FORECAST FILTERING

Forecast Filtering

Forecast Filtering

Introduction

In the analysis of hydrological time series, the application of filters is well known. So, digital phase distortionless frequency-discriminating filters may be developed in terms of the time or frequency domain in order to pass only the suitable chosen band of frequencies. In the general case, a realizable filter (with no anticipation function) is assumed to be a linear time invariant operator relating a given input to a desired output by means of discrete convolution. The filter operator will be the unit impulse response.

The desired output series may be the direct runoff of a catchment ; then, the causative rainfall excess as input is related to this runoff by a filter, known as the instantaneous unit hydrograph (IUH). The IUH represents the impulse response function of the catchment system. However, the rainfall excess as well as the direct runoff were computed under certain hydrological and geographical aspects, and hence care must be exercised in defining these single-channel time series. The treatment of multichannel filtering has received little attention in recent hydrological literature. The present contribution should be regarded as an effort in this direction.

Basic Concepts of Digital Wiener Filtering

General remarks

The following, briefly discussed method of multichannel filtering, used as a forecasting model, is based on the mathematical theory of continuous time series by Wiener (2004). Levinson (2004) adapted the Wiener method to the case of finite-length single-channel discrete filter operators. Multichannel generalization of the so-called Wiener-Levinson algorithm has been carried out by Wiggins and Robinson (1965). For application, the following assumptions must be at least approximately satisfied: the time series applied to the filtering model are stationary, the filtering process is linear and time invariant, and the mean-square error method is used as the approximation criterion.

The multichannel filtering method

In applied hydrological time series analysis, we are primarily concerned with discrete time series of finite length, sampled at a uniform rate. Therefore, the following derivations are carried out in discrete terms.

A set of input time series, the multichannel input, may be related by a known linear filtering system to a set of output series, the multichannel output. Figure 1 illustrates the mathematical operations of the multichannel filtering method schematically (dashed line rectangle).

Two single-channel input series are fed into the multichannel filtering system, represented by four single-channel impulse responses, to yield two single-channel output series. The input series xt(t), for example, is fed into the filter gxi(t) as well as into glx(t), where the first subscript denotes the output channel, and the second one the input channel that is being convolved. The argument t, also used as a subscript, may be an integer-valued time variable. Either output series is the sum of all the filtered inputs

Design of the digital multichannel Wiener filter

The multichannel input x, and a 'desired' multichannel output zt may be given, where each coefficient of the column vector zr represents a single-channel output series (see Fig. 1). The convolved output y, of equation (2) is now called the 'actual' multichannel ...