## 1. Introduction

The comment (Tsonis and Elsner 1998) gives a number of references of which, in particular, the many Tsonis and Elsner papers are worth reading. The content of these papers ranges from a good tutorial on subjects loosely connected to Steppeler (1997a) to analyses following Grassberger and Procaccia (1983) and Vautard et al. (1992). The authors claim that I was not aware of such developments and that some of these would be more basic to Steppeler (1997a) than the papers I had cited. After studying the papers I cannot endorse this statement. Though the papers given by Tsonis and Elsner have a loose connection to Steppeler (1997a), I wish to reaffirm the statement made in the beginning of Steppeler (1997a) and not enter into a discussion of the extensive work made along the lines of Grassberger and Procaccia (1983) and Vautard et al. (1992). These approaches are quite different from the simple, ordered subsystem (SOS) approach, and therefore I felt that it is sufficient to cite these two basic papers and that it would distract from my subject to discuss the follow-up work of such papers extensively. I found it sufficient to discuss the principal oscillation patterns (POPs) and principal interaction pattern (PIP) methods extensively, because these methods have some similarity with SOS. The POP and PIP approaches are based on subsystems, the latter one using a nonlinear approach. So it is quite clear that I did not claim to have invented the ideas of a subsystem, as Tsonis and Elsner (1998) seem to believe. I strongly recommend that the authors of the comment read section 4 of Steppeler (1997a). Also, I cannot endorse the implication that Tsonis and Elsner (1989) have invented the notion of a subsystem. In order to give credit to early references of subsystems I have to go back into the early history of science in section 4.“The simple examples that will be given in section 2 will also make clear that the notion of a subsystem is common and at the basis of all parameterization approaches.” For the point made by Steppeler (1997a) the references given by Tsonis and Elsner do not warrent consideration, though they would be warrented in a review paper on the subject. I do encourage the authors of the comment to write such a review. In such a review many more papers should be considered than given by Tsonis and Elsner (1998). I had available to me a list of three pages of references provided by Hense (1992, personal communication).

I was somewhat surprised to see in Tsonis and Elsner (1998) only papers cited along the lines of approaches that Steppeler (1997a) had already discussed shortly. However, there are a number of approaches that I had deliberately not referenced in order to concentrate on those approaches being most similar to SOS. “In section 3 I will mention some more approaches, which are basically different from those discussed so far. One of those is the current approach to practical (ENSO) prediction. This will make a discussion of the mutual strengths of the different approaches possible.” It will become clear that the different approaches have to be used in concert, rather than in concurrence, as Tsonis and Elsner (1998) seem to believe. Certainly Steppeler (1997a) does not intend to compare the different approaches based on their results. I would not encourage the authors of the comment to do such a comparison now, not even for the special problem of ENSO prediction. Much more work on the different approaches will have to be done before such judgment can be made.

I am grateful to Tsonis and Elsner (1998), as their comments exemplify the kind of misunderstandings that I encountered when first discussing the SOS approach with colleagues. It seems to be difficult for people familiar with one of the existing approaches to the subject to see what distinguishes the SOS approach from others. The notion of dimension as used in Steppeler (1997a) is not recognized, and there seems to be a strong desire to enter into the discussion of side issues.

It may be worthwhile to list the features that are special to the SOS method.

- The phase space is the space of the physical phenomenon (e.g., atmosphere oceans, solid earth) being infinite-dimensional in most cases.
- The subsystem is not the attractor; rather it approximates a small part of it and has a much smaller dimension.
- The subsystem is a nonlinear manifold (defined as smooth). This makes sense even when the attractor is not smooth.
- The states of the manifold need not be known in terms of basis function representations; rather a small number of coefficients relating the state in phase space to measurements and defining the dynamic laws are sufficient.
- The method is applicable to all systems (even biological or economic) that are expected to contain a predictable part. A knowledge of the dynamic law in the (often infinite-dimensional) phase space is not necessary.

These points struck me as quite important and not properly observed in current approaches. The main advantage of SOS is a comparatively low demand on measured data, as exemplified in section 4 of Steppeler (1997a). How such advantages are related to using manifolds with SOS, rather than limiting the approach to spaces, will be further exemplified in section 2. It seems necessary to define the notion of the dimension of a subsystem in section 2, since this is used quite differently by Steppeler (1997a) than in all references given by Tsonis and Elsner (1998), so their conclusions do not apply. For example, Lorenz (1991) raises doubts on dimensional estimates within the Grassberger and Procaccia (1983) approaches and this has nothing to do with Steppeler (1997a), though they apply to Tsonis and Elsner (1989).

The intent of Steppeler (1997a) is to show the advantages of the above-mentioned features of SOS. With this intention it is clear that the claim is not that the model of the Southern Oscillation presented is final. Some indications of necessary further developments have been given by Steppeler (1997a). The improvements suggested by Tsonis and Elsner (1998) to evaluate the impact of other parameters such as global temperature are worth considering in this connection. However, such points should not be discussed in connection with Steppeler (1997a), but are rather a matter of follow-up papers. The great desire to discuss side issues with respect of the main point of Steppeler (1997a), as summarized above, is certainly caused by the great potential range of applications and potential implications to many questions of current interest. When in section 3 some alternative approaches to the prediction of the Southern Oscillation will be discussed, it cannot be avoided that the currently most suitable models will be mentioned, even though such judgment is of no consequence to the key issue of Steppeler (1997a).

To the author of this paper it is quite important to make clear that he does not want to promote the SOS approach as the only possibility. In the past the high-dimensional models of GCM type had an advantage when it comes to practical prediction. It is only suggested that a low-order approach may be worth following as an alternative, in addition to the high-dimensional approaches, such as Zebiak and Cane (1987). In order to show how the different approaches, in particular high- and low-dimensional methods, can be used in concert rather than in concurrence; section 5 gives an example. Since the author is not involved in high-dimensional ENSO simulations, the example will rather concern a climatological application at quite a different scale. It is the project LITFASS (Lindenberg Inhomogeneous Terrain—Fluxes between Atmosphere and Surface: A Long-Term Study Project), aiming at investigating the parameterization of surface fluxes for models of 3-km mesh length using a field experiment and a model of 100-m mesh length. The key assumption of this model is the parameterizability of the surface fluxes, which essentially means that surface fluxes are defined by a low-dimensional manifold. Even though the model used to reproduce the fluxes is high dimensional (scales ∼100 m), the assumption of the existence of a low-dimensional manifold can be used to make optimal use of profiler measurements in order to steer the model.

## 2. Definitions and examples concerning the concept of dimension used with SOS

Some of the low-order methods discussed by Steppeler (1997a) or Tsonis and Elsner (1998) use constructed phase spaces. The phase space *S* used with SOS is the space of the physical system. For example, for the atmospheric system it consists of fields such as temperature, winds, etc. The system is generally infinite-dimensional, since the finite-dimensional discretized computer models are still not accurate enough for many purposes of climatology. The system will have a time development defined by dynamic equations, but this will not be used, as SOS is applicable also to economic systems, which do not have a dynamic equation in the phase space.

Now consider an *n*-dimensional parameter space with elements *r* or a regular subset of it. An *n*-dimensional manifold in *S* is a differentiable mapping associating a vector, **v**(*r*), in *S* with any element, *r.* With the SOS concept we generally do not have an explicit knowledge of **v**(*r*), a functon that because of the infinite dimension of *S* would require a huge number of parameters and measurements to be approximately determined. Any state **v**_{0} of the system could be written as **v**_{0} = **v**(*r*_{0}) + **v**′, with **v**′ being the nonpredictable part of the system. In most phase spaces, least squares procedures can be defined in order to define the value of *r*_{0} corresponding to the above-mentioned partition for a given **v**_{0}, if the manifold **v**(*r*) is given. Since, however, we do not know **v**(*r*) explicitly, we cannot take advantage of this partition.

If we have an observation, *o*(*t*), for example, an observation of temperature, it is a priori clear what this is as a function of **v**_{0}, where **v**_{0} is assumed to be time dependent. For example, if *υ* is a temperature field, *o*(*υ*) is the field value at the measurement site. This defines a predictable part of the observation *o*(*r*) = *o*[*υ*(*r*)], where now we assume that *r* is time dependent. As *υ*(*r*) is not assumed to be known, we do not know *o*(*r*) a priori. It has to be determined making functional assumptions on *o*(*r*) and minimizing a least squares functional using a dataset. The dynamic laws are also obtained from a least squares functional. They provide an equation of motion for *r.*

Since with respect to the parameters *r* to be determined at the initial time, all data are indirect, the distinction between direct and indirect, does not exist. Variational principles, as commonly used in high-dimensional models in order to analyze the initial state, are able to use the nonlinear dynamic interactions in order to determine parameters not directly related to the measurements. In the example of ENSO prediction of Steppeler (1997a) an ocean-related parameter was determined using an atmospheric observation. The use of another measurement, such as ocean temperature, may be expected to increase the accuracy of the initial state.

The merits of basing a method on manifolds, rather than spaces, may not be clear from the above definitions but will become evident considering two simple examples. Unfortunately the triviality of these examples may pose a difficulty to understanding. At a meteorological colloquium at the University of Bonn in 1992, Kraus complained of the triviality of the examples, which made him unable to consider the relevance of SOS to low-order climate evaluation (Kraus 1992, personal communication). This contributed to the reasons that the examples were missing from Steppeler (1997a).

The assumption of linear subspaces for *υ*(*r*), which is implicit in all methods based on EOF implies *υ*(*r*) = *υ*_{1}*r*_{1} + *υ*_{2}*r*_{2} + . . . , with *r*_{1}, *r*_{2}, . . . being the amplitudes belonging to *υ*(*r*), and *υ*_{1}, *υ*_{2}, . . . being the basis functions of the subspace. Many successful applications of subsystems achieve solutions on a very low-dimensional manifold, where the lowest-dimensional space containing this solution is quite high.

The first example is the computation of decrease of sunshine due to shadowing by a planet in the planetary system. The phase space consists of fields *υ*(*x, y, z*), where *υ* contains fields such as density of planetary matter. The subsystem of interest is *c*(*x* − *x*_{0}, *y* − *y*_{0}, *z* − *r*_{0}), where *x*_{0}, *y*_{0}, and *z*_{0} describe the position of the planet casting the shadow and *c* is its characteristic function. Since *x*_{0}, *y*_{0}, and *z*_{0} will be determined as a function of time *t,* we have a one-dimensional manifold as a subsystem (a line). This manifold is strongly curved, not at all a linear space (a straight line). The smallest space approximating this line depends on the resolution for which the result is required, but the dimension is at least of the order *L*/*d,* where *L* is the length of the planetary path and *d* is its diameter. If you prefer to use the model of a phase space of point masses, the manifold is still one-dimensional, and the smallest space containing it has the dimension 3*n,* where *n* is the number of planets under consideration.

The other example is a water reservoir with known inflow and a resistance law for the outflow, which is not necesarily known. The attractor consists of states where all water is at the bottom of the reservoir. This attractor has a high dimension since many velocity fields are consistent with the condition. A high-resolution finite difference model has to be of a high spatial resolution in order to resolve the turbulent fluid motion responsible for the resistance of the outflow. A one-dimensional subsystem is defined if one is interested only in the time development of the height of the water level. The time development of this water level is largely independent of the particular state of the attractor chosen with this water level. Different motions are possible with the same height of the water, but these are of little consequence for the time dependence of the water level. An a priori model can be done if the resistance law of the outflow device has been determined in a wind tunnel and if the shape of the reservoir has been cartographically determined. However, it is not necessary to know the exact shape of the basin. The surface area of the water and the amount of outflow as a function of height is sufficient to determine an equation of motion for *h.* More complicated models are not necessarily the best. The high-resolution model must be such that it simulates the turbulence at the outflow accurately enough in order to create an accurate resistance law. In case of a somewhat insufficient grid resolution or due to numerical errors this may not always be the case. The a priori construction of the model for *h* will require an accurate chart of the reservoir including the water soaking of the soil. If an accurate time sequence of height measurements is available, the simple modeling approach based on a one-dimensional manifold can in some situations perform better than the more elaborate models.

These simple examples should be enough to show the importance of admitting manifolds rather than spaces for approximation. The SOS method is able to catch these solutions easily. It is left to the reader to detail how these trivial solutions are excluded by the basic assumptions made with many of the methods discussed.

## 3. Some approaches to low-order modeling and ENSO prediction

Steppeler (1997a) gave some arguments in order to consider low-order modeling with the aim to produce realistic results as an alternative to existing approaches to forecasting, of which the most successful are still high-dimensional. The approaches refered to by Steppeler (1997a) and Tsonis and Elsner (1998) are not the only ones, and probably not the most successful ones, in terms of operational type forecasts.

A most important class of low-order models are paradigmatic models, which have been analyzed for a long time with the aim of understanding a process in principle, rather than quantitatively. Steppeler (1997a) referenced a paradigmatic model of ENSO. Lorenz initiated the study of low-order models in order to derive their general properties. One could refer to these as low-order conceptional models to explore consequences of nonlinearity. These have turned out to be quite important, and the basic concepts derived seem to be applicable quite generally to problems of nonlinear systems and climate. In referring to the concept of an attractor, Steppeler (1997a) made use of one of these concepts. The papers referenced by Steppeler (1997a) and Tsonis and Elsner (1998) sufficiently document these developments.

Another class of models could be described as low-order basis function models. Here it is assumed that the general form of the fields is known, and only some amplitude and form parameters need to be determined. This and the determination of the dynamic equation can be achieved using the Galerkin method. Typical applications are flow regimes not being too turbulent. Steppeler (1978) computed realistic convection rolls between plates using a three-parameter model, and Steppeler (1979) gave an estimate of the critical Reynolds number of plane Couette flow, from a three-parameter model. In the latter case three parameters were sufficient because a manifold was used for approximation rather than a linear space. Nonlinear instability models using spaces had to use at least 20 parameters.

Steppeler (1997a) argues in favor of the use of low-order models for purposes of practical prediction and makes suggestions in this respect. However, it remains to be admitted that high-order models based on the numerical discretization of the dynamic equations of the system are currently the method of choice for practical prediction purposes. This applies certainly to the problem of weather prediction but also to problems of climate prediction. The latter models have a special difficulty. While the models due to their use of consistent approximations can be expected to simulate the time development with some accuracy for limited periods of time, they do not necessarily produce the correct fluxes and ocean–atmosphere and land–atmosphere interactions. Some results obtained in the past indicated that numerical errors can in fact get out of hand. Full GCM approaches to coupled ocean–atmosphere integrations, which in particular apply to the ENSO problem, are currently under way but need to be tested. Currently, the most promising approach to ENSO modeling (Zebiak and Cane 1987) uses a high-dimensional finite difference model, based on simplified oceanic and atmospheric models. It may be described as a simplified GCM approach. Due to these simplifications, the problem of coupled modeling mentioned above may be less severe.

For purposes of this paper it may be useful to compare the different strengths of the different methods. Paradigmatic models do not apply to realistic forecasting. An attempt to adapt the conceptual model of Lorenz (1984) formally to practical prediction of the circulation index by Steppeler (1990) has failed. Low-order basis function methods may have their narrow area of application; the same may apply to the Grassberger and Procaccia (1983) approach. This latter approach is often used only to obtain estimates of the attractor dimension, rather than for prediction purposes. Therefore only low- and high-order approaches remain to be compared. High-order approaches have the advantage that, when they work, they give a first principle solution, which is applicable also in parameter ranges, which have not been previously experienced. On the other hand, they need many input parameters, which have to be known, and problems of error propagation may be severe. The models are so complicated that not all processes can be properly represented in a limited model development time. For example, the ENSO models presently use mainly the ocean–atmosphere interaction and have the impact of land surfaces only crudely represented. Low-order systems are, for each subsystem, limited to parameter ranges already experienced in the past and are limited to dynamic laws simple enough that they can be determined by a limited amount of data. They may, however, be more accurate in the representation of the bulk impact of a subsystem in cases where the details of this subsystem are unknown. They may also be faster to develop.

## 4. History of the consideration of subsystems

To find the first use of subsystems it is necessary to go back quite a bit in the history of science. An early application has been done by Newton. The subsystem is the center of mass of a planet. In order to define its motion the detailed structure of the planet is unnecessary to know. Newton even deducted the law of gravity using records of data on the planets. A variational scheme for such problems has been used by Gauss.

While in the example above the impact of the detailed structure of the system on the motion of the center of mass can be neglected, it is turbulence parameterization where the subsystem is the laminar part of the flow and the rest, the turbulence, provides the forces acting on the subsystem. Credit for the treatment of such problems probably has to go to Prandtl.

The author is uneasy when discussing issues regarding the history of science, because it is likely that some ancient Greek natural philosopher has already considered subsystems.

## 5. LITFASS, a project using low- and high-order models and observations in concert

The LITFASS project [see Müller et al. (1995)] is a long-term observational campaign to investigate the fluxes in inhomogeneous terrain. It uses a nonhydrostatic model on scales of 100 m to interpolate between the measurements. The basic assumption of this project is the parameterizibility of surface fluxes. This means that the fluxes are determined by parameters of the solid earth and the atmospheric field parameters in the blending height, which is the height where the inhomogeneity of the flow due to differences of surface properties is no longer present. The parameters of the solid earth are determined using a realistic surface model in combination with measured radiation and rain input. The dimension of surface forcing producing realistic surface parameters is not yet known. According to these assumptions the surface fluxes define a low-dimensional subsystem of the atmosphere, the dimensionality being the number of parameters necessary to force the atmospheric model.

A model will be used to compute the subsystem and the assumption of the parameterizibility can be checked. The observations of LITFASS will be used to check on the ability of the atmospheric model to provide realistic surface fluxes due to forcing of the model by observed fields. The assumption of parameterizibility will require that the horizontally averaged atmospheric fields in the blending height are sufficient to produce correct surface fluxes. However, the accuracy of current nonhydrostatic models (see Steppeler 1997b) is not considered sufficient to produce realistic profiles. This is due to the insecurity of radiative and adiabatic warming. It seems necessary to force the whole horizontally averaged profile of the model by observations of profiler instruments. A first appproach would be to identify the vertical profile *T*_{υ}(*z*) with that of the measurement. Below the blending height this might produce a bias due to internal boundaries caused by terrain inhomogenueties near the measurement site. A better approach would be to assume *T*_{υ}(*z*) = *f*[*T*_{m}(*z*)], where *T*_{m}(*z*) is the measured profile and the function *f* is determined by a variational procedure involving differences between the model and LITFASS measurements.

## 6. Conclusions

Low-order modeling is an area with a huge potential range of applications and the questions raised may lead to discussions on many topics of great scientific interest. Steppeler (1997a) tried to avoid entanglement in a discussion of a very wide range of questions by being very concise, limiting the presentation to the technicalities of SOS and a discussion of the low-order approach being most similar to SOS. Obviously, the presentation was too short to make clear the distinguishing features of SOS and the practical advantages expected from these. Some more information in this respect is given in this paper. In discussing some more of the alternative approaches it is still not intended to comprehensively discuss all questions of this very large field of problems. Many of the questions are still open now, for example, which modeling approach is the best. Neither Steppeler (1997a) nor this reply intends to answer these questions.

Tsonis and Elsner (1998) and Steppeler (1997a) seem to agree that low-order modeling is worth trying also for practical forcasting purposes. Concerning the technicalities of such approaches, I believe that the special features proposed in connection with SOS are practically useful. The most important of these features are the use of manifolds with a dimension smaller than that of the attractor and the use of variational principles that do not require a full knowledge of this manifold.

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