Mark

Mark

Mark

The question “is there feedback?” stems from a mathematical model of the system. I maintain that the basic equation used by SB and D is simply that of linear first order system that does not contain feedback. This model can be modified to contain feedback, using the thermodynamic model of internal functionals defined in Judith Curry’s textbook. Although her analysis is steady state, this can easily be handled by making the internal variables ODEs wrt time.

However, the real problem is being able to distinguish between the models using temperature and flux data, given a relatively short period of data and subject to errors?

I am sure that there is feedback, but the equations used by SB and D do not contain feedback.

If you read Judith Curry’s chapter on thermodynamic feedback in her book on atmospheric physics textbook, she defines it correctly to be a sum of internal functionals, which makes sense. My quiblle with this definition is that it is static analysis and, were each each internal functional treated as a complex function of time, dynamic analysis in control theory terms becomes straightforward.

The difficulty, I believe, is distinguishing between non-feedback as implied by SB and D and a system in which the feedback is defined as a set of ODEs relating internal variables from flux measurements.

If a time(or any other variable) signal passes through a linear system, it produces an output.

A liner system is defined in many ways, but it can be described in terms of LINEAR differential equations (which may be partial in systems with more than one variable), as is SB2011 and D2011. A more signal-processing orientated definition is that the output of a linear system is Proportional to the input, is stationary (i.e.: its properties don’t vary with time) and it obeys superposition: if you put in the sum of two waveforms, the output are the sum of the individual outputs of the individual inputs.

The process of an input being modified by a linear system to produce an output is called a convolution. If the signals are treated as a function of time, the system’s behaviour is defined by its Impulse response – i.e.: what the system does when it is presented with an input of infinite magnitude and infinitesimal duration (and an area of 1). In this case the output is calculated from the convolution integral.

A more powerful method of analysing linear systems (and much more effecient computationally) is the use of integral trasforms: Fourier, Laplace, Z… In this case the behaviour of the system is defined in terms of complex frequency. In this representation, the behaviour of the system is known as the transfer function. Convolution in this domain is multiplication of the transform of the input by the transfer function and the output signal can be recovered by inverse transformation.

To calculate the output of a system, defined by a set of linear DEs, in principle one does the following:

The input signal undergoes a discrete Fourier Transform*, the transform is multiplied by the calculated transfer function and the result is the subjected to an inverse DFT to obtain the output.

Deconvolution is identifying the system when one has an input and output signal. The transfer function of the system is calculated by dividing the transform of the output by the transform of the input and the impulse response of the system is the inverse transform of the calculated transfer function.

In SB2011, we know the analytical form of the transfer function because it is defined by their differential equation, and the parameter lambda/Cp is to be estimated.
This can be done, in principle through deconvolution, which I would suggest is a mathematically “purer” and physically more interpretable method than contortions through OLS.

This idea pervades system analysis, communications, electronics etc and there is a huge amount of work on identification of systems in the presence of noise. Good starting points in the field are “Signals and Systems” by Oppenheim,Willsky & Young (Prentice/Hall) or Random Data by Bendat & Piersol (Wiley)

* One has to realise that integral transforms are an analytical idea and include limits of integration between +- infinity. Thus computation of the transform of real, sampled, signal is highly restricted because the transform is not a continuous function.