I D serial modelling (the AR(2) model)

Serial analysis has been employed to describe statistically channel meandering phenomena (Fergusson, 1975). As pool-rifiQe sequences have been likened to meandering in the vertical scale (Keller and Melhom, 1978), these techniques are appropriate to objectively classify and statistically represent their pseudo-cyclic oscillation (Richards, 1976b, 1979). Clifford and French (1998) advocate the use of stochastic modelling to characterise pool-riffle morphology because of its relative objectivity and it views the morphology as a connected upstream downstream sequence.

Serial analysis assumes that an underlying systematic pattern exists within the data, as well as a random noise element (error) which makes the pattern difficult to identify. If a series can only be described though the statistical relationships between sequential terms, it is termed stochastic, and can be modelled with increasing confidence depending on the consistency and magnitude of dependency between the sequential terms. The data, in this case the pool-riffle profile, are assumed to be a realisation of a probabilistic process, and the stochastic model describes the statistical relationships between sequential terms in the discrete data series according to probabilistic laws (Richards, 1979).

The statistical relationships between sequential terms in the data series can he derived through the technique of autocorrelation, which assesses the degree of similarity a sequence has when compared to itself at successive positions or lags (Richards, 1979). Oscillating autocorrelation function (ACF) coefficients indicate that the profile is successively moving in and out of phase with itself when lagged at increasing increments. This periodic behaviour suggests that the profile may be best described by a higher order model incorporating a greater number of preceding data values as they are still statistically related (although to a progressively lesser extent).

A useful method to examine serial dependencies is to examine the partial autocorrelation functions (PACFs). This is an extension of autocorrelation, where the dependence on the intermediate elements (those within the lag) is removed. PACFs are therefore direct measures of the excess correlation not accounted for by lower order models (Richards, 1979). The partial autocorrelation provides a 'cleaner' picture of serial dependencies for individual lags (not confounded by other serial dependencies), and indicates the order of the most appropriate model (parsimony). The PACF plot can be used as a guide to indicate whether or not the modelled data can be considered to exhibit a definite periodic tendency which constantly changes in period and phase due to the disturbance by random shocks. For situations where the autocorrelation process is expected to display pseudo periodic behaviour, (in this case reflecting the possible nature of the generating process) a second-order process provides the best fit, being a combination of periodic and random elements (Melton, 1962; Richards, 1976b; Clifford, 1993b, Carling and Orr, 2000).

A second-order autoregressive model is specified as:

Xn = 01 Xn.i + 02 Xn-2 + Cn (2.1)

where:

X = bed elevation at distance n

01 = a partial regression coefiBcient for a second-order process at lag 1

0 2=a partial regression coefficient for a second-order process at lag 2 Xn_i = previous value (at lag 1)

Xo_2 = previous value (at lag 2) e^ = random component ('shock')

If the undulations in the bed morphology are to be considered as a realisation of an AR(2) process, then the second-order coefficients at lags 1 and 2 must satisfy certain stationarity constraints. If, for example, 0 ; = 1.5 and 02 = 1, two successive positive values in the series would be cumulated in a weighted sum to give the next value, which would deviate more markedly from zero than the initial two values (Richards, 1979). For stationarity, the (AR)2 coefficients must satisfy:

02 + 0] <1

02 - 0] ^ 1

-1< 02 < 1 (2.2)

If 0 < 02 < 1, then the process exhibits different type of behaviour depending on the sign of 0 j. If the stationarity requirements are met and 0] is positive, the process will tend to consist of successive 'runs' of observations of the same sign. This is because cumulating two positive values require a larger random shock value to 'throw* the series across to negative values (Richards, 1979). If however 0 < 02 < 1 and 0 i is negative, the process will tend to exhibit marked 'up and down' behaviour. This is because cumulating positive and negative values will produce a relatively small value which is easily 'thrown' by a random shock value. As a general rule, in cases wfiere -1 < 02 < 0, the ACF exhibits sinusoidal behaviour regardless of the value of 0]. However, if 0 , is also negative (-1 < 0] < 0), the random shock value results in the series oscillating around zero and disappearing. In cases wbere -1 < 02 < 0 and 0 j is positive (0 > 0] > 1), the random component prevents the series from oscillating around zero and disappearing. Specifically, if:

01^ + 402 < 0 (2.3)

To prevent the series from oscillating around zero and disappearing, 02 must also be < 0. Should these criteria be met, the series represents a disturbed periodic model in which the behaviour is pseudo-oscillatory. The frequency of the process can be calculated through:

cos 271 f = 01 (2^- 02)'^ (2.4)

where f is the number of cycles per unit time or distance in this case, and should be interpreted as an average value (Jenkins and Watts, 1968, p i 66). The wavelength is determined by multiplying the period of the process (1/f) by the data sampling interval.

The degree to which the second-order model can explain the variance in the series can be derived from the first-order coefficients 'p’, which are simple regression coefficients at successive lags.

p / - 2p/pz + pz^ X 100 (2.5)

1 - p / where:

p' = first-order coefficient at lag 1 p^ = first-order coefficient at lag 2

In applying second-order autoregressive model to 8 rural bed profiles, Richards (1976) found the variance explained ranged from 40 to 98%, with an average of 70%. Clifford (1993b) found the second-order model to explain 72% of the variability in his rural data set and Milne (1982a) found the second-order model to explain 70% of the variability. The river profiles for this study are modelled by an AR(2) process using SPSS software (Version 9 for Windows, 1998).