The CYGNSS main observable is normalized bistatic radar cross section (NBRCS), ${\sigma }^o$. This is derived from the radar range equation [25]: where $P^g$ is the calibrated power received by the nadir science antenna; $L^{atm}$ are the losses due to attenuation in the atmosphere; $P^T$ is the power of the transmitter, which in this case is a GPS satellite; $G^T$ is the antenna gain of the GPS satellite in the direction of the Earth specular point; $\lambda$ is the wavelength of the GPS signal, which for CYGNSS is the GPS L1 signal at 1575 MHz; $G^R$ is the gain of the CYGNSS nadir science antenna in the direction of the Earth specular point; $R^{tot}$ is the total range loss of the transmission via the Earth specular point; and $\overline{A}$ is the effective scattering area of the specular point.

NBRCS is directly related to the mean square slope (MSS) of ocean wave spectra via [25,26]: where $\theta$ is the angle of incidence of the scattering geometry and $\mathfrak{R}\left(\theta \right)$ is the Fresnel electric field reflection coefficient for the ocean surface at the specular point. CYGNSS retrieves windspeed from ${\sigma }^o$ via an empirical Geophysical Model Function (GMF) [27] that is processed operationally via a stored look-up table (LUT).

In practice, un-normalized bistatic radar cross section $\sigma$ is calculated for every delay and Doppler, the native coordinates of the delay-Doppler Mapping Instrument (DDMI). ${\sigma }^o$ is calculated from the average of $\sigma$ over a 3-by-5 bin window centered at the specular point, and divided by the effective area $\overline{A}$ corresponding to the surface area bounded by the 3-by-5 window [$m^2/m^2$].

After launch of the CYGNSS mission, it became clear that uncertainty in the GPS effective isotropic radiated power (EIRP) would significantly impact calibration quality, [28,29,30]. The EIRP is simply the product $P^T{G}^T$ from Equation 2. Errors are present in both terms: not only is there general uncertainty with the actual (versus published) transmit power of GPS satellites, newer generations of the constellation operate with a flexible power mode and dynamically change transmit power levels across their respective orbits [31]. Further, only a subset of the GPS antenna patterns were published [32] and they were not sufficiently detailed to constrain the error in EIRP for CYGNSS calibration.

To remedy this, Wang et. al. developed a novel calibration technique that utilizes the onboard zenith CYGNSS antenna and empirically derived relationship from measured GPS antenna patterns to estimate GPS EIRP in the direction of the Earth specular point at the time of observation [33]. This new technique has been adopted in the CYGNSS Level 1 calibration algorithm since version 3.0, and modifies the radar equation to become the following [34]: where all the terms are the same as in Equation 2, with the additions of $L^Z$, the transmission range loss from the GPS source to the CYGNSS zenith antenna (written as a loss in the numerator to distinguish from the $R^{tot}$ range losses); the CYGNSS zenith antenna gain $G^Z$ in the direction of the GPS satellite; the power received from the CYGNSS zenith receiver $P^Z$; and the specular-zenith ratio ${\zeta }^E$ for the GPS EIRP as described in [33].

Each of the terms in Equation 4 can potentially be a source of correlated error, but for practical reasons this study only evaluates those sources with the largest error magnitude and therefore the largest potential utility for future data assimilation users. The five largest sources of error are explored as shown in Table 1, representing a 1-sigma error magnitude for a reference windspeed at 10 m/s. The following sections evaluate each of these terms in depth.

The CYGNSS Level 1 correlated error model is constructed to represent the major error sources based on the physical and operational characteristics of the CYGNSS observatories. These errors are correlated over both space and time, and depend on several variables relating to the operation of the CYGNSS constellation.

Construction of the error model is intended to be realistic enough to represent measurable and plausible correlated errors, yet flexible enough to allow for tuning and parameterization. The general modeled structure of correlated error takes the form: where $R_{mod}\left(i,j\right)$ is the normalized correlated error between any two samples $i$ and $j$ with a domain $-1\le R_{mod}\le 1$. There are $n$ component terms that are added together, and each of the $n$th term corresponds to a unique source of error. $K_n\left(i,j\right)$ represents the error covariance between any two samples $i$ and $j$ for a specific component $n$. $\mathcal{N}$ is a normalization constant to ensure that the diagonal terms of the $R_{mod}$ matrix is 1.

Each of the $K_n$ terms can be further generalized as: where $E\left(n\right)$ represents the magnitude of each error component $n$ as calculated in [33,34,36], and can be thought of as the variance of the error; and $Ecor{r}_n\left(i,j\right)$ is a derived function with a domain $-1\le Ecorr\le 1$ that represents the correlation in error between samples $i$ and $j$ for each component $n$.

Each $Ecorr$ function depends on different arguments, depending on the nature of the error source. The individual errors are explored in depth in the following sections. Further, several tuning parameters have been added to assist with validation. The initial values of the tuning parameters are 1 and essentially leave the model output unmodified. However, this model assumes that the relative magnitude of the error components.

One of the primary challenges in constructing the CYGNSS Level 1 correlated error model $R_{mod}$ is identifying a plausible validation scenario. In practice, it can be difficult to disentangle the various sources of correlated observation error structure, e.g. those caused by the inherent behavior and calibration of the instrument and, by the geophysical retrieval and inversion process, and by the representativity errors imposed when observations are gridded and ingested into models. Further, the choice of ground truth may impose an additional source of correlated error, such as when using reanalysis data or another observation source.

In an effort to disentangle these correlated errors and isolate only those due to instrumental sources, this work validates the correlated error structure $R_{mod}$ by matching up near- simultaneous collocated observations made by two different observatories at nearly identical geometries, generating modeled NBRCS ${\sigma }_{mod}^o$ from reanalysis data, and single- and double-differencing the results.

The constructed model $R_{mod}$ is first tuned to match the overall behavior of $\mathcal{R}\left[DD\right]\left(\tau \right)$ as described in Section 2.3.4. With the appropriate tuning parameters, the bulk modeled autocorrelation ${\tilde{\mathcal{R}}}_{mod}$ closely resembles $\mathcal{R}\left[DD\right]\left(\tau \right)$ in most important aspects, including relative magnitude of white noise component, relative rate of decorrelation roll-off, and endpoint behavior at large lags. The final tuned ${\tilde{\mathcal{R}}}_{mod}$ is shown in Figure 3 with $\mathcal{R}\left[S{D}_{obs}\right]$, $\mathcal{R}\left[S{D}_{mod}\right]$, and $\mathcal{R}\left[DD\right]$.

The behavior in Figure 3 is worth discussing in detail. The single-differenced model generated ${\sigma }_{mod}^o$ decays at a much slower rate than the single-differenced observations and double-differenced data suggests that the errors of the slight mismatch in observation location and time are generating errors on a fundamentally different scale than the instrument errors. Further, the fact that the double-differenced decorrelation is almost identical to the observation single-differencing suggests that single-differencing near-simultaneous observations are a reasonable approximation for bulk error correlation.

Evaluating the double difference trace in Figure 3 also reveals important qualities about the CYGNSS error correlation structure. First, the uncorrelated error component accounts for roughly 5% of the total system error. Therefore, assumptions that samples be treated as independent with uncorrelated error are empirically refuted. Further the error decorrelation rolloff is swift: about 50% of the error has decorrelated within 7 seconds of observation. Using a flat-plane approximation of Earth, this corresponds to a distance of approximately 50 km, which is generally the scale of the gridding of modern global weather models, but much coarser than the resolution of state-of-the-art regional models. Finally, the endpoint behavior at large lag times suggest a small, positive correlated error (~2%) at larger timescales. The statistical properties of the lag correlation become less stable at larger lags, but the existence of a long-timescale correlation is not surprising, considering all observations from a single receiver share a common electronics and processing chain.

The fact that our tuned model $R_{mod}$ can approximate bulk error correlation structure ${\tilde{\mathcal{R}}}_{mod}$ with similar features as $\mathcal{R}\left[DD\right]$ validates the assumption that the overall instrument correlated error can be modeled from the fundamental components of the instrument observable, in our case, the radar-range equation (Equation 2). Further, that $R_{mod}$ can be generated between arbitrary samples suggests that an observation error correlation matrix R can be generated dynamically from first-principles given appropriate knowledge about the instrument and retrieval.

The value of the chosen tuning parameters is also worth investigating. The untuned model is defined by having parameters $\alpha =\beta =\gamma =\delta =1$. Figure 3 demonstrates that the modification of tuning parameters can change the overall model behavior significantly to match observed behavior. Figure 3 further suggests that the untuned $R_{mod}$ generally overestimates both the relative magnitudes of uncorrelated error (i.e., white noise, tuned by $\alpha$) and totally-correlated error (i.e., endpoint correlation, tuned by $\beta$). The chosen parameters for tuning are shown in Table 2.

The tuning parameter $\gamma$ is used to vary the relative magnitude of the near-sample correlated error roll-off. In our tuned model, this parameter remains at 1. The tuning parameter $\delta$ is used to enhance or reduce the steepness of the rolloff in correlation, caused by two observations moving farther apart in the nadir and zenith coordinate systems. If $\delta >1$, it enhances the steepness of decay. The fact that our optimized tuning maintains $\delta =1$ suggests that the filtering theory discussed in Lemma C2 is a reasonable model of decay.

The tuned model can be compared to single track autocorrelations $\rho \left(\tau \right)$ of matched-sample double differences via the modeled error autocorrelation ${\tilde{\rho }}_{mod}$ as described in Equation 12. We compare three exemplar tracks in Figure 4.

The single-track autocorrelation behavior shown in Figure 4 is also revealing of the limitations of this work. The single-track error autocorrelation for double-differenced data is highly variable. This may be due to both artifacts in the data (processing, quality control, insufficient data) as well as real behavior of the observation. It is worth articulating that autocorrelation of data is generally less stable with smaller datasets and at longer lags. With our quality control parameters, we flag out significant quantities of data, which decreases the stability of autocorrelations from track to track. We further note the difficulty in exploring the dynamics of this behavior in a statistical sense: our aperture of observation where two CYGNSS assets are in a near-overlap condition is quite rare, and as the mission progresses, the orbit planes of the satellites have drifted, making further matchup scenarios less representative. These behaviors may evolve differently day-to-day or as a function of observed windspeed, but the paucity of data in this configuration makes it challenging to establish sufficient baselines to test for significance.

The model $R_{mod}$ can produce plausible dynamics suggesting broader importance of a realistic dynamic correlated error model. Figure 5 plots two nearby tracks captured nearly simultaneously by the same receiver (but different GPS transmitters).

Because correlated error $R_{mod}$ can be generated for any arbitrary observation using the bottoms-up model, the dynamics of how instrument errors decorrelate can be explored. If the observations were treated as completely independent without any correlated error, the matrix in Figure 5a would contain non-zero elements exclusively along the main diagonal. However, we observe several structural elements. The ‘pixelation’ pattern is largely a result of the $R_{\zeta }$ term and originates from the coarse mapping of the estimated GPS antenna gain pattern as a function of scattering incidence ${\theta }_{inc}$. This phenomenon is explored in depth in Appendix D. The smooth decorrelation roll-off is from the $R_{G^R}$ and $R_{G^Z}$ components of the model, and represent the direct and reflected signals moving about the nadir and zenith the antenna patterns in CYGNSS receivers as the observation is collected along the track. We note that the model allows for cross-track correlation where the two different tracks share a CYGNSS receiver but have a different GPS transmitter. In both the untuned and tuned cases, these tracks appear to have uncorrelated cross-track error. There are residual correlations between the tracks when the scattering geometry is such that the two tracks share similar incidence angles or are near similar antenna coordinates, in addition to the shared receiver noise at lag $\tau =0$.

We can also determine $R_{mod}$ for our matchup tracks where CYGNSS observes similar track geometries with the same GPS transmitter but with two different receivers. The correlated structure for one of the tracks in the previous example (but matched by difference receivers) is shown in Figure 6.

The model $R_{mod}$ allows for correlated error between two different receivers that share a GPS transmitter. We note that observations sharing the same GPS transmitter may contain correlated error due to correlated misestimation of GPS transmit power. This may be an incomplete articulation of the full cross-receiver error correlation. For example, CYGNSS assets in similar orbital regimes may experience similar thermal and environmental conditions that cause correlated errors during our timescales of interest. We assume that correlated error due to misestimation of GPS transmit power is nearly constant for the timescales of interest.

As demonstrated in Figure 6, the cross-track correlation virtually disappears after tuning the model $R_{mod}$. Taken with the data presented in Figure 5, there appears to be negligible cross-track error correlation, either from when two tracks share a CYGNSS receiver or when two tracks share a GPS transmitter. This is a novel result, but challenging to validate in practice, as the double-differencing may eliminate any shared error correlation between two tracks.

The construction of $R_{mod}$ makes a few important assumptions that are worth examining in depth. First, we will describe the general mathematical framework. In general, if $X$ and $Y$ are random vectors of length $N$, then the covariance matrices of $X$ and $Y$ are constructed: where $K_{XX}$ and $K_{YY}$ are the $N$-by-$N$ covariance matrices of $X$ and $Y$, respectively; the bracket operator denotes the expectation operation, and the superscript $T$ denotes the transpose vector. Notably, the covariance matrix is symmetric and positive semi-definite. The covariance matrices can also be constructed in the following manner:

Where $S_X$ is a diagonal $N$-by-$N$ matrix with the standard deviations of $X_i$: and $R_{XX}$ is the correlation matrix with correlation coefficients ${\rho }_{i,j}$:

As standard, $-1\le {\rho }_{i,j}\le 1$, and the diagonals all must equal 1. The cross-covariance matrices can also be defined in a similar fashion:

where $K_{XY}$ and $K_{YX}$ are the cross-correlation matrices and are not generally symmetric. The error magnitudes in Table XX are calculated using a standard error propagation technique as highlighted in Gleason et. al [25]: $E_n$ is the error magnitude of term $n$, $\mathsf{\Delta }n$ is the estimated one-sigma dynamic range of $n$ and $F$ is an arbitrary function. We can propagate these errors as the sums of covariance matrices with the derivation below, inspired from Chapter 9 in Taylor [45].

For an arbitrary function $F$ with random vector arguments X and Y, $F\left(X,Y\right)$, where $X=\left(X_1,X_2,\dots ,X_N\right)$ and Y$=\left(Y_1,Y_2,\dots ,Y_N\right)$ , we can approximate F by its first order Taylor Series expansion about the mean value, assuming that the errors are generally small compared to the arguments: Noting that $〈F〉=F\left(〈X〉,〈Y〉\right)$, and using $F\approx F\left(〈X〉,〈Y〉\right)+\frac{\partial F}{\partial X}\left(X-〈X〉\right)+\frac{\partial F}{\partial Y}\left(Y-〈Y〉\right)$, then, and expanding using Equations A5, A7, and A8,

Substituting $E_n^2={\left(\frac{\partial F}{\partial n}\right)}^2{\mathsf{\sigma }}_{\mathrm{n}}^2$ from Equation A6, The construction of $R_{mod}$ in Equation 7 ignores the cross-correlation between component terms, i.e., we assert $K_{XY}=K_{YX}=0$. This is for two main reasons:

As we further assume that that each component term of the error $E_n$ comes from a wide-sense stationary distribution where the error magnitudes are treated as constants. This means that for each component of $K_{FF}$, and noting Equation A2, For our error correlation model $R_{mod}$, we simply construct $R_{XX}$ analytically for each term, for which we adopt the nomenclature Ecorr to emphasize that that it is an error correlation function. Therefore, from Equation 5 (reproduced here), where $K_n\left(i,j\right)=E_n^2\cdot Ecor{r}_n\left(i,j\right)$.

For this model, F is drawn from Equation 4 (reproduced here) where we use the parameters of with the largest error magnitude, ${\sigma }^o=F(P^g,P^Z,G^Z,G^R,{\mathsf{\zeta }}^{\mathrm{E}})$. The relevant partial derivatives $\frac{\partial F}{\partial N}$ from Equation A10 become: To estimate $E_n$ as shown in Table 1, these partial derivatives are evaluated at the one-sigma value for a reference 10 m/s windspeed using Equation A6. The total error model becomes: where $K_{p^g}$ is described in Appendix B, $K_P^z$ is described in Appendix C, $K_{G^Z}$ and $K_{G^R}$ are described in Appendix D, and $K_{{\zeta }^E}$ is described in Appendix E. $\mathcal{N}$ is a normalization constant that forces $R_{mod}$ to behave like a correlation such that $-1\le R_{mod}\le 1$, and can be thought of as an estimate for the rolled-up variance. Because this model has tunable parameters, it is not necessarily representative of the true variance of ${\sigma }^o$, but rather of the modeled variance from our bottoms-up model construction.

The term $P^g$ represents calibrated received power from GPS signals reflected from Earth’s surface. $P^g$ is calibrated both from pre-launch characterizations as well as on on-orbit blackbody at known temperature, which as of 2022 takes a reading every 10 minutes to re-compute gain that may have changed due to the dynamic thermal environment in orbit [35].

For CYGNSS, $P^g$ is computed by the Level 1A algorithm: where $C$ is the raw counts at delay-Doppler bins where a scattered signal is present and is the measured parameter for an observation, $C_N$ is an estimate of the background noise without any scattered signal, and $G$ is the receiver gain in units of counts/watt. In current processing, $C_N$ is an average of the raw counts at in the delay-Doppler map at coordinates at lower delay values than the specular point [25]. Further, gain $G$ is measured on-orbit by performing reading from an onboard blackbody at known temperature, and calibrated via

$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}{C}_B$ is the counts measured while looking at the blackbody, $P_B$ is the power in watts from the blackbody as estimated from a thermocouple located near the receiver’s low noise amplifier, and $P_r$ is the receiver noise power in watts estimated from a pre-launch parameterization. Equations B1 and B2 can be combined:The magnitude of the errors for the components calculating $P^g$ is displayed in Table B1 and have a similar calculation as before.

$C_B$, $P_B$, and $P_r$ all vary with temperature, and the instrument gain $G$ will fluctuate as the satellite enters different thermal conditions in orbit. The most significant errors will occur just as the satellite crosses the terminator. At that point, CYGNSS will go from a nearly steady-state thermal environment, such as approximately half an orbit of illumination or eclipse, and then quickly enter the opposite state. The fraction of orbit spent illuminated is determined by the orbit beta angle, which varies on scales of weeks to months.

The dominant error term in the Level 1A algorithm is $C_N$, which also varies with temperature. The calibration sequence is designed to correct for this, and we assume that the errors vary slowly with the timescales of interest, which is defined to be on scales of seconds to minutes. Therefore, all errors from $C_N$ are assumed to be 100% correlated in time within a given track of CYGNSS observations, i.e., during when series of samples adjacent in space and time share a GPS transmitter and a CYGNSS receiver. $C_N$ is also very sensitive to radio-frequency interference (RFI), which will present as non-physical signals above the specular point in a delay-Doppler map. We do not aim to model the complex phenomenologies of RFI in this work and assume there are no correlated error structures from RFI.

Errors in $P_r$ occur because of a variety of reasons. The low noise amplifiers were all characterized on the ground prior to launch to establish the relationship of the noise figure with respect to temperature. The values of this relationship were stored in a look-up table (LUT) for processing science data. However, as the amplifiers age, the noise floor characteristics may have evolved, producing errors in this mapping. Further, the thermal environment of the thermocouple may not be exactly same that is experienced by the amplifier itself. For the purposes of this model, we assume all errors due to incomplete or erroneous knowledge of the true receiver noise power are 100% correlated with each other for a given track, as we assume that the errors evolve slowly compared to the timescales of interest.

Therefore, the error correlation terms for $C_N$ and $P_r$ for any arbitrary CYGNSS samples $x_i$ and $x_j$ are the following:

The conditions that must be met are the following: $x_i$ and $x_j$ must share a CYGNSS receiver, a GPS transmitter, and be observed within 10 minutes of each other.

$C$ is the measured parameter, the raw counts of power from a science observation near the region of the specular point. The analog-to-digital processing chain is the primary source of errors, such as quantization errors and non-common-mode interference. We assume these error terms are 100% uncorrelated with each other, that is for every sample, it can be treated as white noise. The error correlation term for $C$ is then:

The error term for counts measured during a blackbody sample $C_B$ is a source of analytically-defined correlated error. Every 10 minutes (earlier in the mission, every 1 minute), the receiver is switched from the nadir science antenna to look at the onboard blackbody source for a period of 4-6 seconds. Science observation processing linearly interpolates the counts between the nearest blackbody looks. When errors are made in estimating $C_B$, those errors are correlated linearly with all adjacent samples due to this interpolation. Correlation due to linear interpolation has an analytical form. Assuming a blackbody look happens at timesteps 0 and $n$, then the correlation between any two samples $x_i$ and $x_j$ at arbitrary timesteps $i$ and $j$ where $0\le i<j\le n$ is: where ${\sigma }_i=\sqrt{\frac{{\left(n-i\right)}^2+i^2}{n^2}}$ and ${\sigma }_j=\sqrt{\frac{{\left(n-j\right)}^2+j^2}{n^2}}$. The actual values of the sampled blackbodies do not matter, as the correlated error is simply a function of how far the samples are from the blackbody looks in time.

Errors in $P_B$ are due to misestimations in of the blackbody’s true noise power, which may be because the thermocouple is measuring incorrectly. We assume that the errors of this nature not only are slowly varying compared to the timescales of interest, but because of the marginal absolute magnitude, factor a negligible and unmeasurable amount in the overall correlated error structure. As such the model ignores this term.

The rolled-up correlated error model from the sources in $P^g$ between any arbitrary samples $i$ and $j$ can be expressed as follows:

Because this model estimates the correlated error in each term that calculates $P^g$, this contribution to the overall correlated is not multiplied by the error $E\left(P^g\right)$. As such, we do not normalize this construction, as it will be normalized when combined with the other constituent terms in $R_{mod}$. $R_{P^g}$ contains two tuning parameters: $\alpha$ is used to size uncorrelated white-noise error, and $\beta$ is used to size the magnitude of totally-correlated errors.

The zenith receiver on CYGNSS works much the same way as the nadir science receiver. However, the zenith receiver does not have an onboard calibration system, and data are processed from pre-flight characterizations of the electronics. The counts of the receiver are converted to watts via a quadratic regression. Because the satellite is subject to the same thermal dynamics as the nadir receiver, one can expect that the zenith power estimate contains errors due to thermally-driven gain variations. We model the $Ecor{r}_{P^Z}$ similarly to the nadir receiver, but with some important distinctions. Because the zenith receiver operates without an onboard blackbody to calibrate against, a number of simplifying assumptions are made. Errors are broken up into just two components, correlated $Ecor{r}_{P_1^Z}$ and uncorrelated $Ecor{r}_{P_2^Z}$. We further assume that correlated error due to the absence of an onboard blackbody are outside the timescale of interest. The correlated error term $Ecor{r}_{P_1^Z}$ is assumed to be totally correlated provided that the matching conditions are met:

The conditions for $Ecor{r}_{P_1^Z}$ are that samples $x_i$ and $x_j$ are made with the same CYGNSS observatory and within 10 minutes of each other. In addition, an uncorrelated component is allowed:

The rolled-up correlated error model from the sources in $P^Z$ between any arbitrary samples $i$ and $j$ can be expressed as follows: with the same tuning parameters $\alpha$ and $\beta$ as in Appendix A. Note that we assume $E\left(P_1^Z\right)=0.18$ dB as suggested in [33]. $E\left(P_2^Z\right)$ is estimated directly from a 24 hours of CYGNSS data as approximately 1% of the magnitude of the signal $P^Z$, therefore this model assumes $E\left(P_2^Z\right)=0.04$ dB.

The CYGNSS observatory has three antennas, one zenith antenna that is used for direct GPS-to-CYGNSS signal tracking, as well as two nadir science antennas that are used to capture the scattered signal from Earth’s surface. Each of the eight spacecraft had all three antennas characterized pre-launch, and values were stored in a lookup table for science processing.

Errors in the antenna gain pattern can arise for a variety of reasons. First, the measurement equipment on the ground is essentially a receiver, but in controlled conditions. This means that while systematic and correlated errors are likely well-constrained, uncorrelated speckle-type error can still occur. To produce realistic antenna gain patterns, the results of the ground characterization were smoothed with various filters and techniques.

Another source of error is the fact that CYGNSS antennas were not characterized while integrated with the spacecraft. This was a cost-saving measure decided by the mission management team. However, the electromagnetic properties of the antenna couple in some fashion with the spacecraft bus, and that will inevitably change the gain patterns.

Initial analysis of CYGNSS data shortly after launch showed significant retrieval performance dependence on the observation azimuthal angle with respect to the CYGNSS body frame, which was later hypothesized to originate in errors in the CYGNSS antenna patterns. To compensate for this deviation from measured patterns, the CYGNSS antenna patterns have been updated at several instances over the mission life via empirical calibration. The nadir antenna patterns are updated by comparing a climatology of CYGNSS measurements of ${\sigma }^o$ (> 2 years) with model-generate ${\sigma }_{mod}^o$ and plotting a scaling factor in the antenna reference coordinate system. ${\sigma }_{mod}^o$ is generated by using modeled reanalysis winds to generate mean-squared slope with the L-band spectrum extension model as described in [38]. However, during the generation of these updated patterns, a number of smoothing filters are applied.

This prompts discussion of a conjecture used extensively for this section:

This insight drives much of this section’s analysis. Smoothing and filtering will necessarily impose a correlation in error between previously uncorrelated error. For white noise, that implies that the choice of filter will add color and structure to the noise.

In particular, a handy lemma allows us to demonstrate that for white noise, the information required to capture correlated error structure is the filtering kernel itself. We will explore this behavior for a one dimensional case, but it is generalizable to higher dimensions in our application, as the two-dimensional filters used for antenna smoothing are separable by construction.

and where $*$ is the convolution operator, $\star$ is the cross-correlation operator, and $n$ is the lag argument. Observe that convolution operations is commutative, and further, that the cross correlation can be written as a convolution by exploiting its symmetry:

Therefore, to evaluate the correlated error imposed by kernel $K\left(t\right)$,

If the arbitrary signal $D\left(t\right)$ happens to be white noise, it is completely uncorrelated, and its autocorrelation collapses to a Dirac delta function centered at $t=0$. Therefore, the entire structure of the correlated error is from the filter itself:

For the purposes of this work’s error model, we have no knowledge of the potential correlated structure in the actual errors in the gain pattern. The nadir antenna patterns are updated after applying a 6-degree boxcar averaging filter in both the azimuthal and elevation in the spacecraft coordinate frame, and then an additional 10-degree two-dimensional smoothing window. We assume that zenith antenna patterns use a similar post-processing technique during their generation.

These filtering kernels act like low-pass filters. All correlated structure on scales ~5 degrees and smaller and uncorrelated error will be strongly influenced by the filtering process and the correlated structure can be estimated from the Filter Lemma. This model assumes that there is no residual larger-scale structure in correlated error in the antenna gain patterns.

The generated filter kernel, which applies to both the nadir and zenith antenna patterns, can be shown in Figure D1. The correlated error is a function of how close any two observations are with respect to the relevant antenna gain pattern coordinates.

For any arbitrary samples $i$ and $j$, we compute the gain pattern coordinates $\left({\theta }_i,{\varphi }_i\right)$ and $\left({\theta }_j,{\varphi }_j\right)$ in the relevant antenna reference frame. To retrieve how correlated the error is, we compute the distance between the two observations in the reference frame:

The error correlation function is computed via a LUT of the filter kernel K:

This error correlation holds if the samples $i$ and $j$ share the same antenna and are on the same spacecraft. If they are on separate antennas or spacecraft, the correlated error is zero. The rolled-up correlated errors for the gain patterns can be expressed as: where two new tuning parameters have been introduced. $\gamma$ is used to the tune the overall magnitude of the correlated error from these components, and $\delta$ is used to scale the decorrelation roll-off rate as the samples spread in antenna coordinates.

Errors in zenith-specular ratio $\zeta$ are defined as a function of specular incidence angle ${\theta }_{inc}$, which is a function of the geometry of a given GPS transmitter, a CYGNSS receiver at any given sample time.

$\zeta$ is used to estimate GPS EIRP and is a derived via as described in [46]: where the angles are defined in the GPS reference frame. For specular geometries, the azimuthal angles in the zenith direction are nearly identical to the specular direction, so ${\varphi }_z={\varphi }_S=\varphi$. In addition, the elevation angles in the GPS antenna reference frame ${\theta }_z$ and ${\theta }_S$ can be estimated from the angle of incidence of specular reflection from Earth ${\theta }_{inc}$:

As a result, $\zeta$ can be expressed as a function of specular incidence angle and azimuthal angle in the GPS antenna reference frame. While GPS antenna patterns are known to exhibit azimuthal dependence, this variation is less significant than the elevation angle and CYGNSS uses the azimuthal average for its EIRP estimate:

The estimated correlated error in $\zeta$ however come two steps of this processing. First is the mapping of Earth scattering incidence angle ${\theta }_{inc}$ to GPS antenna elevation angles ${\theta }_z$ and ${\theta }_S$ in Equations E2a and E2b. This particular mapping is coarse, as even the high fidelity derived GPS antenna maps are plotted to 0.5 degree increments. Because the dynamic range of ${\theta }_z$ only extends to about 15 degrees, that only leaves ~30 data points to map the full dynamic range of scattering incidence angles.

The second aspect has to do with the way in which $\zeta$ is processed and generated and invokes the same logic as in the Filter Lemma. For every GPS satellite, a $\zeta$ LUT is generated as a function of observation incidence angle ${\theta }_{inc}$. To minimize discontinuities, a fourth-order power series is fit. We argue that this smoothing is predominant source of correlated error structure. An example of this is demonstrated in Figure E1.

While linear interpolation itself imparts some degree of error structure, we believe it is the most representative way to express “raw” data in a continuous series for the purposes of exploring correlated error due to the power series smoothing. For each GPS PRN, we calculate the difference between these estimates: