Seeing Analysis Theory

ATST Science Working GroupNov. 17, 2003

The data

High Low

Shabar scintillation cross-correlationraw data

Average over day for 2 seeing layer heights

Basic ideas 1

• Atmospheric turbulence creates variations in n(index of refraction)

• Assuming turbulence in form of a “thin” layer, then wave passing through layer suffers phase shift Φ due to variations in n.

• Important assumptions: Φ depends only on horizontal position and Φ « 1

• Can then represent continuous turbulence as superposition of contributions from thin layers

Basic ideas 2

• Phase fluctuation at base of layer is observed at ground as intensity fluctuations related to amplitude of the wave

• The amplitude has a spatial power spectrum determined by the turbulence in the layer.

• Assuming Kolomogorov turbulence, the spectrum is

• where λ is the wavelength, f is the spatial frequency, h is the height of the layer, δh is the layer thickness, and Cn

2 is the structure function.

Basic ideas 3

• For continuous turbulence, the phase fluctuations add linearly ( ).

• For statistically independent layers, the power spectra add linearly.

• Then for continuous turbulence,

Φ «1

• The scintillation is 4x the amplitude fluctuations.• The real part of the Fourier transform of the power spectrum is the

spatial covariance, B(x) where x is a distance. B(x) is measured by the Shabar.

Single-layer star at zenith example

Extended source away from the zenith

• Equations must be modified to account for fact that Sun is not a point source and not at the zenith.

• Incorporation of non-zenith case is simple: replace height h by line-of sight distance to layer z, with z = h sec ζ

• Then, power spectrum is

Detector geometry

• Effect of extended source depends on detector geometry.

• θ is the solar angular diameter.

• At height h, the rays from the sun to the detector pass through an area defined by the cone in the drawing.

Averaging over the solar disk

• Amplitude from point source case must be averaged over the beam area changing power spectrum to

• Where is the spatial power spectrum of the beam cross-section:

The Kernel

Can replace sin term by its argument, so

The function Qs: detector separation in units of cone diameter at height h and zenith angle ζ

The Kernels for each Shabar detector

Another view of the Kernels

From the theory to the observations

• This is a classic inversion problem• Use method of regularized least-squares

Determining the coefficients

• The numerics for doing this is non-trivial, next talk

• Once the coefficients ai are determined, the function Cn

2 can be computed.• Finally, get r0(h) from

A few limitations

• The Shabar analysis can provide information only near the ground.

• The height range of the current Shabarinstrument is at best 1000 m, but typically much less at non-zero zenith angles.

• In order to correctly constrain the high-altitude contribution to the seeing, it is necessary to include the S-DIMM measurement of r0 .