W. E. Sharp

The hydrology of Lake Bosumtwi, a climate-sensitive lake in Ghana, West Africa

Abstract Lake Bosumtwi, lying in a million-year-old meteor crater, is hydrologically closed in spite of the wet, humid climate in this part of Ghana. To understand this geographical enigma, a water-balance model that describes both the historical lake-level record and the full range of lake levels observed in terrace deposits around the lake was developed. The water-balance model shows that the reception of rainfall by and evaporation from the lake surface are the dominant constituents of the water balance. Prior to 1969, the average rainfall of 1550 mm year −1 was generally equal to lake evaporation, and the average annual rise of 30 cm year −1 was produced mostly by runoff from the small watershed around the lake. Since 1969, with a decreased rainfall of 1380 mm year −1 , the combination of rainfall and runoff has nearly equaled lake evaporation, resulting in a nearly steady lake level. Long-term simulations show that stochastic climatic variations very similar to those observed in this century could produce the full range of lake levels observed in terrace deposits. The low salinity of about 1‰ suggests that dissolved solutes were removed by lake overflow in the recent geological past.

The generation of multidimensional autoregressive series by the herringbone method

The generation of isotropic artificial series in two or three dimensions by the autoregressive process is of considerable interest for the purpose of modeling environmental properties such as ore grade or reservoir porosity. The relations needed to produce bilateral symmetry using the one-sided autoregressive recursion equations have been attained on the square net and on the isometric lattice by an alternation procedure. In the case of the square net, the one-sided autoregressive (AR) process is alternated between the two diagonals of the net, while in three dimensions, the alternation takes place among the four body diagonals of the isometric cell.

Stochastic simulation of semivariograms

The semivariogram obtained from simulated space series formed by an autoregressive (AR) process gives a ready explanation for most of the common types. Linear semivariograms arise from a random walk (Brownian motion) process while the transitive and exponential types are generated by an AR process of first order. The continuous (“Gaussian”) type arises from a process of second order which also accounts for the “hole-effect” when the process becomes pseudo-periodic. The nugget effect can be included by the addition of a moving average (MA) process of first order.

A review of portable random number generators

Abstract Portable random number generators for personal computers can be evaluated quickly and effectively by plotting their two- and three-dimensional serial patterns. An examination of about 40 published generators yielded only a few that had acceptable serial patterns.

Estimation of semivariograms by the maximum entropy method

A wide variety of semivariograms may be represented in terms of a first- or second-order autoregressive (AR) process, and the nugget effect may be included by use of a moving average (MA) process. The weighting parameters for these models have a simple functional dependence on the value of the sill and the semivariance at the first and second lag. These may be estimated either graphically from the semivariogram or directly from the computed values. Improved spectral estimates of geophysical data have been obtained by the use of the “maximum entropy method,” and the necessary equations were adapted here for the estimation of the weighting parameters of the AR and the MA processes. Comparison among the semivariograms obtained for the ideal case, the observed case, and the estimated case for artificial series show excellent correspondence between the ideal and estimated while the observed semivariogram may show marked divergence.

A method for determining the reversibility of a Markov sequence

This paper describes, given a tally matrix with strictly positive entries, a method to determine whether the associated Markov process is reversible, and (for reversible Markov processes) methods to compute the reversibility matrix from the tally matrix. If the tally matrixN is symmetric, then it is shown that the Markov process must be reversible and the reversibility matrixC equalss (R−1NR−1), whereR is the diagonal matrix whoseith diagonal entry is the sum of the entries of theith row ofN (for everyi) ands denotes the sum of all the entries ofN. Because a symmetric tally matrix is of special importance in applications, a χ2 test is proposed for determining, in the presence of experimental errors, whether such a matrix is symmetric.

Unilateral ARMA processes on a square net by the herringbone method

Because multidimensional ARMA processes have great potential for the simulation of geological parameters such as aquifer permeability, it was important to resolve which of two proposed alternative methods should be used for determining the two-dimensional weighting parameter, φ″, for a unilateral ARMA (1, 0) process on a square net. Practical simulations demonstrates that the correct formulation is: φ″=ρ10/(1+ρ102 where ρr,s is the correlation between lattice points at lagsr and s. When the simulations are performed with correlations of 0.8 or more a residual bias was detected which was found to be caused by a difference in the variance between the one- and two-dimensional models. This can be rectified by modifying the two- dimensional model as follows: zij=φ″(zi−1, j + zi, j−1) + λaij whereλ2=1/(1 +ρ102).

Generating random numbers in the field

A pseudo-random number generator which is easy to remember and uses the very simple sequences 321 and 123 can be readily implemented on any available hand calculator. The generator is given by the recursive relation: Ri=321 × Ri−1+123, and is initiated using any 5-digit number for Ri−1and saving the rightmost 5 digits of Rifor the next Ri−1;the two leftmost digits of Rithen yield an acceptable random number between 00 and 99. Alternately, a much better generator is given which can be jotted down in a notebook or taped to the back of your calculator.

Reversible Markov grain sequences in granite

A key feature of an ideal granite is the occurrence of grain sequences which are reversible Markov chains. This property was tested using a χ2 test on a 2 × 6 contingency table consisting of reversible grain pairs for microcline, plagioclase, quartz, and biotite, and on a 2 × 12 contingency table consisting of reversible grain triads. All 28 samples examined from the Pacolet Mills pluton, South Carolina, passed the χ2 test for grain pairs, and all but three of these passed the χ2 test for grain triads. The coefficients of the reversibility matrix were examined for statistical significance after normalization, using a logarithmic transformation. For all three phases of the Pacolet Mills pluton, the average coefficients were in the range 1.08–0.89. Elevated and depressed values of these coefficients suggested possible differences among the three phases of the granite in their crystallization paths.

Quasi-Symmetry and Reversible Markov Sequences in Sedimentary Sections

Quasi-symmetry can be defined as a purely mathematical property of a matrix—that is, any matrix whose entries are strictly positive possesses quasi-symmetry if it can be written as a product of a diagonal and a symmetric matrix. A unique inverse solution for a quasi-symmetric matrix is readily obtained when the nondiagonal elements of the symmetric and quasi-symmetric matrix are set equal. Then it is shown that a Markov sequence is reversible if and only if it has a quasi-symmetric tally matrix. Because a properly counted Markov sequence must have marginal homogeneity, a simple chi-square test for symmetry on the tally matrix is sufficient to determine if an observed matrix is symmetrical and hence whether the Markov chain is reversible. Applications to sedimentary sequences are illustrated by the use of classical examples and with cyclothem data to determine if the sequence conforms to a reversible or nonreversible Markov process. Should the tally matrix lack marginal homogeneity, it is likely that a sampling bias was introduced by the counting procedure. However, a chi-square test for symmetry on a direct inverse of the tally matrix can be used to determine if the sedimentary sequence conforms to a reversible or a non-reversible Markov process.