Donald L. Baker

Ph.D. Soil Physics, M.S. Agricultural Engineering,
M.S. Ocean Engineering, B.S. Electrical Engineering

answering machine:


Selected Experience:

Currently living on disability and working to improve artwork. Can work about half time. Need $20/hr plus health benefits on half time to leave disability, flexible hours due to medical reasons, safe and positive workplace.

Writing research papers and sending out resumes. Jan 2002 to Jan 2003. Wrote and submitted papers on earthen dam sensors, dam breach flow equations and vadose zone flow modeling, and a proposal for a coffee table book. Certified by State of Oklahoma as a person with a severe disability due to injuries delivered by a drinking driver in 1985.

Part time Postdoctoral Associate at the UDSA-ARS Plant Sciences and Water Conservation Research Laboratory, Hydraulic Engineering Research Unit. Soil Scientist GS-0470-12. May 2001 to Jan 2002. Built tensiometer/piezometer sensors to track the rise of the phreatic surface in a dam overtopping experiment. Analyzed the head-discharge data from physical model experiments for over 380 different geometries of simulated dam breach. Developed calibrated discharge equations for the cases of a fully supported jet in a trapezoidal channel and a free falling, fully aerated nappe. Discovered that the transition from free-falling to flow through a partially-supported jet may be expressed as a difference power term with an offset in reservoir entrance head.

Part time Technical Assistant to Dr. H.D. Scott, Agronomy Dept., UAF (Passed away suddenly, 2006, while working at Mount Olive College, NC). May 2000 to present. Calibrated an infiltration model with sparse field and laboratory data sets using a unique minimal-parameterization approach, Fletcher-Reeves optimization and an objective function based on the probability that model estimates fit the mean values of replicate field sensor readings. Some preliminary work available at The calibration report not yet reviewed. Developed www-based instructional materials for students in soil physics on UAF subcontract to USDA grant to Dr. D. Nofziger at Oklahoma State, including a new quasi-analytic exact solution to Richards' equation in 1-D infiltration. The new approach transforms Richards' equation into a 2nd order ODE, solvable by the shooting method, requires only that the diffusivity and conductivity be expressible in a Fortran subroutine, is confirmed for constant-head and constant-flow inflow conditions, and makes itself accessible to students without a background in PDEs. Preliminary work available at Dr. Scott's University of Arkansas web site (no longer existent) and at

Independent research into Darcian means and proposal writing. May 1999 to May 2000. Development of software tools to calculate and utilize Darcian means. Recent papers (see below) demonstrate that common methods of calculating intergrid hydraulic conductivity (water modeling) or relative permeability (oil modeling) in unsaturated flow can often produce violations of mathematical principle. Violations of the min-max principle for elliptic boundary value problems (steady-state flow) can produce extremely non-physical spikes and oscillations under similar conditions in transient flow models. Darcian means alleviate this problem and produce more accurate estimations of flow that converge faster to fine-grid cases. See the research papers on

Part time Technical Assistant to Dr. H.D. Scott, Agronomy Dept., University of Arkansas, Fayetteville, AR. July 1998 to April 1999. Assisted in the review and revision of a new undergraduate textbook on Soil Physics, including writing of a section on modeling heat, water and solute flow with finite differences. Developed a laboratory for Dr. Scott's Soil Physics Lab, teaching the design calibration and use of a wet and dry bulb air temperature sensor, made from common plumbing and electronic parts.

Small Business Innovation Research Award No. DE-FG02-97ER82329, U.S. Department of Energy, "Developing Darcian means to correct order-of-magnitude subsurface flow errors in models". See abstract of "Developing Darcian means..." Report DOE/ER/82329-2, below.

Technical Assistant to Dr. H.D. Scott, Agronomy Dept., University of Arkansas, Fayetteville, AR Responsible for the improvement debugging, calibration and validation of a model predicting the effects of poultry waste fertilizer on the vadose zone. Click here for an unfinished draft of, Scott, H.D., Baker, D.L., Cochran, M.J. and Smartt, J., Modeling the mineralization and volatilization of nitrogen in poultry litter applied to tall fescue. Click here for a draft input/output specification for the poultry litter infiltration model poulit. Provided engineering support to assemble and install weather and subsurface instrumentation and to collect data in a rice field. (I made a neat little instrument shack out of a pickup camper that withstood 70 mph gusts.)

Ph.D., Soil Physics, 1994. Agronomy Department, Colorado State University, Ft. Collins, CO 80523. Wrote five research papers (now published) before Dissertation. Research recognized by the U.S. Nuclear Regulatory Commission (FY 1994 Phase I SBIR Proposal #I-2 Evaluation) as "at the leading edge of the state of the art in the application of numerical methods to flow and transport through variably-saturated soil and rock". GPA 3.74/4 in Ph.D. courses.

M.S. Agricultural Engineering, 1989, Agricultural Engineering Dept. Colorado State University, Ft. Collins, CO, 80523 Developed a new soil probe in support of Advisor's research contract to mark the passage of the wetting front in irrigated agricultural soil. Published one paper with Advisor, Dr. Paul D. Ayers, in peer-reviewed journal. GPA 3.56/4.

Electronics Engineer, GS-855-11, U.S. Naval Oceanographic Office, NSTL, MS Designed a microprocessor-controlled magnetometer data buoy as Principle Engineer on a Defense Mapping Agency development project in support of magnetic mapping surveys, in alliance with Mississippi State University. Designed and built a remotely-tuned preamplifier for the Geometrics G-801 magnetometer. Provided assembly language and hardware design support (Mot. 6800) for the NAVO Shipboard and Airborne Digital Acquisition Recorders.

Electronics Engineer - A, Computer Sciences Corp., NSTL, MS While unsupported by contract money, taught self to use a Tektronix 8002a Microprocessor Development System. Then used it to design and develop most of the hardware and all of the 6800 assembly-language software for the NOAA Data Buoy Office Aids-to-Navigation-Buoy Environmental Sensing System (ANBESS) test package. Published paper on project in Oceans '80 Conference.

Assistant Marine Scientist B, Virginia Institute of Marine Science, Gloucester Point, VA Developed a system to translate 10.2 kHz Omega navigation signals to 2.398 MHz, while preserving phase information, to track drifting buoys under a NASA-NGL grant. Under Bureau of Land Management contracts, procured, integrated, maintained and calibrated state-of-the-art physical oceanography instrumentation in support of scientists on preliminary surveys of the mid-Atlantic coastal oil lease lands. Wrote FORTRAN data post-processing programs to convert raw data into engineering-unit water column profiles. Contributed to final reports. Did similar work on an EPA river study.

M.S. Ocean Engineering, 1976, Civil Engr. Dept., U. Massachusetts, Amherst, MA Worked on Dept. of Interior contract WR-B021-MASS. Did final debugging and field maintenance of an automated weather recording station at Quabbin Reservoir. Wrote FORTRAN programs to process raw weather data for input into reservoir stratification model. Thesis work cited in final report. GPA 3.55/4

Additional studies at the University of Arkansas, Fayetteville. M.S. in EE Control Systems not completed due to a massive collapse of the aerospace industry and resulting layoffs in the early 1970s. Went on to second program at UMass. A later review of mathematics at UAF made studies of modeling partial differential equations at CSU possible.

B.S. Electronic Engineering, 1968, Massachusetts Institute of Technology, Cambridge, MA GPA 4.02/5

Full transcript copies in Adobe pdf format are available by clicking on the GPA numbers.

   This is an ongoing project, begun as a demonstration of research product for a small business, Aquarius Engineering. Although the business is not currently operating, this web site continues to serve as an outlet for my current research and writings. Some pages are better than others, but I take a certain amount of pride in producing work that is rich in graphs and equations. Some of the papers are preprints of peer-reviewed journal articles. All serve to establish professional precedence for my work, and to allow my peers to review it at their leisure.

  In earlier work, the most popular page was a student tutorial in finite difference modeling, uploaded about March 2000, visited by people from all over the Internet world. It then drew from 100 to 150 visits a month. Currently the most popular pages are a set of soil physics tutorials.

Personal Interests

Photography and guitar artwork, thumb-thumping guitars, written satire.


Research and Engineering Publications and Comments (some with abstracts):

Baker, Donald L., 2006, General Validity of Conductivity Means in Unsaturated Flow Models, ASCE Journal of Hydrologic Engineering, Nov/Dec 2006, 11(6):526-538.

Baker, D.L., 2003, Sensing breaches in earthen dams, IEEE Instrumentation & Measurement Magazine, June 2003, pp 13-18.

Baker, D.L., 2002, Equations to fit partially supported jets in models of dam breach.

Head-discharge equations are fitted to a set of physical models of dam breach for the cases where there is a drop below the jet but apparently no aeration. Investigations with simulated annealing and a commercial curve-fitting program suggest that the reductions in flow from that of a free-falling, aerated jet can be fitted with equations in a dimensionless scaling system derived from the Buckingham Pi theorem and integration of the ideal weir equation. A set of linear corrections are fitted to data for a breach width of 0.406m and applied to data for breach widths of 0.203m and 0.813m, to check scaling.

Baker, D.L., 2002, Head-discharge equations to fit a set of physical models of dam breach.

Head-discharge equations are fitted to a set of physical models of dam breach, covering 378 different geometries. The method of fitting consists of three aspects: 1) the integration of the ideal weir equation over the geometric boundaries of the notch, 2) use of the Buckingham Pi theorem to remove data with explainable deviations from the fitting, and 3) the use of simulated annealing to do the fitting, with an objective function of mean absolute relative error. The equations are applied to an additional 60 geometries to check scaling.

Baker, D.L., 2002, Dispersive errors induced by a non-Darcian mean in a model of unsaturated flow. Rewritten version of 1999/2000 work, Dispersive errors induced by a non-Darcian mean in a model of unsaturated flow.

A simple three-point grid test is developed to show that some intergrid hydraulic conductivity means do not have mathematical and physical validity for unsaturated steady-state flow in a homogeneous porous medium. Several commonly-used means are demonstrated to violate the min-max principle for elliptic boundary value problems (steady-state flow). An example using the arithmetic mean shows that the violations of the min-max principle that it causes are associated with non-physical spikes and oscillations in a model of vertical flow. By comparison, a parallel model using the same time and space steps with an analytic Darcian mean demonstrates stable, superior behavior.

Baker, D.L., 2002, Technical note: Application of the Buckingham Pi theorem to dam breach equations.

Before the recent collapse of a major corporation, a Fortune magazine journalist asked a simple question, "How do you make money?" In this business there is an equally simple question, "How do you do math?" Sometimes an assemblage of dimensionless variables is presented as a case of dimensional analysis, misapplying the Buckingham Pi theorem. The theorem, its usage and its limitations are reviewed in the context of water flow through a dam breach model, taken from a set of measurements of flow through trapezoidal notch in a trapezoidal reservoir embankment made of plywood in a flume.

Baker, D.L., 2002, A Class of Exact Numerical Solutions to Richards' Equation in Vertical Infiltration.

Similar to: A new quasi-analytic exact solution of Richards' equation. Richards' equation is transformed by means the Boltzmann variable, B, relative saturation, S, and fractional flow, F, into a second-order ordinary differential equation of F(S). This formulation assumes that a leading edge of the wetting front exists, at and beyond which initial conditions occur. The resulting ODE boundary value problem is solved by the shooting method, using the 4th-order Runge-Kutta-Nystrom numerical method, or better, finishing with high-order polynomial extrapolation of the value of dF/dS at the end point. Except at the boundaries, the solution requires only that the diffusivity and conductivity be expressible as a continuous function of S in a Fortran subroutine. A transformation of dF/dS to B recovers spatial information. The quasi-analytic numerical exact solution results, for constant-head and constant-flow inflow boundary conditions, are confirmed by finite difference models of Richards' equation.

Baker, D.L. and H.D. Scott, 2000, 2001, draft publications on development of a 1-D infiltration model with sparse data, a new quasi-analytic exact solution of Richards' equation, and soil physics tutorials at and

Baker, D.L., 2000, Why Darcian means are not a "first-order" method, unpublished

Baker, D.L., 2000, Program Manual, fkysa, a program to fit pressure-conductivity relations with simulated annealing, unpublished.

Baker, D.L., 2000, Two fitting functions for pressure-conductivity relations with links to Darcian means, unpublished.

The exponential fitting function for pressure-conductivity relations, K = Ks*(a+1)/(a+exp(l·y)), a = exp(l·yd), where Ks, l and yd are fitting parameters, has an algebraic equation for the vertical flow Darcian mean that can be solved by iteration with one or two zeros. The power fitting function, K = Ks/(1+y/yk)h , where Ks, y and yk are fitting parameters, has no such equation, but has Darcian means with some simplifying characteristics that make it favorable for consideration. The exponential function fits conductivity data for coarse-textured soils better and the power function fits data for fine soils better. Pressure-conductivity data for 58 soils in the 1996 UNSODA data set, measured by the double-plate, steady-state and centrifugal methods, were chosen for test data. Five methods, with single or weighted double fitting functions, were fitted by simulated annealing to the measure data sets, by minimizing the mean absolute relative error (mre). The best of the best of five fits produced mre = 0.0058. The worst of the best produced mre = 0.416 on Weld clay #4210, a very noisy set.

Baker, D.L., 2000, Some notes on the summing properties of Darcian means, unpublished.

Thus far, Darcian means have been developed in explicit, analytic form for only the exponential pressure-conductivity relation, and only approximately for other analytic pressure-conductivity relations. If they are to be applied to measured and tabulated physical data, then it would be convenient if a summing property existed. If the weighted sums of constituent pressure-conductivity relations directly produced the weighted sums of the associated Darcian means, then a set of basis functions could easily be developed that could describe in composite any tabulated data set. This paper finds that under current assumptions for Darcian means, the summing property exists for horizontal flow, but exists only in the limit, or for special cases, for flow under the influence of gravity. The errors produced by assuming that the summing property is valid in these circumstances can be significant, but might be less under more favorable circumstances, or less in comparison to other current methods of estimating intergrid conductivity and flow means.

Baker, D.L., 1999, Min-max violations produced by intergrid conductivity means, unpublished.

In the simplest model of vertical ground water flow, with one grid point between two boundary conditions, there are few differences between numerical methods, whether finite differences or finite elements, with either explicit or implicit time-stepping. If a method of estimating intergrid hydraulic flow from conductivity means is inaccurate or invalid in this case, it may reasonably be discredited for use in more complicated cases. This study applies thirteen different intergrid conductivity means to the derived hydraulic parameters for three different porous media associated with the Yucca Mountain nuclear waste site. It provides minimum and maximum errors in predicted pressure and flow in the three-point grid compared to a high-accuracy solution to the steady-state flow equation, violations to the minimum-maximum principle for elliptic boundary value problems that occur, and plots of the relative ratio of predicted flow to true steady-state flow over ten orders of magnitude of boundary condition relative conductivity values. Darcian intergrid conductivity means apply the exact solutions to steady-state flow problems to transient flow problems piecewise in time and space. In this study, an adjusted approximate Darcian integral mean proves superior to all the others.

Baker, D.L., 1999, Dispersive errors induced by a non-Darcian mean in a model of unsaturated flow, , draft available at

A simple three-point vertical grid test is developed to show that the arithmetic intergrid hydraulic conductivity mean does not have mathematical physicality for unsaturated flow in a porous medium with an exponential pressure-conductivity relation in all but trivial special cases. Compared to the analytic Darcian mean for this medium, the arithmetic mean can overestimate hydraulic conductivity for dry-over-wet conditions and underestimate conductivity for wet-over-dry conditions. This can give rise to non-physical separation and clumping of the mass flow, and can produce astable behavior in a model of a long, vertical fracture. A numerical experiment shows that in a vertical unsaturated flow model with space steps significantly larger than the "displacement pressure", yd, the arithmetic mean produces dispersive oscillations in the leading edge of an initial-condition wet pulse and a dry spike of excessively increased matric suction at the trailing edge. The trailing dry spike tends to increase as the logarithm of time in the model, out to approximately 2.63 years of model time, and can be delayed in onset by transient conditions. This error does not bode well for such models attempting to the predict flow of water and solutes on the scale of thousands of years. By contrast, a parallel model using the analytic Darcian mean demonstrates smooth, well-behaved response at all the model time and space step sizes.

Baker, D.L., 1999, Darcian means between saturated and unsaturated modeling points, unpublihsed.

Previously, Darcian interblock hydraulic conductivity means for vertical unsaturated flow were expressed between modeling grid points that were both unsaturated. In this paper Darcian means are extended to circumstances where one of the points has a saturated or positive pressure head. The method used involves a relatively simple Newton-Raphson iteration. Comparison examples are given for an exponential conductivity relation that has analytic solutions, true Darcian means for a Brooks-Corey relation for fracture flow developed by numerical solution, and an analytic approximation to Darcian means for the same Brooks-Corey relation. The characteristics of Darcian means common to both exponential and Brooks-Corey conductivity relations are noted. These additional analytic and iterated solutions for the exponential conductivity relation make it possible for other investigators to study some of the characteristics of Darcian means with commonly available desktop computers in a wider variety of numerical modeling situations.

Baker, D.L., 1999, Some methods to examine the validity of Darcian means, unpublished.

Many current models of vertical unsaturated flow, some of which are used in models of transport of nuclear and hazardous waste, may be deficient or suspect on a very basic level. Fixed, standard interblock hydraulic conductivity means, such as the arithmetic, upstream or geometric, usually have no direct connection to either the porous medium hydraulic properties or model vertical space step size. In conjunction with previous work, this paper demonstrates, for typical fracture flow in a Yucca Mountain, Nevada, medium, that the arithmetic mean is demonstrably nonphysical, even at small grid spacings, both by mathematical and model development. Where the arithmetic mean overpredicts the advance of a wetting front by up to 18.75%, a new approximation to Darcian means is shown to hold the advance to within 0.36% of the fine grid solution over a range of 8 to 470 adaptive grid segments in the wetting front. Thus criticisms that Darcian means are nonphysical and invalid are refuted.

Baker, D.L., 2000, A Darcian Integral Approximation to Interblock Hydraulic Conductivity Means in Vertical Infiltration, Computers & Geosciences, 26(2000):581-590.

Previous work has demonstrated that using non-Darcian interblock conductivity means in mass-conservative models of unsaturated infiltration can produce mass infiltration errors up to twenty times larger than those that mass conservation corrected. But the method to generate true Darcian means from solutions to elliptic boundary condition problems is too computationally intensive to be used effectively in models, especially with variable porous media. A previously introduced approximation did not estimate Darcian means well near saturation for relations, such as a van-Genuchten form, with strong curvature away from the asymptote near saturation. This paper introduces a generally more accurate approximation to Darcian means, using the examples of matrix and fracture flow conductivity relations for Topopah Spring welded volcanic tuff (Yucca Mountain, Nevada, USA), and compares it to the geometric and arithmetic means for matrix and fracture flow, respectively. While the new Darcian Integral Mean approximation may have significant computational overhead of its own, compared to standard means, as well as imperfections, it offers to other investigators a reasonable tool to examine the properties of Darcian means in running models.


Baker, D.L., M.E. Arnold and H.D. Scott, 1999, Some Analytical and Approximate Darcian Means, GROUND WATER, 37(4):532-538, July-August 1999.

In the finite difference modeling of unsaturated infiltration by Richards' equation, standard interblock conductivity means like the arithmetic, geometric and harmonic have been shown to produce flows different from true Darcian by up to orders of magnitude (Warrick, 1991; Baker, 1995a, 1998). This may have significant impact on models used for contaminant transport of nuclear and other hazardous waste. Previously, it has been impractical to calculate Darcian means in a running model. This paper introduces the analytic solution for Darcian means for the case of an exponential relative conductivity relation, and one possible approximation for general use, a piecewise Brooks-Corey Darcian (PBCD) mean. It compares the PBCD mean to the arithmetic mean in a modern mass-conservative numerical model (Celia, et al., 1990) for vertical space steps from 0.8 to 105 cm, using soil parameters from Haverkamp, et al., (1977). The results demonstrate that even in a model with mass balance good to one part in 108, non-Darcian flow errors are alive and well, costing up to 50% of the total mass infiltrated with the arithmetic mean and large model space steps. The new PBCD mean reduces that figure to 0.5% for this example soil.


Baker, D.L., M.E. Arnold and H.D. Scott, 1998, Some comparisons of new time steps for mass-conservative infiltration models

Modern mass-conservative models of unsaturated infiltration of water into dry soil lend themselves to a simple feedback algorithm that adapts the time step size to maintain a given number of iterations to mass balance. Four cross-combinations of methods, fixed and adaptive, first-order and second order Runge-Kutta, are compared for equivalent time step sizes over a range of about 0.1 to 500 s. Measures of the error in total mass infiltrated, the total number of Newton iterations, and the computational efficiency (error times effort) are used to evaluate the accuracy and utility of the methods. The mass infiltration error was found to be roughly equivalent between fixed and adaptive time steps; going from first-order accurate in time to second-order times steps is more effective in reducing the overall cumulative infiltration error. But for a range of small time steps, or maximum iterations, the new adaptive second-order method has an order of magnitude better error times effort figure than the other three methods. There is a similar effect for adaptive first-order over fixed first order, but it is much less pronounced. These results demonstrate that there are still gains to be made in the accuracy and computational efficiency of mass-conservative methods for numerically simulating infiltration.

Baker, D.L., M.E. Arnold and H.D. Scott, 1998, A piecewise Brooks-Corey approximation to Darcian interblock conductivity means for vertical unsaturated flow, unpublished. (This paper was divided into the next two papers above for publication. It contains added information, including the concept of model conductivity state space. - DLB)

In one example of vertical infiltration into dry soil, an iterative numerical solution to Richards' equation, with head-based time derivatives, produces mass balance errors on the order of 2% and errors in the mass infiltrated on the order of 2.5%. The modern, mass-conservative numerical solution to unsaturated flow developed by Celia, et al., (1990) reduces mass balance error to an arbitrarily small amount. But for model space steps, dx, significantly larger than the conductivity relation "displacement head", Celia's method with the arithmetic interblock conductivity mean misplaces up to 50% of the mass infiltrated. It also produces oscillations and some regions of non-convergence. In this example, the oscillations are eliminated and the mass infiltration error is reduced to less than 0.5% by using a piecewise Brooks-Corey Darcian (PBCD) approximation to a true Darcian mean. Darcian means are developed from the implications of the modeling assumption of constant Darcian flow between grid points for the duration of a time step. The Darcian approach shows that as dx increases, the true Darcian mean transforms, from the integral of conductivity with matrix suction, to the conductivity of the upper grid point in a pair, the k2-mean, or up-gravity mean. Although it produces about twice the mass infiltration error of the PBCD mean, its simplicity makes it a viable candidate, and explains the popularity of the upstream mean, which is the same for wet-over-dry conditions. The concept of the model conductivity state space is also introduced as a visualization tool. It can show how the model progresses through interblock conductivity mean errors with time. It considers the positions of pairs of adjacent grid point conductivities in the space of the chosen mean.


Baker, D.L., 1998, Developing Darcian means in application to Topopah Spring welded volcanic tuff, Report DOE/ER/82329-2 to the U.S. Department of Energy under Award No. DE-FG02-97ER82329, 118p. Click here for extracts in html format.

For the proposed Yucca Mountain site for a nuclear waste depository, the prediction of flow for the transport of contaminants is critical. Common standard methods of calculating vertical, unsaturated, single-phase flow between model grid cells can produce predicted flows in error up to six orders of magnitude, in the wrong direction, or with effectively misplaced hydrostatic conditions. This raises valid questions about assuring the eternal safety of any depository through such modeling. Calculating Darcian mean flows between grid cells, derived directly from the modeling assumptions, depending on characteristics of both the medium and the model, addresses this issue. This provides a new mathematical and modeling framework with which to judge the appropriateness of such means and their approximations. Using a presumed composite relative conductivity relation for Topopah Spring welded tuff, this work develops true Darcian means over many modeling regimes, and the viable approximations for general exponential and Brooks-Corey conductivity relations necessary for running models. This demonstrates how very differently matrix and fracture flow must be treated in a model. It illustrates how a composite relation itself may be responsible for poor-posedness in the problem and suggests that separate, linked models for fracture and matrix flow may be in order.


Baker, D.L., 1997, Comment on Modeling variably saturated flow and transport into sandy soil (Journal of Hydrology 186 (1996) 315-325), Journal of Hydrology 199:213-214.

Baker, D.L., 1995, Darcian weighted interblock conductivity means for vertical unsaturated flow, GROUND WATER, May-Jun 1995, 33(3):385-390.

The true Darcian interblock conductivity mean for vertical unsaturated flow, Kv, is derived from Darcy's law, q/Ks = Kv (dH/dx), and the assumption that flow is constant between grid centers. It can be expressed as Kv = (Dx·Kx - Dp·Kh) / (Dx - Dp), where Kx is the gravity flow mean, Kh is the capillary flow mean, and H = x-p (total head = elevation - matric suction). Plotting contours of Kx, Kh and Kv in the plane of adjacent grid center conductivities, (kr1,kr2), shows that a component of Kx behaves like a simple weighted mean, Kw = w kr2 + (1-w) kr1, where w is a nonlinear, monotonic function of the grid separation, Dx. Results from a mass-conservative model of infiltration into dry sand are compared by substituting: 1) the arithmetic mean, Ka = (kr1+kr2)/2, for Kv, 2) Kw for Kv, 3) Ka for Kx in Kv (called Km), and 4) Kw for Kx in Kv (called Kmw), over a range of the ratio of Dx to the conductivity displacement pressure, pk, from 0.01 to 1.6. The simulation for Dx/pk = 0.002 is taken to be the exact solution. Kw and Kmw reduce the gross cumulative net inflow error generated with Ka and Km from 3.84 to 8.84 times. Kmw performs consistently better in terms of gross water content distribution error by up to a factor of three.


Baker, D.L., 1995, Applying higher-order DIRK time steps to the "modified Picard" method, GROUND WATER, Mar-Apr 1995, 33(2):259-263.

The "modified Picard" iteration method, which offers global mass conservation, can also be described as a form of Newton's iteration with lagged nonlinear coefficients. It converges to a time step with first-order time discretization error. This paper applies second- and third-order diagonally-implicit Runge Kutta (DIRK) time steps to the modified Picard method in one example. It demonstrates improvements over the first order time step in rms error and error-times-effort model quality by factors ranging from two to over two orders of magnitude, showing that the "modified Picard" and DIRK methods are compatible.


Baker, D.L., 1994, Improved algorithms for finite difference modeling of Richards' equation, Ph.D. Dissertation in Soil Physics, Agronomy Dept., Colorado State University, Ft. Collins, CO 80523, Summer 1994. (Also available through University Microfilms, Inc.)

Inaccuracies in standard finite difference modeling methods make new laboratory procedures for estimating saturated and unsaturated soil parameters, by calibrating a model to a transient inflow experiment, improbable, if not impossible. Time discretization error in the first-order-accurate Euler implicit (backward-in-time) time step make the minima for objective functions (used in estimation with Powell's conjugate direction nonlinear optimization method) shift radically with very small changes in time step size. Diagonally-implicit Runge-Kutta (DIRK) time stepping methods are developed. In DIRK time steps, a number of Euler implicit partial time steps are taken and combined linearly with the final implicit time step, using Runge-Kutta methods. For each additional implicit time step taken, another power of Dt in the local time discretization error is eliminated, improving the computational accuracy and efficiency of the model substantially. DIRK time steps are also fully compatible with mass-conservative forms of Newton's iteration, and can improve the error-times-effort model quality of a mass-conservative method by factors of from two to over two hundred times, depending on the time step size. Another line of inquiry develops the concept of Darcian interblock conductivity means which requires more strict mathematical consistency with Darcy's law than standard means. Graphic visualization of Darcian means, derived by numerically solving and elliptic boundary condition problem, demonstrates a rich complexity not previously considered. Darcian means for vertical flow depend not only on soil unsaturated parameters, but on the model space step size. Visualization leads to a new first-order approximation for the gravity flow mean in vertical infiltration, a weighted mean where the weight varies monotonically and nonlinearly with space step size. In an example of zero-head infiltration into dry coarse soil, the new approximation reduces gross mass inflow error and sum-absolute mass distribution error up to three times over previously-accepted means. This demonstrates that interblock means closer to strict mathematical consistency with Darcy's law produce better results, and calls into question current methods that rely on simple and conveniently-calculated means.


Baker, D.L., 1993, A second-order diagonally implicit Runge-Kutta time stepping method, GROUND WATER, Nov-Dec 1993, v31(6):890-895.

This paper presents a subset of the family of diagonally-implicit Runge-Kutta (DIRK) time-stepping methods for finite difference models of parabolic (diffusion-like) equations. It includes the first-order-accurate Euler implicit (backward-in-time, DIRK1) and the second-order Crank-Nicolson (DIRK2) methods as special cases. It combines a series of DIRK1 partial time steps so as to eliminated additional power terms of the time step, Dt, in the local error, by the number of partial steps used. This offers a large increase in computational efficiency, going from a DIRK1 to a DIRK2 method, and improves on the Crank-Nicolson method with a better choice of Runge-Kutta parameters. For a linear diffusion example, an optimal-parameter DIRK2 method offers the same accuracy as the Euler implicit method at two orders of magnitude large time step, with an order of magnitude better accuracy than the Crank-Nicolson method. In a highly nonlinear horizontal unsaturated water flow example, using eight simulated medium to coarse soils, a DIRK2 method produces either an average maximum accuracy improvement of 4.7 times over the Euler implicit method, without Newton or Picard iteration, or from 3.8 to 48 times faster computer run times for the same accuracy. (Note: The model in this paper did not use iteration to mass balance, a fact which caused some to dismiss the gains as illusory. Later work (above) demonstrates that time step discretization error is separate from mass balance error. DLB)


Baker, D. L.. 1992. "Comment on 'Estimation of finite difference interblock conductivities for simulation of infiltration into initially dry soils' by J. Zaidel and D. Russo", Water Resour. Res., 29(10):3599-3601.

Baker, D. L.. 1992. "Estimating Pressure-Saturation Parameters with Nested Nonlinear Optimization". The IX Int'l Conference on Computational Methods in Water Resources, Denver, CO, June 9-12, 1992, Vol. 1 : Numerical Methods in Water Resources, T.F. Russell, et al., Ed., Computational Mechanics Publications, Southampton, pp 387-393.

The parameters for the Brooks-Corey (BC), van Genuchten (VG), modified van Genuchten (MVG) and modified Brooks-Corey (MBC) pressure-saturation equations can be estimated with nested, nonlinear optimization (NLO). For the root-mean-square-relative-error in fitted Sw, with the eight Brooks-Corey soils, there is little difference between the original BC, the NLO-BC and NLO-VG parameter fits. The NLO-MVG and NLO-MBC fits are significantly better than the other three at the 0.4% confidence level. Using the Burdine (B) and Mualem (M) relative conductivity prediction equations with the NLO-VG, -MVG and -MBC fits, the MVG-B and MBC-B predictions are significantly better at the 5.5% confidence level than the other four combinations.  

Baker, Donald L.. 1992. "The Linear-K Mean Cell-Face Hydraulic Conductivity". Proceedings of the Twelfth Annual Amer. Geophysical Union HYDROLOGY DAYS, Colorado State University, Ft. Collins, CO, March 31 - April 3, 1992, Hubert J. Morel-Seytoux, Ed., Hydrology Days Publications, Atherton, CA, 94027, pp 15-26.

Baker, D. L., P. D. Ayers. 1990. "A Wetting Front Arrival Time Probe". TRANSACTIONS of the ASAE, 33(1):140-146.

Baker, D. L., P. D. Ayers. 1990. "A Wetting Front Arrival Time Probe". Paper 89-2521, 1989 Int'l Winter Meeting, American Society of Agricultural Engineers, New Orleans, LA, Dec 12-15, 1989, 24pp (presented by Dr. Ayers).

Baker, Donald L. 1989. "Developing a Probe to Relate Wetting Front Arrival Time to Soil Penetration Resistance", M.S. Thesis, Agricultural and Chemical Engineering Department, Colorado State University, Fort Collins, CO 80523, Summer 1989.

Baker, Donald L., Raymond H. Canada Jr. and George D. Prine. 1980. "The ANBESS Test Payload", Oceans '80 Conference Record, pp. 350- 356, IEEE, New York.

A low-power (36 Wh/day), microprocessor-controlled weather data telemetry system was developed to test the concept of using U.S. Coast Guard Aids-to-Navigation buoys as platforms for measuring nearshore weather. The mP, interfaced to a hardware arithmetic processor chip, reduces 128 samples of time series data to a short ASCII-encoded engineering unit message. The message is transmitted hourly via GOES satellite link in a format suitable for nearly direct transmission over the National Weather Service computer net. About 100 days of raw and processed data can be stored on a high-density cartridge tape recorder for evaluation of the satellite link, buoy motion effects and mP performance.  

Ruzecki, E. P., C. S. Welch and D. L. Baker. 1977. "Physical Oceanography and Climatology", Chapter 3 of Chemical and Biological Benchmark Studies on Middle Atlantic Outer Continental Shelf, draft final report of first year study under contract No. 08550-CT-5-42 with the Bureau of Land Management, U.S. Dept of Interior, prepared by the Virginia Institute of Marine Science, Gloucester Point, VA, April 1977.

Munday, John C., R. J. Byrne, C. S. Welch, H. H. Gordon, J. D. Boon III, D. K. Stauble, D. L. Baker, E. P. Ruzecki. 1975. Applications of Remote Sensing to Estuarine Problems, Annual Report No. 3, Grant NASA-NGL-47-022-005, Virginia Institute of Marine Science, Gloucester Point, VA, December 1975, 168pp.

Baker, Donald L. 1975. "Data Collection and Processing for a Reservoir Stratification Model". Thesis for M.S. in Ocean Engineering, Dept of Civil Engineering, University of Massachusetts, Amherst, December 1975, 173pp.

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