A method to overcome the problem of 'slow' sensorsJan H. Schween1, Jochen Reuder2

1Meteorological Institute University Munich, Germany , jan.schween@lrz.uni-muenchen.de

2Geophysical Institute, University of Bergen, Norway

IntroIn general sensors have a certain inertia, they follow a sudden change of the measured quantity with a certain delay. As a result the signal received appears to be smooothed in time. In many cases it is desirable to have a faster response of the sensor.

We are measuring Temperature (T) and Humiditiy (rH) on the cable cars at Zugspitze mountain, Germany (see Poster EGU06-A-01076). Every ride of the cable cars a profile is measured. Due to the inertia of the sensors the measured values of the ride up differ from the ride down resulting in a hysteresis like effect (see fig. 1). We use a numerical method to get around this Problem.

AcknowledgementsWe thank the Bayerische Zugspitzbahn (BZB) for the possibility to equip the cable cars with our instruments.

The Bannerwolken project is granted by the German Research Foundation (DFG)

ResultsFigure 2 shows the result for the temperature profiles of the Vaisala Sensor (V) shown in Figure 1b. The original measured values were smoothed with the filter according to equation [4] and a time constant f = 6s (i.e. the sensor was made artificially slower). Reconstruction of the ambiance Temperature was made using equation [1*] with x = 50s.

The effect of noise is visible in Figure 3: The signal of the Fischer Sensor (F) is noisy and even a filtering with f = 14s leads to strong noise in the reconstructed Profile (x = 162s).

Despite this noise there is general agreement between the two reconstructed Profiles indicating that the reconstruction is a good approximation to reality.

Numerics

1[1]

dyy x

dt

y = sensor valuex = ambiance value = time constant

special solution for x = const. :

0( ) ( ) exp( / ) [2]y t x y x t

1( ) ( ) exp [3]

t s ty t x s ds

The general solution of eq.[1]

A tangent at every point of y(t) reaches after t= the ambiance value x (equation [1*]).

( ) ( ) (1 ) ( ) [4]

exp( / )

y t q y t t q x t

q t

is a low pass filter and can be aproximated by a recursive Filter

Equation [1] and [4] can be solved for the ambiance value x giving:

( ) ( ) [1*]t

dyx t y t

dt

( ) ( )( ) [4*]

1

y t q y t tx t

q

Both solutions are very sensitive to noise in y(t) (derivation of a noisy signal). Accordingly the signal y has to be smoothed, e.g. with eq.[4] and an additional time constant f.

t/

y0

x

Fig.1b: T vs P for different sensors (F and V)

MethodSince the time constant of Thermometers depends on the wind speed at the sensor, it is not constant. Successive down and up rides of the cable car are used to estimate :

smoothingf = 6s [14s]

reconstructionx = 50s [2:42]

22x xup xdn

i

T T

Tdn

Tfdn

Txdn

Txdn (i·P)

Tup

Tfup

Txup

Txup(i·P)

Interpolation 2min x

Fig.2: measured (T), filtered (Tf) and reconstructed (Tx) Profiles of the Vaisala Sensor Fig.3: as Fig.2 for the Fischer Sensor (F)

Fig.1a: cable car data 10.12.2004 T vs time

down up