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McMillan, DG, Constable CG, Parker RL.  2002.  Limitations on stratigraphic analyses due to incomplete age control and their relevance to sedimentary paleomagnetism. Earth and Planetary Science Letters. 201:509-523.   10.1016/s0012-821x(02)00747-1   AbstractWebsite

A major limitation in the analysis of physical quantities measured from a stratigraphic core is incomplete knowledge of the depth to age relationship for the core. Records derived from diverse locations are often compared or combined to construct records that represent a global signal. Time series analysis of individual or combined records is commonly employed to seek quasi-periodic components or characterize the timescales of relevant physical processes. Assumptions that are frequently made in the approximation of depth to age relationships can have a dramatic and harmful effect on the spectral content of records from stratigraphic cores. A common procedure for estimating ages in a set of samples from a stratigraphic core is to assign, based on complementary data, the ages at a number of depths (tie points) and then assume a uniform accumulation rate between the tie points. Imprecisely dated or misidentified tie points and naturally varying accumulation rates give rise to discrepancies between the inferred and the actual ages of a sample. We develop a statistical model for age uncertainties in stratigraphic cores that treats the true, but in practice unknown, ages of core samples as random variables. For inaccuracies in the ages of tie points, we draw the error from a zero-mean normal distribution. For a variable accumulation rate, we require the actual ages of a sequence of samples to be monotonically increasing and the age errors to have the form of a Brownian bridge. That is, the errors are zero at the tie points. The actual ages are modeled by integrating a piecewise constant, randomly varying accumulation rate. In each case, our analysis yields closed form expressions for the expected value and variance of resulting errors in age at any depth in the core. By Monte Carlo simulation with plausible parameters, we find that age errors across a paleomagnetic record due to misdated tie points are likely of the same order as the tie point discrepancies. Those due to accumulation rate variations can be as large as 30 kyr, but are probably less than 10 kyr. We provide a method by which error estimates like these can be made for similar stratigraphic dating problems and apply our statistical model to an idealized marine sedimentary paleomagnetic record. Both types of errors severely degrade the spectral content of the inferred record. We quantify these effects using realistic tie point ages, their uncertainties and depositional parameters. (C) 2002 Elsevier Science B.V. All rights reserved.

Hobbs, UA, Parker RL.  1978.  Limitations on the parameters of the solar wind in modelling lunar electromagnetic induction. Geophysical Journal of the Royal Astronomical Society. 52:433-439.: Blackwell Publishing Ltd   10.1111/j.1365-246X.1978.tb04241.x   AbstractWebsite

Summary. A striking feature of the day-side response of the Moon to periodic fluctuations in the solar wind is the rapid rise, and subsequent fall, in the amplitude of the transfer function as the inducing field frequency increases. This behaviour can be characterized by the amplitude values at the two frequencies 24 and 40 mHz. Before the response of a conductivity model representing the Moon can be calculated at a given frequency, the parameters (ν, θ) (where ν is the solar wind speed and θ is the angle between the solar wind velocity and the magnetic field propagation direction) have to be specified. By applying some results due to Parker (1972) to the above two data points, we have determined constraints on the parameter space (ν, θ). In particular, we determine the region of the (ν, θ) space in which conductivity models may be found that satisfy our data pair. Outside this region, there are no conductivity models satisfying the data pair, and hence many (ν, θ) values are inconsistent with the original data and the model assumptions.

Parker, RL.  1977.  Linear inference and underparameterized models. Reviews of Geophysics. 15:446-456.   10.1029/RG015i004p00446   AbstractWebsite

A version of Backus's theory of linear inference is developed by using a new finite-dimensional space. This approach affords a clear geometric interpretation of the essential role played by a priori model smoothing assumptions and also facilitates the construction of a theory for the treatment of random data errors that is quite different from the treatment of Backus. When the unknown parameters form a (necessarily incomplete) description of the model, it is possible to formulate a special smoothing assumption that is particularly appropriate; in practical examples this strategy often leads to tighter bounds on the model uncertainty than those obtained with previous assumptions. An analysis of the numerical aspects of the problem forces one to the conclusion that the theory is not competitive numerically with conventional least squares parameter estimation, unless one of the large submatrices in the problem possesses a simple inverse. An example of this kind is discussed briefly.