2.3.3 Ramp/step method
Solution of the inverse problem consists of three general steps: (1) choice of an appropriate physical model, (2) parametrization of the theoretical equations, and (3) estimation of the parameters. Naturally, this procedure cannot be started from zero and requires including as much a priori information as possible. Each underdetermined problem needs certain assumptions. More complex model implies the greater number of assumptions and/or amount of additional information for the successful past climate reconstruction. Because of the common insufficiency of the field information, simpler models in some cases can be more preferable.
The ramp and/or step models that are sometimes referred to as the last event analysis (Lachenbruch and Marshall, 1986) represent the most simple, but robust manner of the parametrization. While other modern inversion techniques allow an arbitrary form of the GST history, the step method assumes the GST changes with time according to a three-parameter law
for 0 < t < t and n = 0, 1, The value t represents the duration of the GST change of the amplitude AT, and the power n determines the shape of this change. At n = 0 we have a simple step increase/decrease in temperature, n = 2 represents a linear one, and so on. The solution of this equation for a homogeneous half-space at t = t is (Carslaw and Jaeger, 1962)
T ( z, t ' ) = T0 + z + AT 2n r (n +l\ xinerfc —j= K V2 ) V 2Vkt*
where r(x) is the gamma function of argument x and inerfc is the nth time integral of the error function. Formula (15) thus enables us to calculate the subsurface temperature at the end of the GST change.
Expression (15) corresponds to a simple four-parameter model m = [AT, t*, T0, g0]. In the pioneering work by Lachenbruch and Marshall (1986) quantities T0 and Q0 were defined independently of the parameters AT and t* by the linear fitting to the measured temperature-depth profile immediately below the obviously disturbed near-surface part of the temperature record. In the more recent works unknown parameters T0, AT, and t* are estimated together with the background heat flow Q0. Using this form of parametrization, the inverse problem can be formulated as: given the data T(zk) measured at various depths zk, k = 0, 1, ..., M, find the temperature history characterized by the vector m. To determine the optimum values of the unknown parameters most of the researchers (e.g. Cermak et al., 1992; Safanda and Kubik, 1992) applied the least squares inversion theory proposed by Tarantola and Valette (1982a, b). In this approach, an M-dimensional vector of temperature-depth measurements T is related to parameter vector m by the equation T = g(m), where g is an M-dimensional vector-function whose components are determined by Eq. (15). Inversion is performed by using the iterative procedure. The advantage of the method is the possibility of quantifying our confidence in the a priori estimates of T and m in a form of their a priori standard deviations as well as of obtaining estimates of the a posteriori standard deviations of m. The a posteriori to a priori standard deviations ratio (SDR) is generally used to characterize the reduction of the uncertainty of the estimated parameters.
The ramp/step problem can be extended: (1) for the case of the stratified medium each with constant known thermophysical parameters in the individual layers, and (2) for the multi-step approach
where t* and AT represent the epochs and magnitude of temperature change (e.g. Beltrami and Mareschal, 1991). For the series of steps of equal duration expressed as departures of the mean temperature value and starting at time tl in the past, present temperature at depth z is given as
2^fkt erfc erfc-
The single-step approach may be more suitable in the case when for the area under investigation there is no information on the surface temperature history at all, since it relates the calculated surface change to the conditions averaged for a long prior period. The use of the multi-step model may give a better insight into what really happened. As shown by Putnam and Chapman (1996), a series of step changes can approximate any real surface temperature variation.
2.3.4 Singular value decomposition (SVD) algorithm
Singular value decomposition method was first presented in the works by Beltrami and Mareschal (1991), Mareschal and Beltrami (1992), and Wang (1992). Later it was improved in the work by Bodri and Cermak (1995) by including into the analysis various kinds of additional information and the special technique of the GST discretization that ensures optimal choice of the estimated GST vector. As in the previous case, the problem is formulated as the pure conductive 1-D heat transfer of the surface temperature variations in a layered slab with the constant, known thermophysical parameters in each layer as described in Section 2.2.2. The model vector can be thus formulated as m = [U0,Qm,V,(t)].
The surface temperature history is approximated by a series of unequal intervals of constant temperature
The mean values of temperature for the individual time intervals Vi are the unknown parameters of the problem. At such assumption the integral in the forward problem like Eq. (13) can be easily transformed into the series
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