5.3 Complication of Unsteady Forcing

5.3A. Nature of the complication--

The preceding techniques are based on the assumption that forcing is constant (external forces applied to a system are constant through time). Where this condition is met, the observed history of the system results purely from intrinsic processes or self-organization. Although this condition can be approximated in lab experiments, few geologic systems meet this requirement. If forecasting techniques are applied to a system subject to unsteady forcing, the results may apply to the external forces rather than to the system that is being studied. For example, if the stiffness of the spring in our mass-spring system (c1 in equation 5.2) varied in response to oscillations in temperature, the system would be altered in two important aspects: (1) the observed history of the system would be grossly different (Fig. 5.4), and (2) some of the observed history would reflect the character of the forcing rather than the character the mass-spring system.

Figure 5.4 Unsteady, linear, mass-spring system described by allowing spring stiffness (c₁ in equation 5.1) to vary sinusoidally through time. (a) Time series of spring stiffness and location x. (b) Attractor.

We can use the same approach to modify the Lorenz equations (2.13) to describe convection resulting from unsteady forcing (time-varying temperature differential between the upper and lower surfaces of the fluid). A time-varying temperature differential is incorporated by allowing r (the ratio of the Rayleigh number to the critical value for the initiation of convection) to vary from one time step to the next. As in the mass-spring system, this unsteady forcing results in a time series that reflects both system behavior and forcing (Figs. 5.5 and 5.6).

Figure 5.5 Lorenz system with steady forcing. (a) Time series of Y (temperature difference between ascending and descending fluid) computed using the standard value of r=28.0 in equation (2.13). (b) Attractor of system in (a).

Figure 5.6 Lorenz system with unsteady forcing. (a) Time series of Y incorporating a time-varying r. The mean value of r is 28.6, approximately the same as in Figure 5.5a. The time series of Y is more complicated because of the unsteady forcing. (b) Attractor of system in (a).

Complications due to unsteady forcing are ubiquitous in geologic systems. For example, a time series of sediment transport at a point on a bed of ripples in a tidal flow would have two components: the cyclicity of astronomically induced tidal flow as well as transport variations caused by the passage of the ripples on the bed. The ripples would display a tidally driven cyclic behavior in addition to any self-organized interactions between ripples. If the processes operating in this system were completely unknown (and the periodic forcing was therefore not recognized) an investigator might be misled into thinking that ripples have a intrinsic tidal cyclicity, whereas the tidal cyclicity merely reflects the forcing.

The importance of external forcing in the system described above is so obvious that it may seem absurd to worry about overlooking it. But the effects of unsteady forcing may be much more obscure in geologic data. For example, the structure of a stratigraphic sequence might be influenced by: (1) processes within the depositional environment (analogous to behavior of the Lorenz system under steady forcing), (2) changes in conditions in adjacent regions (analogous to changing the size or shape of the fluid body), or (3) global changes such as climate (analogous to a universal change in the temperature difference between bottom and top of the fluid).

In modern systems, three approaches can be used to work around the problem of unsteady forcing: (1) regulate forcing experimentally, (2) measure forcing as well as system response, and only use data collected at times when forcing is within a narrow range, and (3) measure forcing as well as the system response and use the measured forcing as input in the modeling (input-output modeling of Hunter and Theiler, 1992). In the following examples, the complication of unsteady forcing was resolved by keeping forcing constant (Lorenz example), by choosing a field site where forcing was spatially uniform (wind-ripple example), and by incorporating the unsteady forcing in input-output models (climate and surf-zone sediment transport). In geologic time series, however, the problem of unsteady forcing may be intractable, because only response (not forcing) can be inferred from stratigraphic deposits.

5.3B. Input-output modeling--

Input-output modeling (Hunter and Theiler, 1992) is particularly useful in the earth sciences, where it is usually impossible to regulate the external forces exerted on a system. Instead, input-output modeling utilizes two simultaneous time series: forcing (input) and system response (output). The underlying principle is to use a catalog to learn how the system responds to different forcing events. The technique is computationally similar to the single-series forecasting described in equation 5.5, but it relates response x at time t, to the forcing y measured during a sequence of m steps through time

(5.7)

The response of a system may lag behind the forcing, and forecasting can be used to quantify such a lag. Equation 5.7 can be modified to incorporate such a lag by replacing y_(t-i) with y_(t-(i+n)), where n equals the lag time from the end of the sequence of input forcing values to the time of the model-response output. Equation 5.7 is applied in the same manner as equation 5.5: once to solve for the local linear relation between forcing and response in a learning set, and a second time to predict the response for the testing set. The lag of a physical system is quantified by determining the value of lag n that yields the most accurate forecasts. An example of this approach is given in the discussion of surf-zone sediment transport (Chapter 5.4E).

5.3C. Forecasting in practice--

In applying the forecasting techniques, a researcher might go through the following sequence of operations to characterize the system that produced an observed time series:

1. Use spectral analysis to evaluate the time series for periodic or nonperiodic structure. If the power of the time series is restricted to one or more narrow frequency bands, the time series is largely periodic and linear. Although the time series may still contain a minor nonlinear component, the nonlinear techniques will probably not greatly improve the forecasts. In contrast, if the power of the time series is distributed across broad bands of frequencies, then nonlinear techniques may discover a deterministic nonlinear structure hidden in what the spectral analysis detects as "noise". Although spectral analysis is a useful technique for identifying periodic (linear) structure in a time series, this technique can not identify structure that is nonperiodic or nonlinear. Nonlinear techniques are useful where a time series contains nonlinear structure.
2. Split the time series into a testing set and learning set. To reduce computation time, the testing set should have just enough points to obtain stable error statistics (values that do not change appreciably if the number of points were to be increased); a few hundred points is generally sufficient, provided that the testing set and learning set both are representative samples of the underlying attractor.
3. Choose the first exploratory values for embedding dimension (m), number of neighbors (k), and forecasting time (n), and apply equation 5.5--equation 5.7 for input-output systems--for every point in the testing set. Use equation 5.6 or the correlation coefficient between predicted and observed values to evaluate the performance of that model. Repeat this procedure for other values of m and k, searching for the model that makes the most accurate predictions.
4. To demonstrate nonlinearity, the best nonlinear model should be substantially more accurate than the best linear model (for a given forecasting time). An additional test for nonlinearity is to use surrogate data, as explained above, and as will be illustrated in the wind-ripple example below.
5. The success of modeling efforts depends on how well null hypotheses are formulated and eliminated. In particular, low-dimensional nonlinearity (chaos) can only be demonstrated by formulating and eliminating null hypotheses that include interactions between noise and periodicity as well as the more common null hypotheses of high-dimensional linear and nonlinear systems.

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