Propagation of Instrumental Errors to the SkyTemperature Measurement

Raul Monsalve

SESE, Arizona State University

May 2, 2013

Description

This report studies the impact on the sky temperature Tsky , ofdeviations in different instrumental parameters from theirnominal values.

An idealized instrumental model is assumed, described byequation 13 of the Rogers & Bowman (2012)1 paper.

The analysis is performed in the range 100-199 MHz.

1Rogers, A., & Bowman, J., Absolute calibration of a wideband antennaand spectrometer for accurate sky noise temperature measurements, RadioScience, Vol 47, RS0K06, 2012.

2

Equation for the Sky Temperature

From Equation 13 of the paper:

Tsky =

[1− |Γl |2

(1− |Γa|2)|F |2

]×[

Tamb + Tcal(Pant − Pload )

(Pcal − Pload )− Tu

|Γa|2|F |2

(1− |Γl |2)

− (Tc cosφaF + Ts sinφaF )|Γa||F |

(1− |Γl |2)

](1)

I F =

√1−|Γl |2

1−ΓaΓl

I φaF = phase(Γa · F )

3

MethodsErrors are expected on the following vector parameter:

θ̄ = (Tamb,Tcal ,Tc ,Ts,Tu, |Γa|, φa, |Γl |, φl) (2)

These errors are propagated to the sky temperature using twomethods:

1. Calculus-based approach (Taylor expansion, assumingsmall errors)

2. Perturbation-based approach

Both cases require the use of the nominal value and the errorassigned to each of these parameters.

Computations are performed at a frequency resolution of 1MHz, and in some cases interpolation or model-fitting arenecessary to take the data to this resolution.

The following slides present the nominal values of theparameters along with a brief description.

4

Fiducial Values: Ambient and Excess Temperature

100 120 140 160 180 200frequency [MHz]

0

100

200

300

400

500

Te

mp

era

ture

[K

]

ambient

excess

Figure: Nominal values of 300 K and 400 K are used for the ambient and excess temperatures.

5

Fiducial Values: Noise Wave Parameters

100 120 140 160 180 200frequency [MHz]

0

10

20

30

40T

em

pe

ratu

re [

K]

cosine

sine

uncorrelated

Figure: These values and slopes were obtained from the Rogers & Bowman (2012) paper, figure 8. They wereprojected to the 100-199 MHz range at 1 MHz resolution.

6

Fiducial Values: Spectra

100 120 140 160 180 200frequency [MHz]

0

2.0x10−09

4.0x10−09

8.0x10−09

1.0x10−08

1.2x10−08

Pload

Pcal

Pant

Figure: These spectra were obtained from current EDGES data. Interpolation was used to reduce thefrequency resolution to 1 MHz. Some noise remains after interpolation.

7

Fiducial Values: Antenna and LNA Magnitude

100 120 140 160 180 200frequency [MHz]

−25

−20

−15

−10

−5

0

|Γ| [d

B]

antenna

lna

Figure: The reflection of the ferrite-balun antenna is used, expected to be close to the one in the field. It is ameasurement performed last year, no interpolation or smoothing involved. The LNA profile was provided by Hamdi.The measurement was very noisy, and therefore modeled using a 5th-order polynomial for smoothing purposes.The red trace is the model.8

Fiducial Values: Antenna and LNA Phase

100 120 140 160 180 200frequency [MHz]

−600

−400

−200

0

200

Φ(Γ

) [d

eg

]

antenna

lna

Figure: Same as previous plot.

9

Description of Method 1

I Vector parameter

θ̄ = (Tamb,Tcal ,Tc ,Ts,Tu, |Γa|, φa, |Γl |, φl)

I Sky temperatureTsky = f

(θ̄)

(3)

I Error

σ2sky =

9∑i=1

(∂f∂θi

)2

σ2θi

+9∑

i=1

∑k 6=i

2∂f∂θi

∂f∂θk

σθiθk (4)

10

Description of Method 1: Matrix Form

σ2sky =

(∂f∂θ1

∂f∂θ2

. . . ∂f∂θ9

)σ2θ1

σθ1θ2 . . . . . .

σθ2θ1 σ2θ2

. . .

. . . σ2θ9

∂f∂θ1∂f∂θ2

. . .∂f∂θ9

σ2

sky = J · C · JT (5)

I The Jacobian matrix J is computed symbolically, and thenevaluated at the fiducial vector parameter θ̄0, at eachfrequency.

I The covariance matrix C is generated assuming realisticerrors.

I Errors are assigned for one parameter at a time, in order tosee their individual impact on the sky temperature.

I No covariance is assigned.11

Description of Method 2

In the second method, the propagation of errors is performedas:

∆Tsky = f (θ̄∗)− f (θ̄0) (6)

I f (θ̄) = Tsky

I θ̄0: fiducial vector parameterI θ̄∗: vector parameter after perturbation of one parameter

12

Comparison of MethodsThe following plots show a comparison of the two methods.

In Method 1, the error in parameter θi is characterized by astandard deviation σi . The output error is also a standarddeviation, always equal or larger than zero.

In Method 2, the input and output are differences with respectto the fiducial values. Since the output difference can bepositive or negative, the plots show its absolute value. Thissimplifies the comparison between the methods.

The assigned errors (or perturbations) are constant infrequency. Two values have been chosen for each parameter,as follows:

I Temperatures: 1 K and 5 KI Magnitude of reflection coefficient: 0.000115 and 0.00115

(0.01 dB and 0.1 dB at -20 dB)I Phase of reflection coefficient: 0.1◦ and 1.0◦.

13

∆Tambient = 1 [K]

100 120 140 160 180 200frequency [MHz]

0.0

0.5

1.0

1.5

2.0∆

Tsky [

K]

method 1

method 2

Figure: Results with the two methods overlap.

14

∆Tambient = 5 [K]

100 120 140 160 180 200frequency [MHz]

0

2

4

6

8∆

Tsky [

K]

method 1

method 2

Figure: Results with the two methods overlap.

15

∆Texcess = 1 [K]

100 120 140 160 180 200frequency [MHz]

0.0

0.5

1.0

1.5

2.0∆

Tsky [

K]

method 1

method 2

Figure: Results with the two methods overlap. Noise is propagatedfrom that of the spectra (page 7).

16

∆Texcess = 5 [K]

100 120 140 160 180 200frequency [MHz]

0

2

4

6

8

∆T

sky [

K]

method 1

method 2

Figure: Results with the two methods overlap. Noise is propagatedfrom that of the spectra (page 7).

17

∆Tcosine = 1 [K]

100 120 140 160 180 200frequency [MHz]

0.0

0.2

0.4

0.6

0.8

1.0∆

Tsky [

K]

method 1

method 2

Figure: Results with the two methods overlap.

18

∆Tcosine = 5 [K]

100 120 140 160 180 200frequency [MHz]

0

1

2

3

4∆

Tsky [

K]

method 1

method 2

Figure: Results with the two methods overlap.

19

∆Tsine = 1 [K]

100 120 140 160 180 200frequency [MHz]

0.0

0.2

0.4

0.6

0.8

1.0∆

Tsky [

K]

method 1

method 2

Figure: Results with the two methods overlap.

20

∆Tsine = 5 [K]

100 120 140 160 180 200frequency [MHz]

0

1

2

3

4∆

Tsky [

K]

method 1

method 2

Figure: Results with the two methods overlap.

21

∆Tuncorr = 1 [K]

100 120 140 160 180 200frequency [MHz]

0.0

0.1

0.2

0.3

0.4

0.5

0.6∆

Tsky [

K]

method 1

method 2

Figure: Results with the two methods overlap.

22

∆Tuncorr = 5 [K]

100 120 140 160 180 200frequency [MHz]

0.0

0.2

0.4

0.6

0.8

1.0∆

Tsky [

K]

method 1

method 2

Figure: Results with the two methods overlap.

23

∆|Γant| = 0.000115 (0.01 dB @ −20 dB)

100 120 140 160 180 200frequency [MHz]

0.00

0.02

0.04

0.06

0.08

0.10

0.12∆

Tsky [

K]

method 1

method 2

Figure: Differences between methods are evident for the antennareflection coefficient. They are expected, since equation 1 isnon-linear in this parameter. Shape is similar, however, which servesas validation.24

∆|Γant| = 0.00115 (0.10 dB @ −20 dB)

100 120 140 160 180 200frequency [MHz]

0.0

0.1

0.2

0.3

0.4

0.5

0.6∆

Tsky [

K]

method 1

method 2

Figure: Differences between methods are evident for the antennareflection coefficient. They are expected, since equation 1 isnon-linear in this parameter. Shape is similar, however, which servesas validation.25

∆Φ(Γant

) = 0.1°

100 120 140 160 180 200frequency [MHz]

0.00

0.02

0.04

0.06

0.08

0.10

0.12∆

Tsky [

K]

method 1

method 2

Figure: Differences between methods are evident for the antennareflection coefficient. They are expected, since equation 1 isnon-linear in this parameter. Shape is similar, however, which servesas validation.26

∆Φ(Γant

) = 1.0°

100 120 140 160 180 200frequency [MHz]

0.0

0.2

0.4

0.6

0.8

1.0∆

Tsky [

K]

method 1

method 2

Figure: Differences between methods are evident for the antennareflection coefficient. They are expected, since equation 1 isnon-linear in this parameter. Shape is similar, however, which servesas validation.27

∆|Γlna| = 0.000115 (0.01 dB @ −20 dB)

100 120 140 160 180 200frequency [MHz]

0.00

0.02

0.04

0.06

0.08

0.10

0.12∆

Tsky [

K]

method 1

method 2

Figure: For the LNA reflection coefficient, the results are almostindistinguishable. At some frequencies the difference seems toincrease for larger input error, due to non-linearity.

28

∆|Γlna| = 0.00115 (0.10 dB @ −20 dB)

100 120 140 160 180 200frequency [MHz]

0.0

0.1

0.2

0.3

0.4

0.5

0.6∆

Tsky [

K]

method 1

method 2

Figure: For the LNA reflection coefficient, the results are almostindistinguishable. At some frequencies the difference seems toincrease for larger input error, due to non-linearity.

29

∆Φ(Γlna

) = 0.1°

100 120 140 160 180 200frequency [MHz]

0.00

0.02

0.04

0.06

0.08

0.10

0.12∆

Tsky [

K]

method 1

method 2

Figure: For the LNA reflection coefficient, the results are almostindistinguishable. At some frequencies the difference seems toincrease for larger input error, due to non-linearity.

30

∆Φ(Γlna

) = 1.0°

100 120 140 160 180 200frequency [MHz]

0.0

0.2

0.4

0.6

0.8

1.0∆

Tsky [

K]

method 1

method 2

Figure: For the LNA reflection coefficient, the results are almostindistinguishable. At some frequencies the difference seems toincrease for larger input error, due to non-linearity.

31

Summary

The following table summarizes the results, obtained with method 2. This method is preferred since its output erroris allowed to be positive or negative.

The RMS value is computed in the range 100-199 MHz, in four ways: 1) direct, 2) after removing an offset, 3) afterremoving the best-fit function of the form k · xα (k and α are fit parameters), and 4) after removing the best-fitfunction of the form k1 · xα + k2 (k1, k2, and α are fit parameters).

The results due to errors in reflection coefficients are compared to those presented in Alan’s memo 105, for the casewith no attenuation between the switch and the LNA (first column of his first table). His results represent the range50-200 MHz, but this was the closest configuration I found to compare to my results. His results due to temperatureerrors are not directly comparable to mine, so I do not show them here.

RMS residuals in mK in 100-199 MHzsource error magnitude direct offset scale scale + offset Alan’santenna S11 0.01 dB 36 29 25 22 63antenna S11 1◦ 268 259 227 222 417LNA S11 0.1 dB 201 187 173 140 345LNA S11 1◦ 240 237 217 210 398ambient 1 K 1084 132 106 104excess 1 K 295 291 226 73cosine 1 K 245 210 202 198sine 1 K 127 127 127 127uncorrelated 1 K 107 89 82 76

As the table shows, my results are comparable to Alan’s (first four rows). In fact they are always smaller by a factorof 2 approximately. As expected, the errors decrease as more parameters are used to model them. For equal errorin temperature, the smallest impact is obtained for the excess and uncorrelated temperatures.

32

ConclusionI The impact of instrumental errors on the measured sky

temperature has been studied, although for anot-so-realistic instrumental model.

I Two methods of error propagation were used, for purposesof comparison and validation. They provide very similarresults for the level of errors assigned to the parameters.

I The nominal (or fiducial) values for some terms of equation1 had to be interpolated or modeled, in order to use auniform frequency resolution, and to appreciate the effectof a constant absolute error in the parameters on the skytemperature, with noise remaining sub-dominant.

I The results for the antenna and LNA reflection coefficientsare comparable to those presented by Alan in memo 105,although smaller by a factor of 2.

I My analysis is evolving in order to determine the parameterestimation capabilities of EDGES in the presence of skynoise, instrument noise, and foregrounds.

33