Noise and Sensitivity of RasClic 91

● 10 days of measurement

● frame rate 12.4 Hz

● periods of enhanced

perturbations excluded

(human presence)

● 223 frames (8.4 Mio)

included in analysis

Raw Noise Spectrum of the Signal

● FFT of x and y signal

● step elimination at

data gaps has no

discernible effect

● discard lowest 16

components for lack

of relevance

● 16 component bins

for reduced scatter

(x0.25)

Noise Spectrum in Physical Units

● amplitude in m Hz-1/2

as a square root of a

spectral power density

● white noise at high f

( > 250 mHz )

● 1/f noise at low f

( < 10 mHz )

● broad resonances at 220

and 440 mHz

● some structure at 10—

100 mHz

Current Limits to RasClic Sensitivity

Noise (white noise) and drifts (1/f noise) in the signal:

● white noise: uncorrelated single-point uncertainty = 200 nm

Improvement potential:

➔ up to 100x : better algorithm for image position (Kramer-Rao-limit)

➔ up to 4x on noise amplitude: increase sample rate to 200 Hz

● 1/f noise: random-walk of the image position

Improvement potential:

➔ up to 100x (1000x?) by temperature control (insulation and supervision)

Compare Sensitivity with Seismometers

Limitation as given by empirical noise;

comparison with seismometers requires:●determining RasClic sensitivity as a function of wavelength

●converting wavelengths into seismic oscillation frequencies

●converting noise levels at these frequencies into equivalent seismic

accelerations

●comparing with seismometer specifications

● RasClic Sensitivity as a Function of Wavelength

Assume that the Earth radius is modulated by r() = R0 + a with a = A cos k,

and k the number of periods around the circumference.

The resulting local radius of curvature is (to first order) () = R – (k2–1) a .

For the value x measured by RasClic (the shift of the end point with respect to a

straight line pointing through the start point and the center) with length L follows:

E.g., with L = 91m the sensitivity parameter for the quadrupole mode (k = 2) is:

x/a = 5.1*10–11.

x 2 a 11

k21 cos

k L

2 R

Converting Wavelengths into Seismic Frequencies

Typical seismic waves have phase velocities of 4-6 km/s, corresponding to earth

round trip times of 3 to 2 hours. The graphs show phase velocities and frequency

vs. wavelength of 0S

n modes, used in this analysis.

● Sensitivity as Function of Seismic Frequency

x / a as a function of

frequency for 0S

n modes

comparison for three

different L values:

91 m, 500 m, 20 km.

The maximum value is 4,

when the wavelength

equals L; for shorter

wavelength the

sensitivity oscillates,

becoming zero at integer

fractions of L/2.

● Convert Noise Levels into Equivalent

Accelerations

For comparison with

seismometers, convert

position noise into

accelerations. g – g

dB units refer to the

scale unit m2s–4Hz–1.

E.g., –100 dB means

(10-5m)2s–4Hz–1

These are motions of

the SIGNAL (x), not of

the EARTH (a)!

Noise-Equivalent Earth Surface Acceleration

Dividing accelerations

of signal (x) noise by

the sensitivity

parameter x/a provides

a value of what

seismic acceleration at

a given frequency

would be required to

equal the observed

(status quo) noise.

Comparison RasClic (status quo) – KNMINoise background comparison

RasClic (status quo) vs. state-

of-the-art seismometer (KNMI)

Projected improvements will

result in:

●40 dB white noise reduction by

improved image position analysis

●12 dB white noise reduction by

increased data rate

●40–60 dB 1/f -noise reduction by

temperature control

●every doubling of L (up to the

seismic wavelength) gives an

improvement of 12 dB

Fundamental Noise Limit

Noise limit to sensitivity

determined by: (figures

current projected)

●individual

uncertainty(180 nm 1

nm)

●repetition rate

(12.4 Hz 1 kHz)

●length dependent

sensitivity

(91 m 140 m 500 m

20 km)

There is no intrinsic random walk no intrinsic 1/f

noise