1. Introduction

The physical constituents of climate form a complex chaotic system. This means that it is high dimensional and as a whole not predictable beyond a time t_O, which for the atmosphere is a few days. Other examples of complex chaotic systems are found in biology and economics, and include the widely used general circulation models of our atmosphere.

The assumption of climate predictability implies that it is possible to predict some part of the atmospheric system much beyond t₀. It is tempting to assume this extended predictability be carried by a low-dimensional submanifold of the climate system.

The method of Grassberger and Procaccia (1983) has been widely used to investigate the dimension of complex systems associated with a measurement. It can in principle also be used to make predictions. The EOF analysis performed by Kimoto and Ghil (1993) aims at approximating the atmospheric attractor by a linear space. The singular spectrum analysis (Vautard et al. 1992; Plaud and Vautard 1994) is a method to extract a predictable signal from the climate system. This method is based on the computation of a wide range of timescales for the oscillating phenomena. A discussion of the extensive work that has been done along these lines is beyond the scope of this paper.

These investigations on climate predictability exploit the phase space contraction, which means that a system occupies only a small part of the phase space. In this paper, the submanifold carrying the predictability is assumed to be much smaller than the attractor, which may still have a fairly high dimension. This situation is rather common for turbulent systems.

The idea of the method used for the two examples in the present paper is very simple. Suppose you have a very complex system like a big climate model, with a very high or even infinite dimensional state space X. Suppose now that you wish to construct a simple low-dimensional model that will be useful in predicting some specific functional of the model state, like the outflow rate of some river or the surface pressure at Darwin.

Choose now some low-dimensional control space Y. Assume for the sake of definiteness that Y has the dimension 2. Assume that there is an X-valued function F that takes Y into X. The explicit form of F need not be known. Let the coordinates in Y be y₁ and y₂, and write the dynamics of our simplified model as

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with p₁, . . . , p_n being a parameter space, defining the functions G₁ and G₂. Such parameters could be gridpoint values of G₁ and G₂. Here, y₁ and y₂ define the predictable submanifold of the phase space, which is two-dimensional in this case. Inside this submanifold the prediction is done by defining a trajectory. In Eq. (1) y₁ and y₂ are assumed to be on the trajectory: y₁ = y₁(t), y₂ = y₂(t).

Now suppose that there is some series of scalar observations, z(t), with z(t) = H(y₁,y₂,u₁, . . . , u_r), that is, the relation between the observation and the submanifold is known, with u₁, . . . , u_r being a parameter space. Generalization to vector observations is obvious. Some mathematical definitions on which this approach is based are given in the appendix. Now choose the initial values y₁(0), y₂(0) and the parameters p₁, . . . , p_n, u₁, . . . u_r to minimize the functional Q:

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2. A water reservoir problem

There may be a parameterization of the outflow by the existing volume, D = f(V). This assumes that some kind of turbulence is the resistance mechanism of the outflow.

Hindcast and verification are shown in Fig. 1. There is quite a good forecast for the first two days, and the drop in discharge is also indicated in the forecast, though smoothed out over some time. The characteristic feature of atmospheric forecasts, to lose forecast skill with time, is also shown in this simple, ordered model.

3. Forecast of the surface pressure at Darwin

The simple, ordered model presented in this section aims at forecasting the surface pressure at Darwin. A low-order model made for understanding rather than quantitative forecasting was given by Münnich et al. (1991).

With the simple model given here, no attempt is made to have a system that has a long time behavior similar to that of the real atmosphere. The forecasted fields take on unrealistic values for time becoming large. They are valid only for a short time, which will turn out to be about half a year. Sophistication of the model to meet additional requirements is possible but is beyond the scope of this paper.

To obtain our dynamic model, we assume that there is an atmospheric circulation of importance to the Southern Oscillation that can be described by one parameter only. For a full description of the physical situation see Rassmusson and Carpenter (1982). We assume that P(t), the surface pressure at Darwin, is sufficient to determine this circulation. Therefore, P(t) is one of the two dynamic parameters of our simple, ordered subsystem (SOS). The other parameter B(t) describes the oceanic circulation. Following Münnich et al. (1991), we assume that B(t) describes the structure of the well-mixed layer, that is, the spacial distribution of the height of this layer.

The forecast is performed using only surface pressure information. Thus, the forecast of P is done using only initial values of P, like in many other low-order predictive systems. The variable B here serves to motivate and greatly simplify the nonlinear prediction procedure for P. The SOS method would make it possible to assimilate initial values more directly related to B, such as sea surface temperature or precipitation. There is the possibility that the inclusion of such redundant information could increase the accuracy of the initial state and thus that of the forecast. The simple model presented here has the purpose to show that by using the concept of a (nonlinear) manifold, rather than that of (linear) spaces, very low-dimensional approaches are possible.

The elaboration of the approach necessary to achieve the best forecast possible is beyond the scope of this paper.

The dynamic equations are obtained on the basis of the assumption that the atmospheric and oceanic circulations influence each other and that there is a diffusion term for each of the circulations, which for simplicity is assumed to be linear. The results will indicate that a better simulation would probably have been obtained if the diffusion terms were nonlinear. The dynamic equations are assumed to be of the form

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where P₀ is the time-averaged pressure for each month. The data available were a record of pressure data for Darwin, time averaged for a month. They were available for each month covering the time from January 1883 to December 1984. Here, P₀ is shown in Fig. 2, together with P_min and P_max, the minimum and maximum values observed for each month. As our model does not contain any limitation to stay within a climatological range, the forecasts were used only when they stayed within P_min and P_max. The functional Q was always set up for periods of 4 years. Since P is used directly as dynamical variable and no other observations are used to determine B, there are in this case no parameters u_i to be determined to relate observations to the dynamic variables P and B. Initial values B₁(0), B₂(0), B₃(0), and B₄(0) are assumed for each January of the 4 years considered. The values of P and B for the rest of the years are then known using the dynamic equations (6) for B and the measurements for P. The functional to be minimized is

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where P_i(t) and B_i(t) are the values of P and B for each of the 4 years considered.

The functions f₁(B) and f₃(P) are assumed to be linear splines with four nodepoints fl_1i and fl_3i, respectively. These nodepoints are defined at B = −1, −1/3, 1/3, 1 for f₁(B) and at P − 1000 hPa = 4, 8, 12, 16 for f₃(P). Since P_i(t) are given by measurements, the parameters to be determined are B₁(0), B₂(0), B₃(0), B₄(0), fl_1i, fl_3i, i = 1, 2, 3, 4, f₂, f₄. The rather special choice of the nodepoints of B does not make the approach less general, since the unit of B is not specified. For each of the 4-yr periods, the functional will have to be minimized with respect to these 14 parameters. The time t_k corresponds to the different months, and the time derivative of P is obtained by numerical differentiation from the observational record P(t). Here, B_i(t) is the forecast of B for a year, using the initial value B_i(0) and the dynamic equation (6).

Direct minimization by the conjugate gradient method is efficient. The results obtained by variation for one 4-yr period were used as first estimates for the conjugate gradient method in the next 4-yr period. In this way the convergence improved for the later 4-yr periods. The functions f₁(t) and f₃(t) obtained are shown in Fig. 3 after year 5. The different estimates of these functions show a large coincidence, particularly for the function f₁, which indicates that this function does not depend on the 4-year period used for estimation. The function f₃(P) shows a larger variance of the results, in particular for the extreme values of P. This variance of the results ocurred mainly in the beginning of the analysis cycle, when the convergence was not yet achieved. Figure 4 shows the time-dependent estimates of the nodepoints fl_1k of the function f₁, as obtained in the different 4-yr periods. It appears that after 20 years of assimilation constant values are obtained in good approximation.

The corresponding time diagrams for f₂ and f₄ are shown in Fig. 5. These parameters are less constant, indicating a need to investigate more elaborate representations of friction. The parameter f₂ is antidiffusive. This can make the solutions go astray after initially approximating the measurement.

The analyzed values of B are given in Fig. 6 for a period of 10 years. The January values of B are obtained by variation of the functional Q. They are indicated by squares in Fig. 6. The values for the other times are obtained by integrating the dynamic equation for B, Eq. (6). The December values of B are indicated by a star in Fig. 6. They must be used for forecasts of B, since in order to obtain the January values, the observations of P for the following year must already be known in order to perform the variation. The forecasts of B are indicated by a dashed line in Fig. 6.

An example of 1-yr forecasts of P is given in Fig. 7. There is some skill of the forecast apparent in spite of the simplicity of the approach. In the beginning of the forecast there is a fall of pressure for 1 month. The forecasts do not simulate this signal very accurately, although for later times they seem to be able to do a better job.

No attempt was made to systematically change the parameters of the assimilation. Figure 8 gives results of a case where Q is minimized for trajectories extending over 2 years rather than one. The forecasts are also done for 2 years. Here the deterioration of the forecasts with increasing forecast time can be seen. Also in the beginning the forecasts are somewhat worse than those shown in Fig. 7. For the 1-yr forecasts the rmse taken over the whole range of the forecasts is shown in Fig. 9. The forecast obtained by P₀(t) is referred to as the climatology forecast. For short forecast times, the climatology forecast matches the SOS results, reflecting the difficulty of the latter to correctly predict the pressure fall in the beginning of the forecast. After this, in the time range up to 6 months, the SOS forecast has better skill than the climatology forecast.

According to Fig. 9, the best skill is obtained for the 5-month forecasts. For example for the eight forecasts shown in Fig. 7, the prediction error in hectoPascals is (P_observed − P_predicted) 0, .2, .8, 0, 1.1, 0, −.4, −.6., as compared to the variance of this quantity of several hectoPascals. The bad results in the beginning of the forecast indicate a need for an elaboration of the model. That and a systematic approach are beyond the scope of this paper. The purpose of the simple approach presented is to show the relative ease of the SOS approach in constructing nonlinear models and the low dimension, which can be used to obtain meaningful models.

4. Comparison of SOS and POP

In the following, the SOS method will be compared with the linear method using principal oscillation patterns (POP), as described by Storch et al. (1995). POPs can be obtained from SOSs by, in particular, using linear spaces for the manifold occuring in the SOS and using linear dynamic laws. Storch et al. (1995) report a number of applications of the POP approach. Since it will turn out that for any POP system it is possible to define a nonlinear generalization, we have an indication how to identify more predictable subsystems and a systematic way of defining the equations governing the subsystems by using the linear system as a starting point. Such nonlinear generalizations of a linear POP are worthwhile to pursue. The sytems described by Storch et al. (1995) are all quasiperiodic. With linear dynamic laws they are approximated by a superposition of damped oscillations. A nonlinear dynamic offers the chance to reproduce the nonperiodic behavior.

The linear POPs are simple cases of the nonlinear principal interaction patterns (PIP). Storch et al. (1995) report no full implementation of the PIP concept. As with SOSs, the dynamics of PIPs is obtained using a least-squares procedure. To perform the variation, PIPs need gridded fields or spectral coefficients as data. In the case of the atmosphere these are available for the last 30 years from the analyses performed for the purpose of numerical weather prediction. Truncated EOFs (empirical orthogonal functions), which have been obtained from the gridded fields, can also be used in order to reduce the size of the problem. A PIP can be seen as a special case of an SOS. To obtain the PIP concept from SOS, it is necessary to use gridded analyses (or elements of EOF spaces derived from them) for the observational material and assume the manifolds to be linear spaces. The relation between phase space parameters and data then becomes explicitly known. The function H(y₁, y₂, u₁, . . . , u_r) is linear in y₁ and y₂. The parameters u_k describe the basis functions of the predictable subspace. For each of them the number of u_ks necessary is equal to the dimension of the imbedding phase space (space of gridded fields). This number is quite high, and in addition, the parameters that determine the equations governing the subsystem have to be found by the variational procedure.

The recommended use of SOS is to directly take the observations as data material. The amount of data to be used will normally be several orders smaller than with gridded fields.

In case the subsystems are linear spaces, a complete knowledge of the basis functions of the predictable subspace is not required. It is only necessary to know the value of the basis functions at the positions of the measurements. The number of parameters to be determined in this respect is therefore the number of observational sites times the dimension of the predictable subspace. For the PIP approach it is the dimension of the embedding phase space times the dimension of the predictable subspace. The latter number is much higher, and this is responsible for the relative ease of the SOS approach.

Once a POP is given, there are two ways to generalize it to a nonlinear model. Using the terminology of Storch et al. (1995), the dynamics of a POP is defined by an operator A. Since we have limited our presentation to two-dimensional predictable subsystems, we assume also two-dimensional POPs, for simplicity of the presentation. Here, A operates on y₁, y₂, and y₁, y₂ are amplitudes of a basis function representation in a two-dimensional subspace of a rather large phase space. To treat this POP in the framework of SOS, one has to replace the function G in Eq. (1) by A(y₁, y₂).

1. Even in the linear framework it may be useful to try the SOS using the observations directly. In this way it is possible to carry the analysis to times where observations are too sparse to perform an analysis in order to obtain the gridded fields. The parameters to be varied in the least-squares approach are the real and imaginary parts of the eigenvalues of A and values of the two eigenfunctions at the positions of measurements. The model can then be made nonlinear by replacing A by a nonlinear function A(y₁, y₂). Assuming that an unlimited amount of data exists, no further assumption is necessary to determine A(y₁, y₂). It is only necessary to assume a sufficiently fine gridpoint representation of A over the range of y₁, y₂ and to vary the gridpoints of A. The drawback is that quite a number of parameters have to be determined by variation.

   Unfortunately, there will generally not exist sufficient data in order to determine a nonlinear A in this way. An assumption leading to a rather moderate amount of parameters to be determined by variation is the following. Here A(y₁, y₂) is the sum of one-dimensional functions depending on y₁, y₂, and g(y₁) − y₂. Each of the one-dimensional functions is represented by only a small number of nodepoints. Since we are not assuming normalized basis functions, we can impose the range of y₁ and y₂, for example, to the interval (0, 1), without losing generality. In this interval we may assume equally spaced nodepoints for the one-dimensional functions. These nodepoints are varied for the least-squares functional. The parameters connecting the measurements to the predictable subspace are already known from the linear approach.

   The method indicated above will provide candidates for predictable subsystems. To identify these as such, a verification of the forecasts is necessary, using data that are independent from those used for the tuning of the system.

2. The limitation to linear spaces used above may exclude some very low-dimensional systems. The river outflow system described above is an example where a one-dimensional manifold can be contained only in a high-dimensional space. For phase spaces that can naturally be divided into subspaces, such as atmosphere, ocean, and ice, there is always a simple hypothesis that can be tested with a small dataset. The assumption is that we have a one-dimensional manifold in each of the subspaces. This means that the subsystem is described by one parameter y_i for each of the subspaces. An assumption on the nonlinear function A that is sufficiently simple to be tested is the following. Here A is composed of one-dimensional functions of the y_i and their differences. The one-dimensional functions can be represented by only a few nodepoints. The second example given in this paper uses this approach.

5. Conclusions

A method has been proposed to predict simple, ordered subsystems (SOS) directly, as opposed to using a complex chaotic model and extracting the predictable signal afterward. This involves the analysis of dynamic variables of relevance to the physical problem. In the case of the Southern Oscillation, these dynamic variables were two parameters determining the atmospheric and oceanic circulations. The simple atmospheric subsystems are assumed to form a low-dimensional manifold inside the phase space, and a dynamic law is assumed reflecting the physical situation. The parameters of this dynamic law are determined using a record of data.

The application of this concept to the problem of surface pressure prediction at Darwin resulted in some skill, even though the model used was crude in some respects and did not make use of any direct ocean observations.

The method is multivariate in that it naturally incorporates different observations, even observations of different components like atmosphere and oceans.

The examples given, however, used only one measurement. There was a clear indication that an additional observation related to the ocean circulation might have improved the result.