## 1. Introduction

Radio occultation (RO) using the global positioning system (GPS) was demonstrated by the GPS Met mission in 1995–97, and since then a number of other GPS RO missions have flown, generally in a single-satellite configuration but once in a six-satellite configuration (Anthes et al. 2008). GPS RO soundings can be inverted for profiles of the microwave index of refraction of the atmosphere with a vertical resolution of approximately 100 m. More conventional thermodynamic state variables can be derived from refractive index profiles. [For a description of the GPS RO technique, see Kursinski et al. (1997).] The spatial–temporal distribution of GPS RO is global in extent with a sampling density that is uniform in longitude but highly nonuniform in latitude. The number of soundings per day depends on the tracking capabilities and configuration of antennas on the receiving satellite, ranging from approximately 100 to 650 soundings per day per satellite. A number of GPS RO satellites can be flying contemporaneously, so the total sampling density can vary in number and pattern depending on the types of GPS RO instruments flying and their relative orbits.

Sampling error arises from undersampling of the spatial–temporal variability of the atmosphere. Some atmospheric phenomena occur at such short time scales and small horizontal spatial scales that there can never be enough RO soundings to sample all of the structures; consequently, any global or regional average will be slightly in error. In the case of typical GPS RO distributions, the physical phenomena that are undersampled by GPS RO are those associated with weather—baroclinic eddies and fronts in mid- and high latitudes, barotropic eddies and large-scale convection in the tropics—and so we call this type of variability *synoptic variability*. With the increasing density of GPS RO soundings, more synoptic variability is resolved and then can no longer contribute to sampling error.

While GPS RO has been identified as ideally suited to climate monitoring, it is not without error. Kursinski et al. (1997) gave a thorough description of single-sounding sources of error for GPS RO and showed that errors are at a minimum in the region of the upper troposphere/lower stratosphere (8–30-km height). It is unclear how much of each source of error is systematic and contributes a bias that would affect trends. By way of comparison, Foelsche et al. (2006) showed that sampling error is a major contributor to uncertainty in climatologies of RO data products in this vertical region. The subject matter for our paper is sampling error only and none of the other known sources of error in GPS RO. Recent work in climate monitoring using GPS RO sought to remove sampling error by subsampling the operational analyses of the European Centre for Medium-Range Weather Forecasts (ECMWF) to occultation locations (Pirscher et al. 2007; Steiner et al. 2009; Lackner et al. 2011) and subtracting this climatology less the gridded truth of the analyses from the climatology formed using the GPS RO data. Here we take a different approach because we prefer not to remove sampling error from GPS RO data but instead to form climatologies of GPS RO data that extract as much useful information from the data as possible without recourse to outside information in the form of a model analysis or other data.

While GPS RO does not adequately resolve synoptic variability at present, it can still resolve larger-scale, longer-lived structures in the atmosphere such as the equator-to-pole temperature gradient and long-lived waves. When any structure is resolved, it can no longer contribute to sampling error. A longer averaging period permits more soundings for improved spatial resolution; however, a longer averaging period reduces the ability to resolve short-lived transient structures. It is possible that there is an optimal averaging period for capturing climate averages, and it should depend on the ability of the sampling pattern to resolve atmospheric structures before they fade away. We require an analysis method to determine the optimal period to construct the best possible maps of GPS RO data.

Bayesian interpolation is ideally suited to this purpose (MacKay 1992). This interpolation scheme uses a set of basis functions that are fit to irregularly sampled data that have unknown noise characteristics. Here, we define the unknown noise to be sampling error. What makes the scheme Bayesian is its use of a regularizing function—much like a penalty function—that prevents overfitting of the data without degrading the misfit of the data. (Overfitting is the phenomenon wherein fits to the data exhibit structures on scales smaller than is justified by the density of the data.) MacKay (1992) presented the method and demonstrated it for datasets with a single independent coordinate, showing that there is more evidence for some bases that fit the data than others and that there is more evidence for some regularizers than others.

In the second section of this paper we give a brief overview of Bayesian interpolation, based strictly on MacKay (1992), and will formulate Bayesian interpolation for mapping on a sphere. In the third section we analyze simulated GPS RO data based on simulated Challenging Mini-Satellite Payload (CHAMP) and Constellation Observing System for Meteorology Ionosphere & Climate (COSMIC) data to find an optimal regularizer (to prevent overfitting of the data) and to find optimal spatial and temporal resolutions for capturing climate averages. In the fourth section, we show how strong spatial variations in RO sampling density give rise to systematic sampling error in a standard binning and averaging approach to forming climatologies and how Bayesian interpolation resolves the problem. In the fifth and final section we summarize our findings and offer a discussion.

## 2. A brief overview of Bayesian interpolation

Bayesian interpolation is an implementation of hierarchical Bayesian inference. The first level of inference is least *χ*^{2} fitting of predefined basis functions to data with a prior defined so as to penalize the overfitting of data. The second level of inference selects the optimal balance between misfit of the data and overfit of the data. The result is the most probable fit. When implemented in two dimensions on a sphere, the result is the most probable values of the expansion coefficients that can be used to compose maps. For an accessible and complete introduction to Bayesian inference for the purpose of data analysis, see Sivia (2006). We follow the notation of MacKay (1992) in this brief description.

### a. First level of inference

**t**given basis set

*σ*=

*β*

^{−1/2}, and the basis expansion coefficients are

**w**:We consider the data unbiased in this application because we neither expect a bias caused by a selection effect—such as the “dry bias” engendered by systematic undersampling of the lower tropical troposphere when it is very moist (Foelsche et al. 2003)—nor do we intend to remove such a bias if it exists. We assume that sampling errors are uncorrelated and are uniform across the globe. (Neither assumption applies in reality; they are addressed in sections 3 and 5.) The prior, or the subjective estimate of the expansion coefficients

**w**in the context of a specific regularizer

*α*, is also Gaussian:The absolute value of a vector is its

*L*

_{2}norm, and the absolute value of a matrix is its determinant. The prior distribution [Eq. (2)] serves as a penalty function in that it penalizes overly large values of the expansion coefficients. The matrix

*N*and the dimension of the basis is

*k*. The matrix

**is the expansion of the basis functions at the locations of the data and thus has dimension**

*φ**N*×

*k*.

**w**after considering the data

**t**, is a Gaussian centered about

**w**

_{ML}, the most likely values of the coefficients, with an uncertainty covariance matrix

^{−1}:where

*φ*^{T}

**. This can otherwise be thought of as the least**

*φ**χ*

^{2}estimate of

**w**when

*χ*

^{2}is defined asWhen Eq. (5) is evaluated at the most likely values of the coefficients, the first term on the right is the measure of misfit of the data and the second term is the measure of overfitting of the data. We define

*χ*

^{2}(

**t**,

*α*,

*β*,

**w**=

**w**

_{ML}, and the Bayesian evidence for the data isThe Bayesian evidence is the probability that the data

**t**would have been obtained if the regularizer

*α*and

*β*.

Equations (3) and (4) are at the core of the first level of inference. Given the values of *α* and *β*, they are used to determine the coefficients of the basis and the uncertainty covariance in those coefficients. The Bayesian evidence for the fit in Eq. (6) is important in the second level of inference described next.

### b. Second level of inference

*α*and

*β*, the weights of the regularizer and of the data misfit, respectively. Those values are found by considering the evidence function of the first level of inference [Eq. (6)] as the likelihood function of the second level of inference. The corresponding prior in

*α*and

*β*is flat, or uninformative. The posterior PDF of the second level of inference is

*P*(

*α*,

*β*|

**t**,

*α*and

*β*at the maximum of this distribution, the most probable values

*α*

_{MP}and

*β*

_{MP}, obey the following properties:The number of good parameter measurements

*γ*, also called the degrees of freedom for signal in the literature of inverse problems (Rodgers 2000), is defined asMacKay (1992) gives the derivation of these properties. One can evaluate the probability of the data in light of the regularizer

*α*and

*β*. That integral can be approximated by evaluating Eq. (6) at

*α*=

*α*

_{MP}and

*β*=

*β*

_{MP}and multiplying by the approximate widths of the distribution Δ

*α*and Δ

*β*, which are approximated asUsing Eqs. (7), (8), and (9), the evidence for the data in light of the basis and the regularizer isIn what follows, we apply this equation to evaluate the relative evidence for different basis sets and different forms for the regularizer.

In the procedure of Bayesian interpolation, one begins with nominal values of *α* and *β* and applies the equations of first inference to obtain **w**_{ML}. Next, one applies the equations of second inference, beginning with Eq. (9) (the definition of *γ*). With that value of *γ*, one then updates the values of *α* and *β* using Eqs. (7) and (8), respectively. With the updated values of *α* and *β*, one reevaluates the coefficients **w**_{ML}. The iteration proceeds until *α* and *β* converge to stable values. In practice, we terminate the iteration when updates in *α* and *β* are both within 1% of the previous values. Those stable values are *α*_{MP} and *β*_{MP}, and the evaluation of **w**_{ML} at those values gives the values for the most probable coefficients of the expansion, **w**_{MP}. MacKay (1992) states that there is only one evidence maximum in *α* and *β*, hence the most probable *α*_{MP} and *β*_{MP} are unique. Also, we have never encountered a nonconvergent iteration. Maps of the data can then be constructed using the basis functions of **w**_{MP}.

We note a strong parallel to variational data assimilation of meteorological data. Bayesian interpolation is based on Bayes theorem, but so also is variational data assimilation as shown at length in Lorenc (1986). Both are linearized problems with similar assumptions, and, consequently, one can be recognized from the other. The translation of the language of Bayesian interpolation to the language of variational data assimilation is **t** → **d**, **x** → **w**, ** φ** →

*β*→

**O**

^{−}^{1},

*α*

^{−1}. Comparison shows an implicit assumption of Bayesian interpolation: the prior on the coefficients has zero mean [

**x**

_{b}(=

**w**) = 0; see second term on the right of Eq. (5)]. This assumption is expected to lead to zero-biased estimates of the coefficients

**w**.

## 3. Application to CHAMP and COSMIC distributions

In this section we describe the simulation of CHAMP-like and COSMIC-like distributions of data and investigate which regularizer is most probable, what degree of spherical harmonic expansion is most probable (and thus what basis is most probable), and how sampling error in global and zonal averages of retrieved variables in occultation depends on the regularizer, degree of spherical harmonic expansion, and binning time. It is necessary to use a simulation of data rather than actual data in this study because a simulation enables us to compare to an artificial “truth” to evaluate the performance of this mapping technique. Certainly it is possible to simulate other realistic configurations of GPS RO satellites (e.g., MetOP-1, TanDEM-X, and SAC-C), but we have chosen to simulate just CHAMP and COSMIC separately because of the long record of CHAMP and the massive record of COSMIC.

### a. Simulation of data

Here we describe the simulation of CHAMP-like and COSMIC-like GPS RO distributions of data. The Earth is defined to be an oblate spheroid with *GM* = 3.986 004 4 × 10^{14} m^{3} s^{−2}, an equatorial radius *a _{e}* = 6378.245 km, and oblateness

*o*= 1/298.3. The oblateness relates the polar radius

*a*to the equatorial radius through

_{p}*o*= (

*a*−

_{e}*a*)/

_{p}*a*. All satellites, GPS and low-earth-orbiting receivers, undergo nodal regression with

_{e}*J*

_{2}= 1.08 × 10

^{−3}(Seeber 1993). The orbits are perturbed in no other way.

We prescribe 24 GPS transmitters—4 transmitters spaced approximately 90° in anomaly in each of six different orbit planes separated 60° in ascending node. Their orbits are precisely circular. The inclination of each orbit is set to 55.0°, and the radius of each orbit is set to 26 610.223 km, making the orbital period for the GPS satellites 12 h. For CHAMP we prescribe an inclination of 87.2° and an orbital radius of 6832.245 km (Wickert et al. 2001). Only occultations that are viewed in the antivelocity direction of the orbit are permitted because CHAMP had only an aft-viewing antenna for RO. For COSMIC we prescribe six satellites spaced 30° in ascending node with an inclination of 77.99°, orbital radius of 7178.245 km, and random initial orbital anomaly (Liou et al. 2007). For COSMIC, both forward-viewed (rising) and aft-viewed (setting) occultation events are permitted because the COSMIC satellites carry both forward-viewing and aft-viewing occultation antennas. Occultation events are defined to occur when the straight-line path connecting a GPS satellite to a receiver satellite is tangent to the Earth’s surface. Recorded events are further restricted by the boresight angle: only occultation events in which the GPS transmitter falls within a 60° cone about the directional center of a limb-oriented occultation antenna are permitted. While the distributions of occultations that result are only idealized approximations to CHAMP and COSMIC RO distributions, we nevertheless refer to them as CHAMP and COSMIC distributions throughout this paper. We obtain ≈338 daily soundings for CHAMP and ≈3680 for COSMIC. These are greater than actually obtained by CHAMP and COSMIC because of data rejection in quality control and limitations in the scheduling and tracking software in actual GPS receivers.

For each occultation event we extract a single data value for fitting. That data value is taken from the interim reanalysis of the ECMWF (ERA Interim) (Dee et al. 2011). The archive of ERA Interim we used contains, among many other variables, the geopotential, temperature, and specific humidity 4 times daily on a 37-level pressure grid from 1000 to 1 hPa at 1.5° longitude and latitude resolution. For each 4-times-daily field we have computed dry pressure *p _{d}* as a function of geopotential height

*h*. Dry pressure is a commonly retrieved quantity in radio occultation analysis that approximates pressure in the upper troposphere and stratosphere but is strongly influenced by water vapor in the lower troposphere (Leroy 1997; Leroy and North 2000). In this paper we seek to map the geopotential height of the 200-hPa dry pressure surface, which is roughly 1 m higher than the height of the 200-hPa pressure surface. The 200-hPa surface is of interest because long-term temporal changes in its global mean arise because of thermal expansion of the troposphere and because changes in the pronounced horizontal gradients arise due to evolution of the dynamical jets. We linearly interpolate the logarithm of dry pressure of the ERA Interim gridded product in height and trilinearly interpolate in longitude, latitude, and time. We have simulated CHAMP and COSMIC RO data in this way for a 10-yr period and interpolated ERA Interim fields beginning in January 2000 and ending in December 2009.

### b. The optimal basis and regularizer

*Y*(

_{lm}*λ*,

*θ*) with

*λ*,

*θ*as the longitude and latitude and

*l*,

*m*the spherical harmonic degree and order. We truncate the spherical harmonic expansion at spherical harmonic degree

*l*=

*l*

_{max}. We denote the on-diagonal terms of

*C*and define them asThe coefficient

_{lm}*ρ*penalizes the global mean term. The smaller

*ρ*is, the more freedom the fitting procedure has to fit the global mean. The number of fitting coefficients in

**w**is

*k*= (

*l*

_{max}+ 1)

^{2}; we require that the number of coefficients be less than or equal to the number of data (

*k*≤

*N*) so that

*μ*penalizes higher-degree spherical harmonics in comparison to lower-degree spherical harmonics. The greater

*μ*is, the smoother the fit becomes. Recall that

*l*(

*l*+ 1)

*Y*(

_{lm}*λ*,

*θ*) is the two-dimensional curvature of the spherical harmonic

*Y*(

_{lm}*λ*,

*θ*). The coefficient

*ν*governs the penalty of purely meridional structures with respect to all other structures. We include this flexibility because all atmospheres are expected to exhibit strong meridional gradients because of equator-to-pole gradients in insolation. The smaller

*ν*is, the freer the procedure is to fit strong meridional gradients. In the case of all of these terms, it is possible to underfit the data or to overfit the data if they are not chosen ideally. We define different regularizers according to these terms (

*ρ*,

*ν, μ*).

First, we investigate the most likely values of *l*_{max} and *μ* for the CHAMP and COSMIC distributions. We use only year 2000 of the simulation, collect the data into 5-day bins, and set *ρ* = *ν* = 10^{−3}. We evaluate the sum of the logarithms of the posteriors *P*(**t**|

Figure 1 clearly shows the disadvantage of underfitting data. When the set of basis functions (spherical harmonics) is made too small, it becomes less possible to fit easily identifiable spatial structures in the data. This is reflected in the log evidence in the two plots becoming increasingly negative at low maximum spherical harmonic degree *l*_{max}. When the basis set becomes too large though, the same figures show that the data can also be overfit. This is seen most clearly for CHAMP at *l*_{max} = 15 and greater.

The effect of underfitting can also be seen in increasing the exponent *μ*. For the CHAMP distribution, the maximum log evidence is found for *μ* = 1.5 and *l*_{max} = 14. The log evidence asymptotes in shape but decreases rapidly for exponents greater than ~1.5. For these larger exponents, smaller-scale atmospheric structures are not resolved because their corresponding spherical harmonics are overly penalized. For the COSMIC distribution, the maximum in log evidence occurs for *μ* = 2.0 and *l*_{max} = 18. Because there is little difference in log evidence for *μ* = 1.5 and *μ* = 2.0 for the CHAMP distribution, we find that the optimal exponent for the regularizer is *μ* = 2. Also, we take as the optimal spherical harmonic expansion degree 14 for the CHAMP distribution and 20 for the COSMIC distribution. These results are contingent on the binning period being 5 days.

Second, we investigate the most likely values of the global mean penalty coefficient *ρ* and the meridional gradient coefficient *ν*. The daily average log evidence, calculated the same way as for Fig. 1, is shown in Fig. 2 based on the simulation of the CHAMP distribution. The plot has *ρ* ranging from 10^{0} to 10^{−8} and *ν* ranging from 10^{0} to 10^{−3}. Of the family of curves of constant *ρ*, the one with *ρ* = 10^{−1} lies above the others. The log evidence is nearly flat between *ν* = 10^{0} and *ν* = 0.3. The results are the same for the COSMIC distribution but not shown. There is little to be gained by relaxing the penalty in the global mean or by relaxing the penalty for purely meridional structures. Based on log evidence, we take *ρ* = *ν* = 0.3 for the optimal regularizer.

In Fig. 3, we show example fits for the CHAMP and COSMIC data when conditioned on the optimal regularizer described here. The CHAMP data are far less dense than the COSMIC data, but the fits obtained by Bayesian interpolation show very strong similarities.

### c. Sampling error

Sampling error is the error incurred by undersampling the atmosphere in space and time. We can evaluate its statistics by examining the differences between maps obtained by Bayesian interpolation and the gridded truth of ERA Interim, from which the simulated data are taken. We examine the statistics of sampling error after varying the exponent *μ* of the regularizer, the maximum degree of the spherical harmonic expansion *l*_{max}, and the fundamental binning time to provide another look at the performance of Bayesian interpolation.

In theory, the uncertainty covariance for the fitted coefficients **w** is just ^{−1}, from which the uncertainty in derived quantities such as maps of the data, estimates of global averages, and spatial gradients can be calculated. In practice, though, the theoretical estimate of the uncertainty covariance will be incorrect because of the assumptions that were made concerning sampling error (see section 2a). Therefore, it is more appropriate to evaluate the performance of Bayesian interpolation by direct computation of sampling errors after fitting than by evaluation of ^{−1}.

### d. Monthly averages

We anticipate that the greatest interest in interpolating and mapping RO data is for the sake of climate, and ultimately the researcher will be interested in monthly averages. As discussed in the introduction, there is a potential advantage in forming monthly averages by better resolving temporal structures. Therefore, we examine the sampling error in monthly average fits after first dividing the data into many smaller bins in time.

To illustrate the general pattern of sampling error, we show mean-square sampling error for monthly average maps when using 5-day bins for the CHAMP and COSMIC distributions. To form true monthly averages, each month is divided up into 5-day bins but with enough 6-day bins included to account for the proper number of days in each month. Fits for each of those bins are averaged together after being weighted by the number of days in the bin to form a monthly average map. We then compute the mean-square monthly sampling error over 10 years of simulations. Results for the CHAMP and COSMIC simulations are shown in Fig. 4.

Four phenomena are readily apparent in Fig. 4: sampling errors decrease with increasing degree of the spherical harmonic basis, COSMIC has less sampling error than CHAMP, the greatest sampling error lies in the mid- to high latitudes between 30° and 70°, and a minimum *l*_{max} is necessary to resolve atmospheric structure. COSMIC captures monthly climate averages better than CHAMP because of the greater density of COSMIC soundings. Even though Bayesian interpolation is a sophisticated procedure, the statistics of the result nevertheless follow closely the statistics of random processes, which have error decreasing in inverse proportion to the square root of the number of soundings. Higher-degree spherical harmonic bases yield smaller sampling error because they are better able to resolve fine spatial structure in variability. Sampling error is greatest in midlatitudes because synoptic variability of the height of the 200-hPa surface is greatest in midlatitudes. Greater sampling density in the midlatitudes offsets the greater synoptic variability, but only somewhat. The banding structure seen most pronouncedly at the spherical harmonic basis of maximum degree *l*_{max} = 10 is caused by underresolution of the meridional structure of the atmosphere. The banding structure becomes less pronounced at *l*_{max} = 14 and especially at *l*_{max} = 18. Finally, no significant reduction in sampling error is gained by resolving CHAMP data at *l*_{max} = 18 rather than at *l*_{max} = 14. This is an indication that CHAMP offers no significant information at spatial scales smaller than those resolved at maximum degree 14.

Quantitatively, CHAMP maps, when resolved at *l*_{max} = 18, obtain monthly averages with the standard deviation ranging from 4 to 8 m in the tropics with a wavenumber-4 oscillatory pattern in longitude peaking at 120°W, 30°W, 60°E, and 150°E longitude. The cause of these peaks is uncertain and the reason for their positions remain unknown to us, but the reason for their existence must pertain to the underresolution of variability by CHAMP sampling density. COSMIC maps, when resolved at the same resolution, obtain standard deviations of 2–3 m in the tropics without any noticeable longitudinal structure. In the midlatitudes, CHAMP maps show standard deviations of 10–15 m with minor longitudinal structure while COSMIC maps show standard deviations of 5–6 m without longitudinal structure.

We show an analysis of the effects of the regularizer on monthly sampling error in Fig. 5. The analysis is performed by region: northern midlatitudes (40°–70°N) in Fig. 5a, the tropics (20°N–20°S) in Fig. 5b, and southern midlatitudes (70°–40°S) in Fig. 5c. Clearly, the sampling error is least in the tropics, greatest in the northern midlatitudes, and somewhat less in the southern midlatitudes, a consequence of synoptic variability of the height of the 200-hPa dry pressure surface in midlatitudes being much greater than in the tropics. The dependence of monthly average sampling error on the exponent of the regularizer [*μ* in Eq. (13)] shows substantially different behavior for the CHAMP distribution than for the COSMIC distribution. For the COSMIC distribution, sampling error is almost perfectly independent of the exponent *μ*. For the CHAMP distribution though, *μ* = 1 shows evidence of overfitting as the order of the spherical harmonic basis increases. While the sampling error for *μ* = 2, 3, 4 asymptotes at higher degrees of the basis, it asymptotes to different values for CHAMP and COSMIC. Generally, *μ* = 2 asymptotes to the lowest values, consistent with the finding in section 3b that *μ* = 2 is a strong choice. Overall, COSMIC should be able to establish monthly averages of the height of the 200-hPa dry pressure surface with a sampling accuracy of 6 m in midlatitudes and 2 m in the tropics; CHAMP should be able to do the same with a sampling accuracy of 12–13 m in midlatitudes and 6 m in the tropics. The statistics are pointwise, meaning that the standard deviation is relevant for every longitude–latitude position in the respective region. Extended regional averages should exhibit a smaller sampling error.

### e. Optimal binning time

Synoptic variability not only contributes to sampling error because of its structure in space but also because of its structure in time. The above analysis has demonstrated that improving the spatial resolution of the fit—by increasing the number of spherical harmonics that define the basis—reduces sampling error until overfitting sets in for the sparse CHAMP-like density of soundings. Those statistics are based on fits of 5-day periods of data. Reducing the fitting time to 2-day periods ought to resolve more temporal structure in synoptic variability but at the expense of resolving spatial structure.

In Fig. 6 we show a comparison of the sampling errors of monthly averages based on 2- and 5-day binning for both CHAMP and COSMIC distributions. We have set *μ* = 2, *ρ* = *ν* = 0.3 in composing Fig. 6. In all three regions and for all maximum degrees of the spherical harmonic expansion, monthly averages based on 2-day binning exhibit less sampling error than those based on 5-day binning. Instead of there being a trade-off between temporal and spatial resolution, binning in shorter time intervals permits the resolution of more spatial structure rather than less for both CHAMP and COSMIC. The gain is fairly minimal though, and it is almost nonexistent in the tropics. In the northern midlatitudes, for example, the CHAMP-like distribution can determine pointwise monthly average height of the 200-hPa dry pressure surface with a sampling error of 13 m when based on 5-day binning and with a sampling error of 12 m when based on 2-day binning.

### f. Interpolation at shorter time scales

We investigate the utility of mapping radio occultation data at time scales much shorter than a month. Because sampling error is generally random, it is expected that sampling error at shorter time scales should be greater by a factor proportional to the inverse square root of the sampling interval. In Fig. 7 we show the sampling error for 5- and 2-day maps of the CHAMP and COSMIC distributions. For every given sampling time and maximum degree of expansion, the sampling error of the CHAMP distribution is greater than that of the COSMIC distribution, in keeping with the argument given above. For each mission and maximum degree of expansion, the sampling error for the 2-day maps is greater than the sampling error for 5-day maps, also in keeping with the above argument. The figure also shows that, for a given mission and sampling time, sampling error is reduced with increasing spatial resolution (or maximum degree of expansion), but only up to a point.

For CHAMP, improvement in sampling error is attained up to *l*_{max} = 14 in all regions but not significantly with greater *l*_{max}. For COSMIC though, sampling error is reduced up to *l*_{max} = 20 with a suggestion of little improvement at greater *l*_{max}. This behavior is a consequence of the smearing of synoptic variability with binning time. The longer the averaging time of the atmosphere, the more likely it is that small spatial structures should be smeared out by atmospheric motion. The larger the spatial structure, the longer it should take for wind to smear it out. The effect is more noticeable in COSMIC than in CHAMP because COSMIC is better able to resolve small spatial-scale features to begin with.

## 4. Systematic error and resolution

The sampling pattern of GPS RO gives rise to a systematic sampling error when an algorithm of binning and averaging is used. In binning and averaging, bins of finite spatial size and temporal duration are specified, individual satellite-based soundings are collected into those bins, and the climatology is the average of all soundings in each bin. Because GPS RO sampling patterns are nonuniform in latitude though, binning and averaging skews climate averages toward the portions of each bin where the sampling is densest. Because the sampling pattern contains weak singularities, the effect is especially pronounced, and it is impossible to eliminate sampling error completely by specifying arbitrarily small bins. (It is generally not in the climatologist’s interest anyway to specify too small a bin because the decreased number of soundings in each bin increases averaging error.)

To illustrate the phenomenon, we compute the sampling error of binning and averaging using the CHAMP and COSMIC distributions. The results are shown in Fig. 8. The simulated data were first binned in 5-day temporal bins and 15° longitudinal bins before forming annual, zonal averages. The latitudinal resolution of the density histograms is 0.25° to highlight the weak singularities in sampling density at 21° and 49° in both hemispheres for both CHAMP and COSMIC. In fact, it can be shown that the singularities occur when the occultation transmitter is at maximum declination: each singularity corresponds to the singularity in the probability density function of the transmitter’s orbital declination near its maximum and minimum declinations. (The maximum declination of a satellite is the same as the absolute value of its orbital inclination.) With the GPS orbits and the Earth’s shape as specified in section 3a, the precise locations of the singularities are 21.14° and 48.84° north and south latitude.

Figure 8 shows the sampling density in the 5° latitude bins for both the CHAMP and COSMIC distributions for each of the 10 years of the simulation. The weak singularities are clearly visible. In the latitude bins that contain those singularities, binning and averaging does in fact skew the averages toward the value of the climatological background at the location of the singularity. Figure 8 also shows the sampling error in zonal and annual averages for each year of the 10 years of simulation. The sampling error varies from year to year, but there is a systematic error over all the years. It is more apparent in the COSMIC simulation than in the CHAMP simulation because of the greater density of COSMIC soundings. We generate a theoretical estimate of the systematic sampling error by first convolving the sampling density with the climatological background in each latitude bin and subtracting the average background in each bin, the result being the colored curves in Figs. 8b,d. Nonuniformity of sampling density is definitely the cause of the interannual systematic sampling error, especially in those bins that contain the weak singularities.

Bayesian interpolation is expected to negate the problem of systematic error. Because Bayesian interpolation fits data using basis functions of position and time, it effectively recognizes where there are anomalous clusters of data. In its internal error analysis, it determines that sampling error as can be computed from ^{−1} [cf. Eq. (4)] is much smaller where the data are densest. Where data are sparse, the data that are available will be fit without being overfit if the regularizer is defined appropriately.

In Fig. 9 we demonstrate that Bayesian interpolation solves the problem of systematic error incurred by nonuniformity in sampling density but that it incurs systematic error by underresolving meridional structure in the Southern Hemisphere. In this figure we have used *μ* = 2, *ν* = *ρ* = 0.3, and *l*_{max} = 14 for CHAMP and *l*_{max} = 20 for COSMIC. The theoretical systematic error for binning and averaging is reproduced from Fig. 8. The sampling error curves that result after analysis by Bayesian interpolation show no influence from the systematic error that results from sampling nonuniformity. On the other hand, the sampling error in the Southern Hemisphere exhibits a sinusoidal systematic error caused by underresolution of the atmospheric background. Moreover, the size of this systematic error depends on sampling density: because COSMIC sampling is so much denser that CHAMP sampling, COSMIC data can be resolved at maximum spherical harmonic degree 20 but only at 14 for CHAMP, and the smaller basis for CHAMP means less of the background structure can be resolved. Nevertheless, the systematic sampling error of Bayesian interpolation in zonal and annual averages does not exceed 1 m in the Northern Hemisphere. In the Southern Hemisphere, the systematic sampling error does not exceed 3 m for CHAMP or 1 m for COSMIC. Anomalous behavior exists over the south polar region, south of 85°S. In this region, CHAMP and COSMIC exhibit a systematic error that peaks at −6 m and at −2 m, respectively.

## 5. Summary and discussion

We have shown how Bayesian interpolation can be applied to map GPS radio occultation data, a method ideally suited because it is intended to determine the most likely fits to data that are randomly but nonuniformly distributed and that have unknown error. In the case of GPS RO, the error is associated with unresolved synoptic variability with only a minor contribution from sounding error. The method was introduced thoroughly by MacKay (1992), and we have given a brief overview here that is sufficient for implementation of GPS RO.

We have incorporated a spherical harmonic basis for interpolation, and hence mapping, on a sphere. We have also presented a form for the regularizer that corresponds to the spherical harmonic basis with three tunable parameters. We have simulated idealized distributions for CHAMP and COSMIC RO data and generated an artificial dataset by interpolating ERA Interim to the locations of the simulated distributions. In this paper we have chosen to work with the geopotential height of the 200-hPa dry pressure level as the data field because of its importance to numerical weather prediction and climate research. We have evaluated the influence of the tunable parameters in the basis and the regularizer by examining the probability—or Bayesian evidence—of fits and sampling error, which is the difference between the fits to data and the gridded truth. We have analyzed sampling error primarily in monthly averages that were divided into 2- or 5-day bins before forming monthly averages.

We have found that the optimal binning time of CHAMP and COSMIC data is 2 or fewer days and that the optimal spherical harmonic maximum degree is 14 for CHAMP and 18–20 for COSMIC. Truncating the spherical harmonic basis has the primary effect of underresolving the background atmosphere (cf. Fig. 4). While we posed a trade-off between resolving spatial structures and resolving temporal structures in the introduction, we found that resolving spatial structures actually requires resolving temporal structures because of the finite lifetime of small spatial features; thus, fitting degrades with longer binning times (cf. Fig. 6). We have found that a penalty that is proportional to the square of the curvature [*μ* = 2 in Eq. (13)] is well suited to Bayesian interpolation. Even though the atmosphere contains large means and large equator-to-pole gradients in most meteorological fields, there is no improvement in performance by relaxing the penalty in global means or in purely meridional structures. Consequently, we have chosen *ν* = *ρ* = 0.3 for fitting RO data.

In keeping with the statistics of random processes, sampling error worsens with shorter integration times (cf. Fig. 7).

One assumption made in Bayesian interpolation is that sampling variability is uniform across the globe, but an analysis of postfit residuals shows otherwise. Bayesian interpolation obtains fits that exhibit far less sampling error in the tropics than in the midlatitudes, which is in keeping with less synoptic variability in dry pressure in the tropics than elsewhere. COSMIC should be able to obtain pointwise monthly averages with sampling errors of 2 m in the tropics and 6 m in midlatitudes; CHAMP should be able to obtain 6 m in the tropics and 13 m in the midlatitudes (cf. Fig. 5). Nonetheless, Bayesian interpolation still fits data adequately enough for the purpose of forming climatologies of RO data.

Another assumption made in Bayesian interpolation is that the postfit residuals **t** − *φ*w_{MP} are uncorrelated. This is almost certainly not a valid assumption. Soundings that occur within the correlation length and time of synoptic eddies should be correlated, and COSMIC soundings are dense enough to satisfy this criterion regularly. Incorporating an appropriate correlation should serve to improve Bayesian interpolation by not overweighting toward those regions where soundings are far more dense than the atmospheric eddies they are sampling. What the appropriate correlations should be is a topic for future investigation, but it is already encouraging that Bayesian interpolation without correlation between soundings performs as well as it does.

Yet another assumption made in Bayesian interpolation is that the prior for the fitted coefficients is zero with finite width. The consequence should be zero-biased values for the fitted coefficients. Nonetheless, the global bias for the height of the 200-hPa dry pressure surface is just −0.04 m for CHAMP and −0.02 m for COSMIC. If more sophisticated basis functions, such as those that approximate the true dynamics of the atmosphere, are used instead of the regularizer here, this method of fitting data should approach the methods of variational data assimilation used in numerical weather prediction.

Finally, we have found that a systematic error incurred by binning and averaging GPS RO data is negated by using Bayesian interpolation instead, but Bayesian interpolation incurs systematic error by underresolving meridional structures. A systematic error in binning and averaging arises because of weak singularities in GPS RO sounding density at 21° and 49° latitude in the Northern and Southern Hemispheres (cf. Fig. 8). By fitting to basis functions, that systematic error is eliminated. On the other hand, because the spherical harmonic basis is truncated to 14 for CHAMP and 20 for COSMIC, meridional gradients in the Southern Hemisphere are incompletely resolved, thus introducing a different systematic sampling error. The greater the density of sampling though, the larger the spherical harmonic truncation is permitted and the better resolved the background atmosphere is (cf. Fig. 9). In the end, CHAMP can determine the zonal and annual average height of the 200-hPa surface to less than 2 m in the Northern Hemisphere and 4 m in the Southern Hemisphere. COSMIC does the same, finding less than 1 m in the Northern Hemisphere and 2 m in the Southern Hemisphere.

It is possible that fields other than the height of the 200-hPa dry pressure surface perform differently when Bayesian interpolation is applied. Thus, it is advisable to subject other fields to the type of analysis presented here before implementing Bayesian interpolation operationally. On the other hand, for pressure and temperature in the midtroposphere and higher, we find it very likely that the tunable parameters given above for the height of the 200-hPa dry pressure surface should perform adequately. Because lower-tropospheric water vapor is associated with much smaller spatial and temporal scales than pressure and temperature, it should require fundamentally different Bayesian interpolation. First of all, the domain should be restricted to limited regions and thus a different set of basis functions should be used. Research would be necessary to determine the best performing regularizer associated with that basis.

We note that Bayesian interpolation is well suited to determining geostrophic winds. GPS RO has absolute position as its true independent coordinate, which is easily converted to geopotential energy per unit mass and hence geopotential height. With independent knowledge of water vapor, pressure can be retrieved from RO data above the midtroposphere. In the lower troposphere, GPS RO provides more useful information on water vapor than on pressure and temperature (Kursinski et al. 1995). Spherical harmonics have simple analytic gradients, and so maps of the height of a constant pressure surface composed by Bayesian interpolation can easily be subjected to a gradient and geostrophic winds inferred. COSMIC data can thus be used to determine important components of the cross-tropopause Eliassen–Palm flux (Leroy and Anderson 2007).

Previous work has addressed sampling error in GPS RO data by correction using operational weather analyses. For example, Steiner et al. (2009) undertake a binning and averaging approach to forming climatologies and remove sampling error from GPS RO data by computing it using the actual locations of RO data in conjunction with operational analyses of the European Centre for Medium-Range Weather Forecasts (ECMWF), much the same as we have done using simulated distributions and ERA Interim. They then removed this sampling error from the binned and averaged GPS RO data. In principle, this should remove sampling error, but it does not afford an error analysis and will introduce new inhomogeneities in the derived record every time the operational analysis center changes its approach to assimilating GPS RO data. We instead have chosen to work independently of outside information to circumvent such potential problems.

We expect the results of this work to give a strong indication of the value of RO data in anchoring atmospheric reanalysis. Modern reanalysis has incorporated variational bias correction to account for instrument error and forward model error in the process of data assimilation (Derber and Wu 1998; Dee and Uppala 2009), so it can drift unless some “anchor” data are assimilated without bias correction. (When the reanalysis drifts, it does so in the direction of a combination of an inaccurate data type that is treated as an anchor and the natural state of the inevitably biased physical model at the core of data assimilation.) RO is justifiably treated as one of the anchor data types in the ERA Interim reanalysis of ECMWF (Poli et al. 2010), but RO data are sparse in comparison to other data that are assimilated. The analysis presented here gives an indication of the accuracy provided to reanalysis by RO data in addition to the systematic errors of RO discussed in Kursinski et al. (1997). Reanalysis must rely heavily on the denser bias-corrected data types to infer any more accuracy.

## Acknowledgments

This work was supported in part by NASA Grant NNX11AD01G and by a grant from the NASA Jet Propulsion Laboratory’s Director’s Research and Development Fund. Work performed by C. Ao and O. Verkhoglyadova was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.

## REFERENCES

Anthes, R. A., and Coauthors, 2008: The COSMIC/FORMOSAT-3 mission: Early results.

,*Bull. Amer. Meteor. Soc.***89**, 313–333.Dee, D. P., , and Uppala S. , 2009: Variational bias correction of satellite radiance data in the ERA-Interim reanalysis.

,*Quart. J. Roy. Meteor. Soc.***135**, 1830–1841.Dee, D. P., and Coauthors, 2011: The ERA-Interim reanalysis: Configuration and performance of the data assimilation system.

,*Quart. J. Roy. Meteor. Soc.***137**, 553–597.Derber, J., , and Wu W. , 1998: The use of TOVS cloud-cleared radiances in the NCEP SSI analysis system.

,*Mon. Wea. Rev.***126**, 2287–2299.Foelsche, U., , Kirchengast G. , , and Steiner A. , 2003: Global climate monitoring based on CHAMP/GPS radio occultation data.

, C. Reigber, H. Luehr, and P. Schwintzer, Eds., Springer, 397–407.*First CHAMP**Mission Results for Gravity, Magnetic and Atmospheric Studies*Foelsche, U., , Gobiet A. , , Steiner A. , , Borsche M. , , Wickert J. , , Schmidt T. , , and Kirchengast G. , 2006: Global climatologies based on radio occultation data: The CHAMPCLIM Project.

*Atmosphere and Climate: Studies by Occultation Methods,*U. Foelsche, G. Kirchengast, and A. Steiner, Eds., Springer, 303–314.Kursinski, E., , Hajj G. , , Hardy K. , , Romans L. , , and Schofield J. , 1995: Observing tropospheric water-vapor by radio occultation using the Global Positioning System.

,*Geophys. Res. Lett.***22**, 2365–2368.Kursinski, E., , Hajj G. , , Schofield J. , , Linfield R. , , and Hardy K. , 1997: Observing Earth’s atmosphere with radio occultation measurements using the Global Positioning System.

,*J. Geophys. Res.***102**, 23 429–23 465.Lackner, B., , Steiner A. , , Kirchengast G. , , and Hegerl G. , 2011: Atmospheric climate change detection by radio occultation data using a fingerprinting method.

,*J. Climate***24**, 5275–5291.Leroy, S., 1997: Measurement of geopotential heights by GPS radio occultation.

,*J. Geophys. Res.***102**, 6971–6986.Leroy, S., , and North G. , 2000: The application of COSMIC data to global change research.

,*Terr. Atmos. Oceanic Sci.***11**(1), 187–210.Leroy, S., , and Anderson J. G. , 2007: Estimating Eliassen-Palm flux using COSMIC radio occultation.

,*Geophys. Res. Lett.***34**, L10810, doi:10.1029/2006GL028263.Liou, Y., , Pavelyev A. , , Liu S. , , Pavelyev A. , , Yen N. , , Fluang C. , , and Fong C. , 2007: FORMOSAT-3/COSMIC GPS radio occultation mission: Preliminary results.

,*IEEE Trans. Geosci. Remote Sens.***45**, 3813–3826.Lorenc, A., 1986: Analysis methods for numerical weather prediction.

,*Quart. J. Roy. Meteor. Soc.***112**, 1177–1194.MacKay, D., 1992: Bayesian interpolation.

,*Neural Comput.***4**, 415–447.Pirscher, B., , Foelsche U. , , Lackner B. C. , , and Kirchengast G. , 2007: Local time influence in single-satellite radio occultation climatologies from Sun-synchronous and non-Sun-synchronous satellites.

,*J. Geophys. Res.***112**, D11119, doi:10.1029/2006JD007934.Poli, P., , Healy S. , , and Dee D. , 2010: Assimilation of Global Positioning System radio occultation data in the ECMWF ERA-Interim reanalysis.

,*Quart. J. Roy. Meteor. Soc.***136**, 1972–1990.Rodgers, C., 2000:

*Inverse Methods for Atmospheric Sounding: Theory and Practice*. World Scientific, 238 pp.Seeber, G., 1993:

*Satellite Geodesy: Foundation, Methods, and Applications*. Walter de Gruyter, 531 pp.Sivia, D., 2006:

*Data Analysis: A Bayesian Tutorial*. Oxford University Press, 246 pp.Steiner, A. K., , Kirchengast G. , , Lackner B. C. , , Pirscher B. , , Borsche M. , , and Foelsche U. , 2009: Atmospheric temperature change detection with GPS radio occultation 1995 to 2008.

,*Geophys. Res. Lett.***36**, L18702, doi:10.1029/2009GL039777.Wickert, J., and Coauthors, 2001: Atmosphere sounding by GPS radio occultation: First results from CHAMP.

,*Geophys. Res. Lett.***28**, 3263–3266.