Detrending time series

Short definition of Detrending time series
Detrending typically refers to removing the low-frequency content of the time series

This is the common definition for Detrending time series, but other definitions maybe discussed within the wiki.

In the context of analysing turbulence observations, the resulting detrended signal should be stationary and contain mostly contributions from turbulence and surface waves (if present). However, frame interference (wakes), vibrations and measurement noise may also contaminate the detrended signal.

Measured velocities at 4 Hz from an Acoustic-Doppler Velocimeters are highlighted against trends obtained from three different techniques. Empirical modal decomposition (EMD) , linear trend, and a 2nd order low-pass Butterworth filter. A cut-off period of 10 min was targeted by both the filter and EMD. Detrended signal would be the difference between the original and the trend

There is no exact definition for what consists of a "trend", nor any set algorithm for identifying the trend. The following techniques can be used for detrending :

1. Linear trend removal
2. Low-pass linear filters
3. Empirical modal decomposition

The first two methods presume the original time series is stationary and linear, while the third is adaptive and applicable to nonlinear and non-stationary time series. Note that chopping a time series into smaller subsets, i.e., segments, is essentially a form of low-pass (box-car) filtering.

Implications for estimating [math]\displaystyle{ \varepsilon }[/math]

Velocity point measurements

The choice of detrending technique has little influence on the estimation of [math]\displaystyle{ \varepsilon }[/math] from the spectral fitting of the inertial subrange as it influences only the lowest wavenumber of the spectra, which most algorithms identity as outliers. Detrending becomes an important consideration when performing other turbulence calculations such as estimating Re stresses [math]\displaystyle{ \overline{u'w'} }[/math] or turbulent kinetic energy.

512 s segment of the measured velocities after applying different detrending methods

Example velocity spectra of the short 512 s of records before and after different detrending techniques applied to the original 6h time series. The impact of the detrending method can be seen at the lowest frequencies only