Roberto J. Acosta and James A. Nessel

Glenn Research Center, Cleveland, Ohio

David D. Morabito

Jet Propulsion Laboratory, Pasadena, California

Data Processing for Atmospheric 

Phase Interferometers

NASA/TM—2009-215599

April 2009
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Roberto J. Acosta and James A. Nessel

Glenn Research Center, Cleveland, Ohio

David D. Morabito

Jet Propulsion Laboratory, Pasadena, California

Data Processing for Atmospheric 

Phase Interferometers

NASA/TM—2009-215599

April 2009

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
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Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2009-215599 1

Data Processing for Atmospheric Phase Interferometers 

Roberto J. Acosta and James A. Nessel 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

David D. Morabito 

National Aeronautics and Space Administration 

Jet Propulsion Laboratory 

Pasadena, California 91109 

Abstract 

This paper presents a detailed discussion of calibration procedures used to analyze data recorded from 

a two-element atmospheric phase interferometer (API) deployed at Goldstone, California. In addition, we 

describe the data products derived from those measurements that can be used for site intercomparison and 

atmospheric modeling. 

Simulated data is used to demonstrate the effectiveness of the proposed algorithm and as a means for 

validating our procedure. A study of the effect of block size filtering is presented to justify our process for 

isolating atmospheric fluctuation phenomena from other system-induced effects (e.g., satellite motion, 

thermal drift). A simulated 24 hr interferometer phase data time series is analyzed to illustrate the step-bystep calibration procedure and desired data products. 

Introduction 

The phase measurements made by an interferometer allow for observations of differential delay paths 

through the atmosphere as a function of time. The parameters that principally affect the differential delay 

path at Ka-Band are, in order of significance: water vapor, temperature, and pressure. Since most 

atmospheric water vapor is confined to the troposphere (<10 km) with an exponential scale height of 2 km 

these measurements are primarily affected by the variability in the precipitable water vapor (PWV). The 

water vapor that is exposed to temporal turbulent air flow conditions in the troposphere generates an 

irregular (random) distribution of PWV. This results in temporal variation of the refractive indices of the 

atmospheric irregularities, which leads to variations in the effective electrical path length (phase) of an 

electromagnetic wave propagating through the atmosphere (ref. 1). 

In this paper, we provide a detailed analysis of the effect of block size filtering in detecting and 

estimating the energy of the atmospheric induced phase fluctuations. The accuracy of the proposed 

algorithm is then quantitatively described through simulated data with known characteristics. 

Interferometer Raw Phase Time Series 

The NASA atmospheric phase interferometer (API) has been deployed at the NASA Deep Space 

Network (DSN) tracking complex at Goldstone, California, since May 2007 (ref. 2). The API records the 

raw phase data in intervals of 1 sec (86400 points per day). This data collection architecture will be 

adopted by the proposed algorithm and is the starting point for the analysis described herein. 

Besides atmospheric disturbances, the raw phase time series contains “2π wraps”, phase fluctuations 

due to the orbital motion of the satellite, slowly varying instrument drifts due to temperature changes in 

transmission lines and receivers, phase noise in the receivers, and errors in the electronic data collection 

(bad points). Before further analysis can proceed, this satellite motion, as well as other slowly varying 

system-induced variations must be removed from the raw phase time series data. 
NASA/TM—2009-215599 2

Root Mean Square Phase Integration Time 

The most fundamentally important quantity that is to be derived from the interferometer phase time 

series (after calibration) is the root-mean-square (rms) value of the phase as a function of time. The rms

provides a direct estimate of the energy of the atmospheric-induced phase fluctuations on a given time 

scale. 

An interferometer will directly detect atmospheric-induced phase fluctuations on time scales ranging 

from 0 to (b/v) or larger, where b is the physical baseline distance and v is the average wind speed aloft 

(ref. 3). The term (b/v) is called the crossing time. This term is defined as the time for a water vapor 

frozen screen to travel from one antenna to the next assuming a constant wind speed v (ref. 4). In order to 

accurately compute the rms of the phase, it is necessary to integrate over many crossing times 

(e.g., > ~ 40 x b/v). 

The physical baseline distance is a known quantity (e.g., 256 m for the NASA API), but the average 

wind speed aloft is not an easily obtainable quantity. In order to estimate crossing times for the NASA 

API we need to make an assumption for the average wind speed aloft at Goldstone, California. A 

reasonable approximation is to assume a factor-of-two increase with respect to the average surface wind 

speed (a measurable quantity). The average surface wind speed for one year of measurements (2007 to 

2008) at 90 percent of the time was approximately 8 m/s (ref. 5). Given our 256 m baseline, the average 

crossing time is estimated to be approximately 16 sec. Using the “rule-of-thumb” given above, we require 

that at least 600 sec be used as the interval for estimating the rms of phase fluctuations. Over a 600 sec 

period, however, there are significant phase variations caused by satellite motion and slowly varying 

system thermal effects. In order to ensure a minimal contribution of these effects in the rms computation 

of atmospheric phase fluctuations a quadratic fit is utilized for the removal of those effects at each 

integration period. 

Data Calibration Procedure 

The Goldstone API stores the raw interferometer phase daily. Each data set contains 86,400 data 

points (1 record per second). The 24 hr data sets are calibrated by using the following step-by–step 

process detailed below. 

Step 1 

The 24 hr time series of phase data is first unwrapped (eliminating all 2π turns). The unwrapped data 

is segmented into smaller blocks of data. The block size is determined by the rms integration time. If we 

choose 600 sec as the integration time, the 24 hr data set will be segmented into 144 blocks, where each 

block contains 600 points. 

Step 2 

A second-order polynomial is fit (least mean square criteria) to each block of data. The polynomial fit 

is subtracted from the raw data points within each block. The resulting data points form a calibrated block 

time series. The calibrated blocks are free from satellite motion and any slowly varying system effects. 

Notice that this step does not eliminate any rapidly varying system effects (e.g., phase noise from 

oscillators, etc.). 

Step 3 

Check for bad data blocks. Bad data blocks are defined in our procedure as blocks with missing 

points, bad data points within the block, and other known instrument or site anomalies. We have removed 

those entire blocks from any statistical computation of the phase rms. 
NASA/TM—2009-215599 3

Description of Data Products

The primary data products that can be determined from the calibrated data blocks include: 

1. The block rms. The block rms is the most fundamentally important quantity in the analysis of the 

API phase. The block rms is a direct estimate of the energy of the atmospheric-induced phase fluctuations 

in the block time scale (integration time). 

2. The block temporal phase structure function. This is an indirect way to estimate the phase 

fluctuation rms on different block time scales. The maximum resolvable time scale in this product is the 

block time duration. The minimum resolvable time scale corresponds to the data sampling rate 

(e.g., 1 sec). 

3. The block temporal power spectrum. The block temporal power spectrum provides a display of the 

power of the atmospheric-induced phase fluctuations as a function of frequency. The area under the curve 

of the power spectrum plot provides an estimate of the energy of the fluctuation or rms. 

4. The cumulative distribution function (CDF) and probability density function (PDF) of the rms. 

The CDF is a measure of the probability that a certain phase rms will be present at any given time. This 

CDF is normally utilized for site comparison and for system design of a widely distributed antenna 

system. 

Data Calibration Applied to a Simulation Case 

In order to confirm the validity of the methodology, we applied the procedure to a simulated raw 

phase data set with known parameters. 

The simulated raw interferometer phase is described by the following equations: 

( ) 2.2689 radians

86400

2

sat 20 2 1 cos ⎥ →

⎦

⎤ ⎢

⎣

⎡ ⎟

⎠

⎞ ⎜

⎝

⎛ +

π φ n = − N (1.a) 

( ) radians

300

2 sin

100

2 sin

10

2 sin

180 atm 5 2 ⎥ →

⎦

⎤ ⎢

⎣

⎡ ⎟

⎠

⎞ ⎜

⎝

⎛ π ⎟ +

⎠

⎞ ⎜

⎝

⎛ π ⎟ +

⎠

⎞ ⎜

⎝

π ⎛ π φ n n n n (1.b) 

( ) [ ] radians

180 noise Gaussian 86400, , 0, 2 →

π φ n = μ σrms → μ = σrms (1.c) 

φrawphase ( ) n = φsat(n) + φatm (n) + φnoise (n) → radians (1.d) 

The first term (Eq. (1.a)), simulates the effect of satellite motion as detected by the interferometer. 

The second term (Eq. (1.b)), models the atmospheric-induced phase fluctuation by using three sinusoidal 

terms. The last equation (Eq. (1.c)) models the system random phase noise by using a Gaussian 

distribution with zero mean. The three terms are added (superposition applies) to form the basis for 

simulating the raw interferometer phase time series. 

Figure 1(a) displays a simulation (wrapped version of Eq. (1.d)) of a typical phase time series 

recorded by an interferometer in a 24 hr period. Notice the many 2π turns in the raw data throughout the 

course of a day. In order to further analyze the data, it is necessary to remove these 2π wraps (Step 1 in 

the data calibration process). 
NASA/TM—2009-215599 4
NASA/TM—2009-215599 5

After the unwrapping function is applied to the data in Figure 1(a) the resulting interferometer raw 

phase time series will not have any 2π discontinuities, as shown in Figure 1(b). 

The next step in the procedure requires that the un-wrapped (Fig. 1(b)) phase time series data be 

segmented into blocks. We assume an rms integration period of 600 sec. This corresponds to 144 blocks 

in a given 24 hr period (one day). A second-order polynomial is fit to each block (144 blocks) and the 

difference between the block raw phase and the block polynomial results in a calibrated block (satellite 

motion removed). The resulting phase time series for the first four calibrated blocks are shown in 

Figure 1(c). 

Effect of Polynomial Fit and Block Size 

The process of fitting a second order polynomial to each data block for removing the satellite motion 

component may add an additional bias or error to the computation of the block rms. The error 

contribution of this procedure can be investigated and simulated by only considering the satellite motion 

component (Eq. (1.a)). Figure 2 shows the maximum error (unwanted) as function of block size. Notice 

that as the block size increases, the maximum error in each block also tends to increase. This can be 

explained by the departure of a satellite motion’s shape from second-order. Nevertheless, for block sizes 

ranging from 600 to 3600 sec, the error contribution from the polynomial fit is negligible. 

Effect of Block Size in the Computation of rms 

In theory, the minimum block size required for detecting the energy of the atmospheric-induced 

fluctuations is determined by having an rms integration time that is much greater than the crossing time. 

This situation can be numerically investigated by simulating the atmospheric-induced fluctuations by 

using several sinusoidal terms of fixed frequencies (see Eq. (1.b)). Since the atmospheric-induced 

fluctuations are known in the simulation, we can compute the error in detecting and estimating the energy 

that each sinusoid contains as a function of rms integration time or block size. 
NASA/TM—2009-215599 6

For the simulation, we selected three sinusoids with periods of 10, 100, and 300 sec, respectively. 

Each sinusoid has 5° rms amplitude. The simulation will not contain any random system noise 

(Eq. (1.c)). Figure 3 shows the average block rms as a function of block size. Notice the average block 

rms converges to the exact rms value (√3×5°) as we increase the block size time. 

Practically speaking the rms converges for a block size time of ~3600 sec, which is a factor of twelve 

times of that of the largest sinusoidal period (300 sec) in the simulation. Notice that at 600 sec, the block 

size error is small (0.4°), which is a factor three smaller than the phase noise for the interferometer. 

Derived Products—Simulation Case 

The 24 hr rms block time series is obtained by computing the standard deviation (rms with zero 

mean) of each calibrated block (144 blocks). The resulting rms time series for the entire 24 hr period is 

presented in Figure 4(a). This product depicts the variation of the energy of the fluctuations (fixed 

integration time) as a function of time. The random temporal variation of the fluctuations can be easily 

seen. 

In addition to the block rms time series, we can also derive the average (over 144 blocks) block 

temporal structure function and the average (over 144 blocks) block phase power spectra. Figure 4(b) 

shows the block temporal structure function. Notice that at large time scales the average rms is 

approximately equal to the average block rms computed from the block rms time series. The periodic 

structure seen for time scales longer than 100 sec is characteristic of that due to sinusoids. The null 

structure and peaks of the variation for large time scales is predictable (ref. 6). 

The average block power spectrum is depicted in Figure 4(c). Notice that the power levels of the three 

sinusoidal components show up as spectral peaks with equal strength (as expected). The effectiveness of 

the block filtering can be indirectly seen in this plot by noticing that the energy at low frequencies is 

decaying very rapidly below any significant levels. The area under the curve in this plot also provides a 

measure of the rms (the energy of the fluctuations) in any frequency band, therefore it can be also be used 

to determine the frequency (period) of the fluctuations. 
NASA/TM—2009-215599 7
NASA/TM—2009-215599 8

Concluding Remarks 

In this paper we described the rationale and validation for calibrating the raw phase time series 

measured by a two-element atmospheric phase interferometer (API). We discussed the impact of block 

size (block filtering) in determining the energy of the atmospheric phase fluctuations through simulation. 

The simulation results showed that a second-order polynomial fit is very effective in removing any slowly 

varying system and satellite motion on time scales ranging from 600 to 3600 sec. The impact of selecting 

a block time of 600 sec for the computation of rms was found to be at least three times smaller than the 

random phase noise power found in a typical two-element API. 

References 

1. Thompson, A.R., Moran, J.M. and Swenson. G.W.: “Interferometry and Synthesis in Radio 

Astronomy,” 2nd Edition, John Wiley & Sons, 2001. 

2. Acosta, R.J., Frantz, B., Jirberg, R., Nessel, J.A., and Morabito, D.D., “Goldstone Site Test 

Interferometer,” Proceedings of the 13th Ka and Broadband Communications Conference, 

Sep. 2007, Turin, Italy. 

3. Holdaway, M.A., Radford, S.JE., Owen, F.N., and Foster, S.M.: “Data processing for site test 

interferometers,” ALMA Memo No. 129, Jun. 1995. 

4. Tatarski V.I.: “Wave Propagation of the Turbulent Medium,” Dover: New York, 1961. 

5. Nessel, J.A, Acosta, and Morabito, D.D., “Phase Fluctuations at Goldstone Derived From One-Year 

Site Testing Interferometer Data,” Proceedings of the 14th Ka and Broadband Communications 

Conference, Sep. 2008, Matera, Italy. 

6. Liu, Q., Nishio, M., Yamamura, K., Miyazaki, T., Hirata, M., Suzuyama, T., Kuji, S., Iwadate, K., 

Kameya, O., and Kawano, N.: “Statistical Characteristics of Atmospheric Phase Fluctuations 

Observed by VLBI System Using a Beacon Wave from a Geostationary Satellite,” IEEE Transactions 

on Antennas and Propagation, vol. 53, no. 4, Apr. 2005. 
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14. ABSTRACT

This paper presents a detailed discussion of calibration procedures used to analyze data recorded from a two-element atmospheric phase 

interferometer (API) deployed at Goldstone, California. In addition, we describe the data products derived from those measurements that can 

be used for site intercomparison and atmospheric modeling. Simulated data is used to demonstrate the effectiveness of the proposed 

algorithm and as a means for validating our procedure. A study of the effect of block size filtering is presented to justify our process for 

isolating atmospheric fluctuation phenomena from other system-induced effects (e.g., satellite motion, thermal drift). A simulated 24 hr 

interferometer phase data time series is analyzed to illustrate the step-by-step calibration procedure and desired data products. 

15. SUBJECT TERMS

Interferometer; Deep Space Network (DSN); Goldstone; Phase stability analysis; Array; Propagation; Atmospheric effects 

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