codes

steady infiltration flows with sloping boundaries general theorems are established that enable immediate solution of the quasilinear steady infiltration equation for any distribution of line and point sources at or beneath a sloping upper soil surface or on or above a sloping impermeable base the required solutions are readily deduced from the known simple solutions for buried line and point sources corollary to the theorems similar relations hold for downslope and cross slope flux density components illustrative solutions are presented for line and point sources in the various configurations distributions of both potential and stream function are mapped for line sources and also for point sources at or beneath a horizontal surface for other point sources potential is mapped in the normal plane through the source for sources beneath a surface distortion due to the surface is relatively small even for dimensionless normal source depth as small as 1 on the other hand for sources above a sloping base the base strongly skews the distributions of both potential and streamlines far downslope the flow is purely line sources or primarily point sources parallel to the base upper bounds exist on total source strength beyond which a saturated region emerges perched on a horizontal base consistent with earlier results and theorems physically acceptable steady flows do not exist above the base

minimal permutation sets for decoding the binary golay codes corresp for permutation decoding of an e error correcting linear code a set of permutations which move all error vectors of weight leq e out of the information places is needed a method of finding minimal decoding sets is given along with minimal sets obtained with this method for the binary golay codes

constant rate rainfall infiltration a versatile nonlinear model 1 analytic solution analytic solutions are presented for a nonlinear diffusion convection model describing constant rate rainfall infiltration in uniform soils and other porous materials the model is based on the darcy buckingham approach to unsaturated water flow and assumes simple functional forms for the soil water diffusivity d and hydraulic conductivity k which depend on a single free parameter c and readily measured soil hydraulic properties these d and k yield physically reasonable analytic moisture characteristics the relation between this model and other models which give analytic solutions is explored as c the model reduces to the weakly nonlinear burgers equation which has been applied in certain field situations at the other end of the range as c 1 the model approaches a green ampt like model a wide range of realistic soil hydraulic properties is encompassed by varying the c parameter the general features of the analytic solutions are illustrated for selected c values gradual and steep wetting profiles develop during rainfall aspects seen in the laboratory and field in addition the time dependent surface water content and surface water pressure potential are presented explicitly a simple traveling wave approximation is given which agrees closely with the exact solution at comparatively early infiltration times

stochastic analysis of unsaturated steady state flow through bounded heterogeneous formations the effect of heterogeneity in saturated hydraulic conductivity ks and in a pore scale distribution parameter on unsaturated steady state flow in bounded domains is studied the properties are assumed to be random space functions having stationary means and covariances an analytic model based on small perturbation approximation in ks and for predicting the spatial moments of the pressure head from the moments of the input variables is developed the solution applies to the entire unsaturated domain without invoking the unit mean hydraulic gradient assumption under conditions of small scale variability the solution simplfies considerably resulting in closed form expressions for the moments of the pressure head these expressions are tested in synthetically generated random fields of and ks having prescribed statistical properties

the liouville equation with singular data a concentration compactness principle via a local representation formula abstract for a bounded domain r 2 we establish a concentration compactness result for the following class of singular liouville equations u e u 4 j 1 m j p j in where pj j 0 and pj denotes the dirac measure with pole at point pj j 1 m our result extends brezis merle s theorem comm partial differential equations16 1991 1223 1253 concerning solution sequences for the regular liouville equation where the dirac measures are replaced by lp data p 1 in some particular case we also derive a mass quantization principle in the same spirit of li shafrir indiana univ math j 43 1994 1255 1270 our analysis was motivated by the study of the bogomol nyi equations arising in several self dual gauge field theories of interest in theoretical physics such as the chern simons theory self dual chern simons theories lecture notes in physics vol 36 springer verlag berlin 1995 and the electroweak theory selected papers on gauge theory of weak and electromagnetic interactions world scientific singapore

stochastic analysis of unsaturated flow in heterogeneous soils 1 statistically isotropic media steady unsaturated flow with vertical mean infiltration through unbounded heterogeneous porous media is analyzed using a perturbation approximation of the stochastic flow equation which is solved by spectral representation techniques the hydraulic conductivity k is related to the capillary pressure head by k ks exp where ks is the saturated conductivity and is a soil parameter a general formulation is presented for the case with ks and represented as statistically homogeneous spatial random fields in part 1 solutions are developed assuming is constant and representing ks variability by one dimensional and three dimensional isotropic random fields results are obtained for head variances and covariance functions effective hydraulic conductivities variances of the unsaturated hydraulic conductivity flux variances and variance of pressure gradient when the parameter is relatively large corresponding to coarse textured soils the head variance decreases and all of the results demonstrate a trend toward gravitationally dominated one dimensional vertical flow the effective conductivity is dependent on the correlation scale of ln ks and the mean hydraulic gradient

energy quantization for a singular super liouville boundary value problem in this paper we develop the blow up analysis and establish the energy quantization for solutions to super liouville type equations on riemann surfaces with conical singularities at the boundary in other problems in geometric analysis the blow up analysis usually strongly utilizes conformal invariance which yields a noether current from which strong estimates can be derived here however the conical singularities destroy conformal invariance therefore we develop another more general method that uses the vanishing of the pohozaev constant for such solutions to deduce the removability of boundary singularities

a compactness result for periodic multivortices in the electroweak theory we derive a priori uniform bounds for solutions of an elliptic system of liouville type equations first analyzed by j spruck and y yang comm math phys 144 1992 1 yielding periodic multivortices in the classical electroweak theory of glashow salam weinberg our proof is based on a concentration quantization result in the same spirit of brezis merle comm partial differential equations 16 8 9 1991 1223 and li shafrir indiana univ math j 43 4 1994 1255 for mean field equations on riemannian compact 2 manifolds

harmonic maps from degenerating riemann surfaces we study harmonic maps from degenerating riemann surfaces with uniformly bounded energy and show the so called generalized energy identity we find conditions that are both necessary and sufficient for the compactness in w 1 2 and c 0 modulo bubbles of sequences of such maps

multidimensional linearized nonsteady infiltration with prescribed boundary conditions at the soil surface the present work explores a class of analytical solutions of moisture movement in unsaturated porous media characterized by an exponential dependence of the hydraulic conductivity and the moisture content on water pressure the green s function method is used to derive a general analytical model pertaining to multidimensional nonsteady infiltration in a semi infinite flow domain with arbitrary initial conditions boundary conditions and root uptake forcing functions and for various simple source geometry the general solution is expressed in integral form from which particular analytical solutions pertaining to cases of surface and subsurface irrigation evaporation root uptake and moisture redistribution can be easily deduced from the general analytical model the model offers the analyst significant flexibility in deriving results and analyzing infiltration phenomena of practical interest new explicit solutions have been obtained for one dimensional infiltration under various prescribed time dependent flux boundary conditions and for two and three dimensional moisture redistribution for constant initial or boundary conditions the multidimensional solution is essentially the product of two or three time dependent terms with each term being a function of only one space variable

asymptotic solutions of the seepage exclusion problem for elliptic cylindrical spheroidal and strip and disc shaped cavities asymptotic roof boundary layer solutions are obtained for the quasi linear seepage exclusion problem for elliptic cylindrical and spheroidal cavities of arbitrary aspect ratio h the remarkably simple result for max the maximum dimensionless potential at the cavity wall is max 2 f0 h f1 h f2 h 2 where k 1 is sorptive number k is apical total curvature and the fn are polynomials in h 2 of degree n 1 the results agree as they should with previous results for the circular cylinder and sphere h 1 and the parabolic cylinder and paraboloid h connections noted in earlier studies between apical sharpness and the variation of potential along the cavity wall are generalized and made quantitative the singular h 0 strip and disc are of especial interest they indicate behavior near the critical upstream stagnation point on cavities and impermeable obstacles with flat strip and disc shaped roofs but otherwise within some limits arbitrary for the strip max s1 4s12 9 36 s12 and for the disc max s1 2s12 6 27 s12 where s1 l1 l1 is semiwidth or radius the central importance of max is that seepage water enters or is excluded from the cavity depending on whether k0 k1 k0 is downward steady seepage velocity far from the cavity k1 is saturated hydraulic conductivity exceeds or does not exceed max 1

clean two and three dimensional analytical solutions of richards equation for testing numerical solvers 1 this technical note derives clean analytical solutions of richards equation for three dimensional unsaturated groundwater flow clean means that the boundary conditions and steady state solutions are closed form expressions and the transient solutions have relatively simple additional fourier series terms two dimensional versions of these solutions are also given the primary purpose for the solutions is to test linear and nonlinear solvers in finite difference volume element computer programs for accuracy and scalability using architectures ranging from pcs to parallel high performance computers this derivation starts from the quasi linear assumption of relative hydraulic conductivity varying exponentially with pressure head and the separate approximation that relative hydraulic conductivity varies linearly with moisture content this allows a transformation to be used to create a linear partial differential equation separation of variables and fourier series are then used to obtain the final solution physically reasonable material properties are also used a total of four solutions are given in this technical note steady state and transient solutions for two different boundary conditions of the sample problem

a geostatistical approach to the trickle irrigation design in heterogeneous soil 1 theory a geostatistical approach was used to investigate the spatial variability of two measured soil hydraulic parameters the saturated hydraulic conductivity ks and d log k h d h where k is the hydraulic conductivity and h is the soil water pressure head using a linearized solution to steady infiltration from a circular shallow pond the spatial distribution of the midway soil water pressure head between emitters hc for a given trickle discharge rate q and spacing between emitters d was calculated using the spatially variable soil parameters ks and the spatial distribution of the crop yield y was calculated from the spatially variable hc by using a quadratic expression for the y hc relationships it was found that the spatial variability of each of the measured input parameters ks and or the calculated output variables hc and y is expressed through the variogram and can be characterized in terms of the coefficient of variation cv and the integral scale j in the case of log ks for which the sill is reached at relatively small range two different were tested results showed that the kriged values of log ks as well as the resultant values of hc hc ks d q were relatively insensitive to the selected the spatial variability of hc reduced the average over the field crop yield that would have been obtained when hc was uniform throughout the field the spatially variable soil parameters ks and were used to calculate the spatial distribution of the spacing between emitters d required to achieve for a given trickle discharge a prespecified value of hc uniform throughout the field it was found that the use of a spatially variable d increased the field crop yield by 3 relative to the yield that was obtained when a single valued d uniform throughout the field was used the spatial distribution of d was combined with hydraulic principles to derive the spatial distribution of lateral diameter and length for engineering design requirements

on minimal decoding sets for the extended binary golay code minimal decoding sets consisting of fourteen permutations have been found for the 24 12 8 extended binary golay code it is shown that by properly sequencing the permutations of one such set the average number of permutations required to decode a random received word can be minimized

analytical solutions for unsteady multidimensional infiltration in heterogeneous soils appendix is available with entire article on microfiche order from american geophysical union 2000 florida avenue n w washington dc 20009 document w91 003 2 50 payment must accompany order the problem of unsteady multidimensional infiltration in heterogeneous soils is approached analytically by assuming that the hydraulic conductivity is an exponential function of the soil matric potential and a linear function of the soil moisture the governing flow equation is linearized for a special variation of the saturated hydraulic conductivity in space by using integral transform methods analytical solutions to the one and two dimensional flow problem are derived and shown to be generalized versions of known results for homogeneous soils the development aims at obtaining quantitative criteria for the validity of the one dimensional approximation which is commonly used in hydrologic applications

existence of conformal metrics with constant q curvature given a compact four dimensional manifold we prove existence of conformal metrics with constant q curvature under generic assumptions the problem amounts to solving a fourth order nonlinear elliptic equation with variational structure since the corresponding euler functional is in general unbounded from above and from below we employ topological methods and min max schemes jointly with the compactness result of 35

steady unsaturated seepage above a sloping impermeable base we establish the nonexistence of steady unsaturated flows in regions infinite in the horizontal and bounded below by a horizontal impermeable base asymptotic analyses are given of two and three dimensional seepage above a sloping impermeable base valid far from the water supply source s the solutions depend only on total source discharge details of source configuration are irrelevant we require only that the total discharge be positive and finite and that the sources be contained in a finite region two dimensional results are found both for quasi linear and for general nonlinear flows the general results apply also to mixed saturated unsaturated flows the asymptotic three dimensional quasi linear results involve the same functional forms as analogous two dimensional flows in unbounded regions an upper bound on unsaturated total discharge exists for two dimensional systems use of the potential a line integral of the kirchhoff potential effectively reduces by one the dimensionality of the problems treated in one instance the dimensionality is reduced by two after a second integration

modeling of transpiration reduction in van genuchten mualem type soils 1 we derive an analytic expression for the matric flux potential m for van genuchten mualem vgm type soils which can also be written in terms of a converging infinite series considering the first four terms of this series the accuracy of the approximation was verified by comparing it to values of m estimated by numerical finite difference integration using values of the parameters for three soils from different texture classes the proposed four term approximation showed an almost perfect match with the numerical solution except for effective saturations higher than 0 9 including more terms reduced the discrepancy but also increased the complexity of the equation the four term equation can be used for most applications cases with special interest in nearly saturated soils should include more terms from the infinite series a transpiration reduction function for use with the vgm equations is derived by combining the derived expression for m with a root water extraction model the shape of the resulting reduction function and its dependency on the derivative of the soil hydraulic diffusivity d with respect to the soil water content is discussed positive and negative values of dd d yield concave and convex or s shaped reduction functions respectively on the basis of three data sets the hydraulic properties of virtually all soils yield concave reduction curves such curves based solely on soil hydraulic properties do not account for the complex interactions between shoot growth root growth and water availability

a geostatistical approach to the trickle irrigation design in a heterogeneous soil 2 a field test in a heterogeneous field in which the soil water properties vary under a deterministic uniform trickle irrigation system the midway soil water pressure head hc and the yield of a crop also differ from place to place these differences may in turn reduce the average over the field yield relative to the yield that would be obtained if the soil was uniform throughout the field a field experiment was conducted to test the hypothesis that this yield reduction may be eliminated by using a spatially variable trickle irrigation system twenty five plots 200 m2 each were established on a 30 m2 grid half of each plot was equipped with a standard trickle irrigation system with constant spacing between emitters of d 50 cm control plots and the other half was equipped with a trickle irrigation system for which the spacing between the emitters was selected by using the pertinent hydraulic properties the saturated hydraulic conductivity ks and the soil parameter according to the procedure of bresler 1978 as described in paper 1 russo 1983b values of hc measured at different times as well as the total fruit yield y of bell pepper capsicum frutescens var maor were used to estimate the seasonal and the spatial distributions of hc and the spatial distribution of y and their moments the variograms of hc and y were calculated and used to estimate their integral scales it was found that the use of a spatially variable d relative to the use of a uniform d did not change the seasonal behavior of hc but reduced the spatial variability in hc and y by 35 and 11 respectively and increased the integral scale of hc and y by 30 and 10 respectively but increased the average total fruit yield by only 1 9 the use of a spatially variable d reduced the dependence of y on hc this indicates that when the emitters are properly spaced it is not the water but other factors that most influence yield when a constant d was used the dependence of y of hc decreased with time this and the relatively good agreement between the values of hc measured at the initial stages of the growing season and those calculated in paper 1 demonstrate that the concept of hc is important in the early stages of the plant s growth when the root system is not fully developed both the theoretical paper 1 and the experimental results showed that although ks and as well as hc varied considerably in the field the spatial variability of the crop yield was relatively small this explains why the use of a spatially variable d essentially was not an improvement over the fixed d it is suggested that this study will be considered as a methodological one which can be adapted to solve practical problems associated with field spatial variability

large time solutions for groundwater flow problems using the relationship of small p versus large t 1 an approximate solution is useful if the corresponding analytical solution is complicated and difficult to accurately evaluate in the past the relationship of small p versus large t was commonly applied to the laplace domain solution and could successfully obtain a large time solution in the groundwater area the large time solution usually has a simpler form than the analytical solution and is much easier for estimating the transient behavior of the groundwater flow system however chen and stone 1993 pointed out that the use of this relationship might fail to yield the correct solution in calculating the wellbore flux for the constant head test problem later mathias and zimmerman 2003 indicated that a poor result was obtained by gerke and van genuchten 1993 when using the relationship of small p versus large t to derive the water transfer coefficient for the dual porosity media problem this note is to show the detailed mathematical derivations involved in the issues that chen and stone 1993 and gerke and van genuchten 1993 addressed and to ensure that the relationship of small p versus large t is correct to obtain a large time solution for transient groundwater flow problems

the quasilinear approximation for unsaturated porous media flow this paper gives a review of the quasilinear approximation an exponential relationship between hydraulic conductivity and moisture potential and the uses this has been put to since the time of its creation one of the major attractions of this approximation is that under steady regimes the highly nonlinear richards equation governing unsaturated flow can be linearized this simplification has enabled analytic and semianalytic solutions to be found for many unsaturated flow problems these are reviewed and the current research areas in which the quasilinear approximation is being used are discussed

state dependent anisotropy comparisons of quasi analytical solutions with stochastic results for steady gravity drainage anisotropy in large scale unsaturated hydraulic conductivity of layered soils changes with the moisture state here state dependent anisotropy is computed under conditions of large scale gravity drainage soils represented by gardner s exponential function are perfectly stratified periodic and inclined analytical integration of darcy s law across each layer results in a system of nonlinear equations that is solved iteratively for capillary suction at layer interfaces and for the darcy flux normal to layering computed fluxes and suction profiles are used to determine both upscaled hydraulic conductivity in the principal directions and the corresponding state dependent anisotropy ratio as functions of the mean suction three groups of layered soils are analyzed and compared with independent predictions from the stochastic results of yeh et al 1985b the small perturbation approach predicts appropriate behaviors for anisotropy under nonarid conditions however the stochastic results are limited to moderate values of mean suction this limitation is linked to a taylor series approximation in terms of a group of statistical and geometric parameters two alternative forms of the taylor series provide upper and lower bounds for the state dependent anisotropy of relatively dry soils

time dependent linearized two dimensional infiltration and evaporation from nonuniform and nonperiodic strip sources using a linearized equation describing unsaturated homogeneous and isotropic porous media flow a general two dimensional time dependent mathematical model is presented for infiltration and or infiltration evaporation from nonuniform and nonperiodic strip sources located at the soil surface the analysis is based on an exponential relationship between the unsaturated hydraulic conductivity and the soil water pressure head and also assumes a constant value for the derivative of unsaturated hydraulic conductivity with respect to water content in the mathematical analysis laplace transform and fourier analysis techniques are used simultaneously and a general equation in an integral form for the distribution of matric flux potential has been obtained the result of warrick and lomen 1976 for a single strip source is shown to be a special case of this general model the solution for infiltration from two strip sources is presented as another special case also a third special case is presented for infiltration from two strip sources and evaporation from another strip located in the middle part the horizontal and vertical flux components for these cases are presented all results are expressed in integral forms and calculated with a computer using a numerical integration method the solutions predict the matric flux potential and flux components as functions of space and time examples that illustrate the applicability of some of the solutions are presented these solutions are of interest in the design of both sprinkler and furrow irrigation systems the results may also be used for different purposes in civil and agricultural engineering applications

a multigrid model for steady flow in partially saturated porous media a numerical model for steady flow in a partially saturated porous medium is presented the model uses the multigrid solution algorithm for solving the system of equations resulting from the discrete approximation to the partial differential equation the model solutions checked favorably with an existing analytical solution used for a hypothetical problem with realistic soil characteristics the model has been shown to be very efficient for problems with the number of unknowns on the order of 10 000 and more specifically the multigrid scheme was shown to be 22 times faster in terms of cpu time than the line successive overtaxation method for the particular problem chosen in addition the required cpu time is shown to be a linear function of the number of unknowns within the solution domain this indicates that the convergence rate of the multigrid solution algorithm is independent of the grid resolution because of the significant improvement in computational efficiency the multigrid model can be used to simulate field scale partially saturated flow problems with very fine grid resolution as is often required by highly nonlinear soil characteristics further studies and application of the multigrid solution technique are suggested

the one dimensional approximation for infiltration in heterogeneous soils appendix is available with entire article on microfiche order from american geophysical union 2000 florida avenue n w washington dc 20009 document w91 004 2 50 payment must accompany order by using analytical solutions to the problem of one and two dimensional linearized unsteady infiltration from a strip source in a soil with saturated hydraulic conductivity varying exponentially in space explicit criteria for the validity of the one dimensional approximation are derived in terms of parameters describing the soil type the uniformity of the infiltrating source at the surface and the soil heterogeneity numerical solutions for the linearized infiltration problem are developed for any variation of hydraulic conductivity in space extensive simulations suggest that for a sizable strip source and moderate variation of hydraulic conductivity a set of independent neighboring soil columns represents well the multidimensional infiltration problem the possibility of defining sufficient parameters controlling the accuracy of the one dimensional approximation is discussed

a quantization property for blow up solutions of singular liouville type equations abstract motivated by the study of self dual vortices in gauge field theory cf vortices and monopoles birkhauser boston 1980 solitons in field theory and nonlinear analysis monographs in mathematics springer new york 2001 we consider a concentrating solution sequence u k satisfying u k z 2 k v k z e u k in b 1 z 2 k v k z e u k z 0 weakly in the sense of measures in b 1 where i k r k ii v k c 0 1 b 1 0 a v k b and v k a in b 1 we prove that necessarily 8 n 8 1 8 z the result is sharp as shown by explicit examples and should be compared with that obtained by li shafrir ind univ math j 43 4 1994 1255 concerning the situation where k 0 k n

a comparison of picard and newton iteration in the numerical solution of multidimensional variably saturated flow problems picard iteration is a widely used procedure for solving the nonlinear equation governing flow in variably saturated porous media the method is simple to code and computationally cheap but has been known to fail or converge slowly the newton method is more complex and expensive on a per iteration basis than picard and as such has not received very much attention its robustness and higher rate of convergence however make it an attractive alternative to the picard method particularly for strongly nonlinear problems in this paper the picard and newton schemes are implemented and compared in one two and three dimensional finite element simulations involving both steady state and transient flow the eight test cases presented highlight different aspects of the performance of the two iterative methods and the different factors that can affect their convergence and efficiency including problem size spatial and temporal discretization initial solution estimates convergence error norm mass lumping time weighting conductivity and moisture content characteristics boundary conditions seepage faces and the extent of fully saturated zones in the soil previous strategies for enhancing the performance of the picard and newton schemes are revisited and new ones are suggested the strategies include chord slope approximations for the derivatives of the characteristic equations relaxing convergence requirements along seepage faces dynamic time step control nonlinear relaxation and a mixed picard newton approach the tests show that the picard or relaxed picard schemes are often adequate for solving richards equation but that in cases where these fail to converge or converge slowly the newton method should be used the mixed picard newton approach can effectively overcome the newton scheme s sensitivity to initial solution estimates while comparatively poor performance is reported for the various chord slope approximations finally given the reliability and efficiency of current conjugate gradient like methods for solving linear nonsymmetric systems the only real drawback of using newton rather than picard iteration is the algebraic complexity and computational cost of assembling the derivative terms of the jacobian matrix and it is suggested that both methods can be effectively implemented and used in numerical models of richards equation

the one dimensional quasi linear intercept on cumulative infiltration graphs the asymptotic time behavior of one dimensional infiltration of water into a class of idealized soils is determined theoretically first we assume that all flows are one dimensional and investigate transient behavior between a steady initial state and another final steady state second soil properties are idealized by making the quasi linear assumption of j r philip 1969 that relative permeability is related exponentially to capillary pressure which linearizes the flow terms third and least importantly we also assume later that changes in volumetric water content vary as a power of relative permeability the transient flow equations then reduce to a nonlinear parabolic equation with relative permeability as the primary variable one dimensional cumulative infiltration is then asymptotically a linear function of time whose gradient depends on the saturated soil conductivity and whose intercept equals the change in field capacity or near surface volumetric water content divided by the quasi linear inverse length parameter the equations for both the gradient and intercept for this asymptotic linear function are independent of time and independent of the relationship between volumetric water content and relative permeability finally the three assumptions above imply that the time constant representative of the duration needed to attain near steady conditions essentially equals the sum of a term independent of the initial conditions plus a term depending only on differences between the initial and final conditions

exact solutions of the richards equation with nonlinear plant root extraction the richards equation commonly used to model water flow in unsaturated soils is highly nonlinear thus making it very challenging to solve analytically for situations meaningful in practical applications the inclusion of realistic forms of root water uptake rates in this equation adds complications in deriving exact solutions this study provides for the first time analytical solutions of the richards equation with a sink term nonlinearly dependent on soil water content these solutions are applied to irrigation furrows using cartesian coordinates and irrigation from a circular plate in cylindrical coordinates

aspects of quasilinear infiltration from surface sources especially the case α 0 three recent papers on steady two dimensional infiltration from spatially variable surface sources may lead to confusion about the quasilinear analysis the papers assert that putting 0 in quasilinear solutions for the stream function gives solutions for saturated flows in fact putting 0 yields the precisely opposite case of purely capillary driven unsaturated flows the argument of the papers is that for 0 the quasilinear equation for the stream function is reduced to laplace s equation which is obeyed by the stream function for saturated flows the kinematics of all two dimensional solenoidal irrotational flows is however such that their stream functions obey laplace s equation and the physical meaning and indeed the physical existence of any flow can be determined only from supplementary dynamic information about the velocity potential

multidimensional steady infiltration with prescribed boundary conditions at the soil surface the present work explores a class of analytical solutions of steady moisture movement in unsaturated porous media characterized by an exponential dependence of the hydraulic conductivity the green s function method is used to derive a general analytical model which can handle multidimensional steady infiltration problems in a semi infinite medium with arbitrary boundary conditions and root uptake forcing functions and for various simple source geometries the power of the green s function method is such that a number of new problems were amenable to analysis and solutions to problems already solved were obtained in a form better suited for numerical computation of special importance is the capability of the analytical model to handle the difficult problem of mixed boundary conditions in an approximative fashion the general solution is expressed in integral form from which particular analytical solutions pertaining to cases of surface and subsurface irrigation root uptake and isothermal evaporation can be easily deduced new solutions include the case of surface sources under a prescribed exponential head and flux distribution an inverse power flux distribution and periodic root uptake the inverse power flux distribution allows the approximate modeling of constant potential sources with evaporation at the nonwetted surfaces it is shown that all two and three dimensional infiltration solutions are essentially integrals of two functions one dependent on the geometry and distribution of the source and the other dependent on the boundary condition at the land surface near field and far field approximations have also been obtained and simple algebraic equations have been derived for the flux and potential beneath the centerline of the source the proposed model offers the analyst significant flexibility in deriving results and analyzing infiltration phenomena of practical interest in civil and agricultural engineering

gromov s compactness theorem for pseudo holomorphic curves we give a complete proof for gromov s compactness theorem for pseudo holomorphic curves both in the case of closed curves and curves with boundary

multidimensional linearized nonsteady infiltration toward a shallow water table the green s function method is used to derive a general analytical model pertaining to multidimensional nonsteady infiltration toward a shallow water table with arbitrary initial conditions boundary conditions and root uptake forcing functions and for various simple source geometries the general green s function solution allows the derivation of particular analytical solutions pertaining to cases of surface and subsurface irrigation evaporation root uptake and moisture redistribution the model assumes that the hydraulic conductivity function is an exponential function of the water pressure and a linear function of the moisture content one dimensional solutions are obtained for time dependent distributions of the flux at the land surface and of root uptake in the subsurface two and three dimensional solutions are also derived for point and line sources in the domain for a strip a rectangle and a disc source at the land surface and for two and three dimensional moisture redistribution and root uptake most of the solutions are a product of two terms one term is a function of depth while the other term is a function of time and the lateral coordinates numerical results show the effect of the water table and soil type on the two dimensional moisture movement from a strip source

blow up analysis for solutions to neumann boundary value problem abstract in this paper we will analyze the blow up behavior of solution sequences satisfying a conformal invariant equation defined on a compact 2 dimensional surface m g with boundary we will provide some accurate point wise estimates for the profile of these sequences

multidimensional steady infiltration to a water table we address the question of the applicability of the body of solutions for quasi linear multidimensional steady infiltration with perfect drainage to cases with a water table or by inference a sloping impermeable base at a finite depth beneath the water supply source the exact solutions for steady quasi linear point and line sources over a water table are our point of departure with their aid we construct approximate solutions to the general problem of three dimensional axisymmetric and two dimensional steady infiltration from finite water supply sources at fixed moisture potential to a water table at finite depth we take as illustrative solutions disc and strip shaped water supply sources when the depth from the source to the water table or the base is of order 4 sorptive lengths perfect drainage solutions yield discharge estimates of ample accuracy when the depth is less our approximate analysis gives in favorable cases the requisite correction in less favorable cases it gives the order of magnitude of the correction the study is facilitated by a method of exponentially weighted images applicable to the quasi linear flow equation

analytical solution to transient richards equation with realistic water profiles for vertical infiltration and parameter estimation a general analytical model for one dimensional transient vertical infiltration is presented the model is based on a combination of the brooks and corey soil water retention function and a generalized hydraulic conductivity function this leads to power law diffusivity and convective term for which the exponents are functions of the inverse of the pore size distribution index accordingly the proposed analytical solution covers many existing realistic models in the literature the general form of the analytical solution is simple and it expresses implicitly the depth as function of water content and time it can be used to model infiltration through semi infinite dry soils with prescribed water content or flux boundary conditions some mathematical expressions of practical importance are also derived the general form solution is useful for comparison between models validation of numerical solutions and for better understanding the effect of some hydraulic parameters based on the analytical expression a complete inverse procedure which allows the estimation of the hydraulic parameters from water content measurements is presented

error analyses of simplified unsaturated flow models under large uncertainty in hydraulic properties the functional relationships between soil hydraulic properties k and are needed for analyzing unsaturated water flow in soils and are generally of a nonlinear form these relationships estimated from field measured data usually show large variability because of measurement errors and natural spatial heterogeneity exhibited by many field soils a probabilistic approach is often employed to account for this uncertainty in parameters whereby the coefficients in the resulting partial differential equation are treated as random quantities usually characterized by their moments the objective of this analysis is to evaluate the errors resulting when simplified solutions based on simple forms of the hydraulic properties are used rather than nonlinear and numerically complicated models as an example of a parametric functional relationship for k we adopt the brooks corey formulation in which a single parameter is considered as random to emulate the spread encountered in practice for data that show a large enough spread and when one is interested in the mean behavior of k or simple flow models such as the green ampt model and the linear model are proposed as alternatives to the more accurate but numerically demanding richards equation to compare models scaled error definitions are introduced and are expressed as a percentage of the saturated hydraulic conductivity of the soil these time dependent errors are then computed under various boundary conditions in general the green ampt solution is found to yield smaller errors than the linear model when compared with solutions from richards equation it is concluded that the use of simplified models is justified when field data exhibit sufficient spread and a tight bound cannot be placed on the parameter estimates

prescribing curvature on compact surfaces with conical singularities we study the berger nirenberg problem on surfaces with conical singularities i e we discuss conditions under which a function on a riemann surface is the gaussian curvature of some conformal metric with a prescribed set of singularities of conical types

steady two and three dimensional flows in unsaturated soil the scattering analog an exact analog exists between steady quasi linear flow in unsaturated soils and porous media and the scattering of plane pulses and the analog carries over to the scattering of plane harmonic waves numerous established results and powerful techniques such as the watson transform are thus available for the solution and understanding of problems of unsaturated flow these are needed in particular to provide the asymptotics of the physically interesting and practically important limit of flows strongly dominated by gravity with capillary effects weak but nonzero this is the limit of large s with s a characteristic length of the water supply surface normalized with respect to the sorptive length of the soil these problems are singular in the sense that ignoring capillarity gives a totally incorrect picture of the wetted region in terms of the optical analog neglecting capillarity is equivalent to using geometrical optics with coherent shadows projected to infinity the paper deals specifically with steady infiltration from circular cylindrical and spherical cavities the asymptotic methods prove remarkably accurate even far from the limit the results replace and explain previous semiempirical estimates of the limiting behavior one notable result is that the depth of the penumbra effectively wetted region for the cylinder is 128 times the depth for the sphere confirming and supplementing previous studies an odd byproduct is that we correct a long standing classical result in scattering theory the scope for extending these methods to flows in other geometries to heterogeneous soils and generally to linear convection diffusion processes is indicated briefly

linearized unsteady multidimensional infiltration the parameters in the linearized multidimensional unsteady equation for unsaturated flow in homogeneous soils can be chosen so that small time flow rates are given correctly with the quasi linear steady solution as the large time limit matched in this way the linearized equation offers insight into the time course of multidimensional infiltration the laplace transform reduces the equation and its governing conditions into forms similar to those for quasi linear steady flows the problems of unsteady infiltration from buried spherical and cylindrical cavities are solved in many cases the solutions are approximately reducible to the product of the steady solution and a function only of time and radial coordinate theorems show that approximate product solutions apply not only to these configurations but generally to multidimensional cavities of arbitrary shape multidimensional infiltration from finite supply surfaces involves two characteristic lengths and two characteristic times one set deriving from the capillarity gravity interaction and the other from the capillarity geometry interaction we explore the influence of the two characteristic times on the time scales of approach to steady moisture distribution near the supply surface and to steady discharge the analysis provides explanation of the relatively rapid approach to the steady state observed in three dimensional infiltration from small supply surfaces

vanishing pohozaev constant and removability of singularities conformal invariance of two dimensional variational problems is a condition known to enable a blow up analysis of solutions and to deduce the removability of singularities in this paper we identify another condition that is not only sufficient but also necessary for such a removability of singularities this is the validity of the pohozaev identity in situations where such an identity fails to hold we introduce a new quantity called the pohozaev constant which on one hand measures the extent to which the pohozaev identity fails and on the other hand provides a characterization of the singular behavior of a solution at an isolated singularity we apply this to the blow up analysis for super liouville type equations on riemann surfaces with conical singularities because in the presence of such singularities conformal invariance no longer holds and a local singularity is in general non removable unless the pohozaev constant is vanishing