Long-lasting, deep effect of Saturn’s giant storms

VLA observations

VLA observations of Saturn were obtained on two dates: 24 to 25 January 2015 when the array was in its compact configuration (C&B), and 28 to 29 May 2015 with the array in its more extended configuration (B&A). These are hybrid configurations in which the antennas on the east and west arms have already been moved to the more compact configuration, but those on the north arm remain extended. Such array configurations are ideal to observe sources at relatively low declinations. In January (C&B), the planet was observed in the Q (39.8 to 47.6 GHz), K (18 to 26 GHz), U (12 to 17 GHz), and X bands (8 to 11.8 GHz), and in May (B&A) in S (2 to 2.5 GHz), C (4 to 4.2 GHz), X (8 to 11.2 GHz), and U (12 to 17 GHz) bands.

These data were presented previously in the analysis of Saturn’s ring system (18), with a summary of observations provided in their table 1. All data were initially processed and calibrated using the internal VLA calibration pipeline, followed by using the MIRIAD software to self-calibrate, clean, and map the data (19). On the basis of a preliminary model fitting of Saturn’s disk (18), we determined the disk-averaged brightness temperatures in each band, as summarized in table S1.

Figure 1 shows current and previous observations of Saturn’s disk-averaged radio emission, together with a model spectrum for comparison. This disk-averaged model assumes a dry adiabatic atmosphere and a uniform distribution of ammonia vapor up to the cloud base, referred to as the homogeneous isentropic model. Above the cloud base, the ammonia vapor follows its saturation value uniquely determined by the temperature. We use the Voyager radio occultation profile (20) for temperatures at altitudes above the 1-bar pressure level and use an adiabatic extrapolation for temperatures below the 1-bar level. The VLA channels do not probe deep enough to see through the condensation level of water, so we used a nominal value of 10 times the solar water abundance (21), estimated from the dynamics of Saturn’s giant storm (6). The disk-averaged brightness temperatures of the C and S bands are sensitive to the ammonia concentration above the water cloud level near 20 bars. The χ² fitting of both bands reduces to χ² < 1 for an ammonia abundance of around 200 parts per million by volume (ppmv), which corresponds to an enrichment factor of 1.5 compared to the solar N abundance (see the Supplementary Materials, section A, for a study on the ammonia concentration). However, this value may not be truly representative of Saturn’s deep ammonia abundance because the S band is not long enough to distinguish various deep ammonia abundances [see the comparison of two other models using 150 and 400 ppm deep ammonia abundance (22, 23)]. Nonetheless, Saturn’s disk-averaged emission can be fitted adequately by a homogeneous isentropic model, while Jupiter’s atmosphere shows substantial depletion of ammonia vapor in the upper layers compared to the deep atmosphere (24–26).

Figure 2 shows images of Saturn in different projections for all six radio bands with the ring contribution removed. The left column displays residual disk images as viewed on the sky with a limb-darkened disk subtracted (based on disk-averaged parameters as shown in table S1); the middle column projects the residual disk image into a cylindrical projection, with rotational smearing of about 60° in longitude; the right column zooms in on the polar region northward of 50°N. We obtain the limb of the disk by tracing the sharp edge of the full disk image and overlaying that on the residual disk image. We focus our analysis on the northern hemisphere where the line of sight was free of ring materials.

Clear signatures of bright and dark latitudinal bands are found from the residual disk images at wavelengths shorter than 10 cm (C, X, U, K, and Q bands). At the S band, the banded structure is barely visible except for the shadow of Saturn’s ring in the southern hemisphere. However, this may be due to the low spatial resolution of the S-band image (⁓10° in latitude). The C-band image starts to show one bright latitudinal band between 20°N and 40°N and a relatively bright equator. The dark band in the southern hemisphere is due to the ring occulting the emission of the planet.

The X-band image features an extremely bright latitude band between 40°N and 50°N. The brightness temperature of this latitude band is about 15 K warmer than the neighboring bands, showing that ammonia vapor must be substantially depleted in the atmosphere. Looking at the polar projection map in the right column, we also find a bright ring surrounding the north pole, which we attribute to the deep signature of Saturn’s hexagon, although the hexagonal shape is unclear at the resolution of this image.

The highest spatial resolution is achieved in the U-band images, in which the bright 40°N and 50°N band breaks into two distant narrow bands with one at 40°N and the other at 44°N. This dual-band structure is only present in the U-band images and is realistic as it consistently shows up at all longitudes in the cylindrical projection map. The bright ring–like structure surrounding the pole near 78°N now takes a hexagonal shape in the U-band image. Since its position is colocated with the visible image of the hexagon, we interpret it as being the deep root of Saturn’s polar hexagon. At a higher altitude (⁓100 mbar), the Cassini/CIRS instrument found warmer temperatures (⁓4 K) than the neighboring latitudes (27). This pattern resembles our finding at the U band. Although the altitude probed by Cassini/CIRS is higher than in our radio map, it is likely that the hexagonal jet remains warm in temperature and extends deep (at least ⁓2 bars) into the lower troposphere.

The 44°N bright feature in the U-band image also exists in the K-band image, but the 40°N bright feature disappeared, probably due to a decrease in spatial resolution. The bright equatorial zone is the most prominent feature in the K-band image, accompanied by a broad dark band centered near 20°N. The Q-band image is noisy and the banded structure is almost indistinguishable, although at these frequencies we probe similar pressure levels to that of the U band.

Zonal-mean residual emission

Figure 3 shows the zonal-mean latitudinal scans of the radio emission across spectral bands, after a limb-darkened disk was subtracted from each image (Fig. 2). The noises are illustrated as gray ticks in each band. Since the emission angle changes substantially across the disk, we average over 60° in longitude around the central meridian of the disk image. For the S band, we increase the width of the latitude bin to obtain at least eight pixels to have a robust estimate of the noise. Thus, the uncertainty reflects the noise in the radio map, which is much smaller than the calibration uncertainty. Any longitudinally dependent features are smeared out because these scans are zonally averaged. But since these scans cover several spectral bands, they reveal changes in brightness temperatures over a range of altitudes.

These zonal-mean scans at the different wavelengths form, in essence, a spectral data cube; we evaluate these at the emission angle between the local zenith and an Earth’s observer. Multiple bright features, such as the 40°N and 44°N bands, are discernible from the spectral scans and can be correlated across spectral space from top to bottom. In general, a feature appears bright (high brightness temperature) when the ammonia vapor, the main microwave absorber, is depleted compared to the average, or the temperature is high. For example, the brightness temperatures are higher at 45°N in the Q, K, U, and X bands but lower than average in the C band compared to the neighboring latitudes, suggesting two end-member scenarios: (i) The ammonia concentration is depleted at higher altitudes (sensed by Q, K, U, and X bands) and enriched at lower altitudes (sensed by C band), or (ii) the temperature is higher at higher altitudes and lower at lower altitudes compared to the neighboring latitudes. The VLA observation alone cannot resolve the ambiguity, but these two scenarios are always related: Evaporation/sublimation of ammonia rain droplets/ice crystals would simultaneously cool the atmospheric temperature and enrich the ammonia vapor concentration.

To have a quantitative assessment of the physical properties of Saturn’s atmosphere, we perform a beam deconvolution followed by a differential spectral inversion to recover the ammonia concentration anomaly (see Materials and Methods). Fitting the brightness temperature anomaly rather than its absolute value bypasses the 3 to 5% (28) calibration uncertainty and leverages the precision of the measurements. Figure 4 shows the deconvolved spectral scans in orange lines and the width of the Gaussian beam in red bars. Deconvolution has the largest effects on K and U band data due to the high precision, small beam width, and large spatial variation of the brightness temperatures. After the deconvolution, the peak amplitude of the brightness temperature anomaly is raised and the phase (position of the brightness temperature peaks) of the spectrum is shifted. Deconvolution has a relatively small effect on Q, X, C, and S bands given their relatively smooth variation and/or large beam size.

The variation of the brightness temperatures in Fig. 4 reflects the variability of Saturn’s atmospheric composition or temperature on top of an imprecisely determined baseline. From Fig. 1, we find that the brightness temperatures of a homogeneous isentropic model atmosphere could adequately explain the disk-averaged brightness temperatures of Saturn. Therefore, we choose the baseline model of the differential fitting to be the simple homogeneous isentropic model. This choice also provides an additional benefit such that the anomaly map can be compared both vertically (altitude) and horizontally (latitude) since the baseline model has a uniform distribution of ammonia and is isentropic. We further fix the temperature profile and only examine the distribution of ammonia because the observed brightness temperature anomaly is far greater than the increase in temperature due to the giant storm in 2010 (2.4 K) (29) or the latitudinal variability of temperature at the 0.5-bar pressure level (⁓2 K) measured by the Cassini/CIRS instrument (30) based on the collision-induced continuum absorption of hydrogen at far infrared. The Cassini/Visible and Infrared Mapping Spectrometer instrument provides somewhat deeper observation of thermal emission at near infrared, but deriving the meridional variability of temperature is impossible because of the degeneracy with PH₃ and aerosols (31). The formula and details of the differential fitting are provided in Materials and Methods.

Anomalous ammonia concentration map

Figure 5 shows the ammonia anomaly map with Saturn’s zonal wind profile in System III (32) for comparison. We find three distinct patches of ammonia anomalies, between 0°N and 12°N, between 25°N and 45°N, and between 55°N and 72°N. There are six occurrences of giant storms on Saturn in the past 150 years: the 1876, 1933, and 1990 storms erupted between 0°N and 12°N; the 1903 and 2010 storms erupted between 35°N and 42°N; and the 1960 storm erupted near 58°N. We find ammonia anomaly signatures for all mid-latitude eruptions and an equatorial mixture with their long-time evolution. Specifically, we find that the ammonia anomaly propagates meridionally in the increasing direction of the zonal wind, meaning that the meridional velocity is associated with the vorticity of the flow. The propagation of the ammonia anomaly is usually stopped at the position of another zonal jet. On the basis of these observations, we suspect that there could be another storm that has occurred near 70°N latitude that was before 1876, the first recorded storm.

Another finding is a division between atmospheric layers at ⁓3 to 5 bars caused by the giant storms: At altitudes above 3 to 5 bars, the ammonia vapor is depleted, and below that level, the ammonia abundance is enriched. During the eruption of the 2010 storm, the Cassini/Radar observed highly depleted ammonia gas at roughly the 1- to 2-bar level (12). But where did the ammonia vapor go? If ammonia is hidden within the ammonia-rich precipitation (14), it would reevaporate and enrich the ammonia concentration in the deep atmosphere. This is exactly what we have seen using longer-wavelength radio observations: After a giant storm, ammonia gas is indeed transported from the upper atmosphere to deeper layers. The layering of the atmosphere indicates stable stratification, and the simplest form is that the atmosphere is divided into two distinct layers. On the basis of the observation of the layering, we estimate the convective mass flux of the 2010 storm using an energy balance calculation.

The last 2010 giant storm increased Saturn’s emitted power in the stormy band by ∼9.2%, which translates into a 2.4 K increase in physical temperature (29). From the ammonia anomaly map, we find that the transition pressure level is at about p_b ∼ 3 bars. Therefore, we posit that at least 3 bars of atmospheric mass is warmed up either by the direct thermodynamic response to the latent heat release or by secondary circulation such as the compensating subsidence through adiabatic heating (6). The amount of latent heat released per unit area per unit time is equivalent to the product of the mass flux of the upward motion and the mass mixing ratio of water, multiplied by the latent heat of water. Equating that to the increase of column temperature yields

where the heat capacity of the hydrogen-helium atmosphere is c_p = 1.3 × 10⁴ J/(kg K); the increase in temperature is ΔT = 2.4 K. The upward convective mass flux is denoted as ${\dot{m}}_u$. The spatial area covered by the storm is S = 3 × 10¹⁵ m² based on the 300,000 km circumference of the planet and the 10,000 km width in latitude, and t = 6 months is the duration of the storm. Saturn’s gravity at 40°N is g = 11 m/s². The latent heat of water is L = 2.5 × 10⁶ J/kg, and the mass mixing ratio of water participating in the convection is q_w. If we use the deep water mixing ratio, which is about 10% of the total atmospheric mass (6), the convective mass flux is

Assuming that the detrainment level of the upward convective mass flux occupies one scale height at the physical temperature of 100 K, H = RT/g = 34 km, where the atmospheric pressure is p = 0.4 bar and the atmosphere density is ρ = p/(RT) = 0.1 kg/m³. The associated cloud expansion rate of the giant storm in 2010 is thus

This value is very close to the measured cloud expansion rate of $\dot{A}=200$ km²/s based on the sequence of cloud images over the whole lifetime of the storm (3, 4). The division levels of other storms are located at lower altitudes than that of the 2010 storm, indicating potential subsidence over the years following a giant storm. The subsidence could be driven by radiative cooling and convective adjustment. For example, the downward propagation of a stable interface separating a convective interior and a stable upper atmosphere was simulated by a one-dimensional convective adjustment model (6). In addition, as described in the theory, the downward propagation is stopped at the water condensation level, which is around 20 bars assuming a 10 times solar water abundance.

The third finding revealed by the ammonia anomaly map is that a mixing barrier seems to form where the zonal wind velocity crosses zero. This is most evident near the two zero-crossings of the zonal wind profile flanking the 40°N westward jet. Specifically, the post-eruption evolution of the 2010 storm is intriguingly split into two distinct components. The northern one propagates northward and the southern one propagates southward, leaving a gap in between.

The gap is located at 43 ± 0.4°N based on the high-resolution U-band image of Saturn. The location is very close to the zero-velocity crossing at 41.8°N using the System III rotation period (10 hours, 39 min, 24 s) determined by Voyager’s measurement of Saturn’s kilometric radiation in 1978 (32). Saturn’s rotational period is uncertain because Saturn’s dipole magnetic field is not tilted relative to its rotation axis (33). Other indirect measurements based on atmospheric dynamics, response to gravity, and ring seismology generally place Saturn’s rotation period in between 10 hours 32 min 35 s and 10 hours 34 min 13 s (34–36), which corresponds to a zero-crossing latitude near 44.2°N. It is unknown what causes the separation and the formation of the gap. We raise a conjecture that the gap could indicate where the zonal wind is zero. This can be tested by a numerical model studying the long-term evolution of the post-eruption tropospheric dynamics.

VLA data acquisition and calibration

The VLA data were acquired in January and May 2015 and presented previously with the analysis of Saturn’s ring system, referred to here as Zhang et al. (18). As discussed in Zhang et al. (18), all data were initially processed and calibrated using the internal VLA calibration pipeline. They subsequently used the MIRIAD software to self-calibrate, clean, and map the data (19). Zhang et al. (18) combined all eight channels in each spectral window, as well as several spectral windows to increase the signal-to-noise ratio (SNR). In U and X bands, the final bandwidth per window was 1 GHz; in the K band, it was 4 GHz; in the Q band, all windows were combined with a resulting bandwidth of almost 8 GHz. In C and S bands, all spectral windows were combined as well, resulting in a bandwidth of 0.25 GHz in each band. For X and U bands, data were obtained in two different array configurations. To increase the sensitivity for atmospheric study, we combined the data obtained on different dates by scaling the flux density of the source to match both observations. This flux density is integrated over the solid angle extended by the radio source, and thus captures the changes in the change of Saturn’s diameters on different dates.

The flux density scale was calibrated on the radio source 3C286, with the standard VLA flux calibrator scale (28, 37). The radio source J1558-1409 was used as the complex gain calibrator (phases). Internal and absolute uncertainties in the flux densities are believed to be better than ∼3% at most frequencies and 5% at K and Q bands. We adopt these, somewhat conservative, uncertainties in our work during the fit to the disk-averaged brightness temperatures. These uncertainties only affect the precision of the baseline inversion and do not affect the accuracy of the retrieved ammonia anomalies through the differential spectral inversion.

Zhang et al. (18) used a “planet + ring” model to self-calibrate the data and determine the best-fit parameters for both the planet and Saturn’s rings. The planet’s brightness temperature T_b of the unocculted portion of the planet’s disk was modeled as T_b = T₀ + T₁〈μ_p〉, where T₀ and T₁ are model parameters of the uniform and limb-darkened disk, respectively, and μ_p is the cosine of the emission angle. The rings were modeled using the Simrings code (38). Modeling the rings this way enabled Zhang et al. (18) to determine the ring properties of Saturn’s main rings, as, e.g., their optical depth, particle size distribution, composition, and porosity. In our work, we use final calibrated UV data from Zhang et al. (18) and subtract the model for the rings to construct the best possible data for Saturn’s atmosphere. Rather than using the combined C&B and B&A array configurations for the X and U data, we instead focus on only the B&A configuration for these bands, although we use the more compact array to calibrate the flux densities.

On the basis of Zhang et al.'s model (18) fits of Saturn’s disk, we determined the disk-averaged brightness temperatures in each spectral window, as summarized in table S1. Since Saturn blocks out the cosmic background radiation (CMB), and the flux density is measured relative to the background, we add the temperature corresponding to the CMB to the disk-averaged brightness temperature. We further used a disk-averaged value of 0.667 for the cosine of the emission angle of the disk when combining the T₀ and T₁ temperatures into a disk-averaged value. While we derive here (table S1) the disk-averaged brightness temperature in each spectral window, to increase the SNR in the maps, we combined all spectral windows per frequency band, resulting in a total of six maps.

Deconvolution algorithm

All radio maps presented here have different resolutions, i.e., they are all “smoothed” by their respective beams, which differ from wavelength to wavelength. To obtain a quantitative assessment of the physical properties of Saturn’s atmosphere, we need to convolve models of Saturn’s atmosphere with their respective beams in an iterative process to determine the cause of observed variations in brightness temperature with latitude. To speed up this process and to mitigate the effect of the beam picking up cold sky signals when close to the limb, we instead deconvolve the latitudinal scans in each wavelength band and then use a differential spectral inversion to obtain an ammonia anomaly map, i.e., the difference in ammonia profiles relative to the nominal (200 ppm) value as a function of latitude that best matches the data.

We denote the beam-convolved zonal-mean brightness temperature anomaly as ${\tilde{T}}_b(x)$, where x is the projected distance normal to the observer’s line of sight, referred to as the sky coordinate. The corresponding point-wise brightness temperatures are denoted as T_b(y). We approximate the VLA beam pattern as a Gaussian function, denoted as G(y; s), where s represents the full width at half maximum (FWHM) of the beam and is related to the SD of the Gaussian function by $s=2\sqrt{2\ln 2}\mathrm{\sigma }$. Using the sky coordinate x, the beam convolution gives

To deconvolve the observed brightness temperature, we use the Fourier transform and regularized optimization. The Fourier transform of Eq. 4 is

where w is the angular frequency and the product is performed element-wise at each frequency.

In principle, one can just divide the Fourier transform of the convolved beam, $\mathcal{F}\{{\tilde{T}}_b\}$, by the Fourier transform of the Gaussian beam, $\mathcal{F}${G}(w_i), to get the Fourier transform of the pointwise brightness temperature, $\mathcal{F}${T_b}(w_i), and then invert it for the pointwise brightness temperature. However, this method suffers from numerical instabilities for high-resolution maps. On the one hand, high-resolution observations contain more information, but on the other hand, it introduces more Fourier modes to invert. Usually, the increase of the information content is not enough to constrain the additional Fourier modes, and therefore, a direct division yields an unstable solution. We use regularized optimization to overcome the difficulty. In discretized frequency domain, w_i, Eq. 5 becomes

Now, we cast the equation above into a matrix multiplication form such that

where G_ij = $\mathcal{F}${G}(w_i)δ_ij are the coefficients of a diagonal matrix. Instead of solving Eq. 7 exactly, we try to obtain a regularized least-square solution that demands vanishing amplitude for high-frequency modes such that

subject to small $\mathcal{F}${T_b}(w_j) for large w_j. The way to do it is to add a penalty term for the amplitude of $\mathcal{F}${T_b}(w_j) as a function of its w_j. By doing experiments with triangle functions (see the Supplementary Materials, section B), we find that a penalty proportional to frequency squared works well for the deconvolution. Mathematically, the cost function of the minimization is

where λ is a tunable parameter governing the strength of the regularization. Written in matrix notation, the least-square solution to Eq. 9 is

where W is a diagonal matrix with its elements being W_ij = w_iδ_ij. The component form of Eq. 10 is

Compared to Eq. 6, the λ² term in the denominator of Eq. 11 provides the stability of the solution. For low-frequency modes, where $\mathcal{F}${G}(w) ≫ λ²w², Eq. 11 reduces to Eq. 6. For high-frequency modes, their amplitude would decay as w⁻². The cutoff frequency, w_c, occurs at $\mathcal{F}${G}(w_c) = λw_c. It is useful to define a cutoff wavelength, $L_c=\frac{2\mathrm{\pi }}{w_c}$, to denote the spatial resolution that can be recovered from the deconvolution. Spatial variations at wavelength smaller than L_c will be suppressed by the deconvolution, and variations greater than L_c will be preserved. Note that Eq. 11 bears resemblance to the Wiener deconvolution, which has been used extensively in the imaging processing community to sharpen a noisy image (39), such as restoring the Hubble Space Telescope (HST) images (40). However, the primary goal of the Wiener deconvolution is to clean up a distorted image due to random noise, and our goal is to recover the underlying pointwise brightness temperature from a (beam-convolved) observation. For a Gaussian-shaped beam, its Fourier transform is

We calculate the discrete Fourier transform of the convolved brightness temperature, $\mathcal{F}\{{\tilde{T}}_b\}$, using fast Fourier transform (FFT) at discrete frequencies w_i. Then, we use Eq. 12 to calculate the Fourier transform of a Gaussian beam at the same discrete frequencies and apply the result to Eq. 11 to get the spectral amplitudes of the pointwise brightness temperature. Last, inverse FFT is performed to recover the deconvolved brightness temperatures.

To examine the performance of the deconvolution, we reconvolve the beam using Eq. 5 and compare that to the original (beam-convolved) brightness temperatures. The results are displayed in fig. S2. The choice of the penalty strength λ² depends on the desired cutoff wavelength L_c, which is related to the width of the radio beam. From the synthetic study of a triangle signal function (Supplementary Materials, section B), we conclude that the spatial sampling resolution can be between one-quarter and one-half of the beam’s FWHM. For S, C, and X bands, we use one-half of the beam’s FWHM as the cutoff wavelength, and for U, K, and Q bands, we use one-quarter of the beam’s FWHM as the cutoff wavelength. The effect of cutoff wavelength can be illustrated by looking at the distribution of the Fourier amplitudes. For example, fig. S2 shows the real part and the imaginary part of the Fourier amplitudes of the beam-convolved brightness temperatures (blue circles and blue crosses). The cutoff wavelength of the S-band data is visually identified at L_c = 0.1 L, where L is the diameter of Saturn’s disk, and that for the X band decreases to about 0.02 L. The beam’s FWHM at the S band is about 1/5 L, and therefore, any variations of wavelength smaller than 1/10 L are suppressed. Deconvolution does not have an effect over wavelengths greater than 1/10 L. Large-scale (compared to the beam width) features are preserved in the deconvolution.

Differential spectral inversion method

After the deconvolution, we perform a differential fitting using the algorithm below. Let T_b(y, μ; λ) be the brightness temperature at latitude y, at cosine emission angle μ = cosθ, and at wavelength λ. We decompose T_b into a baseline component $T_b^0$ subject to a calibration uncertainty and a fluctuation component $T_b^1$ subject to a measurement precision such that

where μ₀ is the averaged emission angle, ɛ(λ) is the random variable representing the calibration uncertainty as a function of wavelength only, and δ(y, λ) is the random variable representing the measurement precision, which is a function of both wavelength and latitude. We assume that $T_b^0\gg T_b^1$ and the calibration uncertainty is much larger than the precision of the measurement: ɛ ≫ δ. To represent the fluctuation, we request the average of $T_b^1$ to be zero. Therefore, $T_b^0+\mathrm{\epsilon }$ can be obtained by spatial average and $T_b^1+\mathrm{\delta }$ is the residual after subtracting the mean.

Suppose that the radiative transfer equation is written as

where q represents the vertical profile of ammonia vapor, which is a function of latitude. Suppose that there exists an unknown profile, $\hat{q}$, satisfying

we use the mean value theorem to obtain

where q_a(y) is between q(y) and $\hat{q}$. Upon matching the terms in Eqs. 13, 15, and 16, we find that

On the other hand, if Eq. 15 can be approximately fitted by a known profile, q₀, within the calibration uncertainty such that

Then, for any profile q(y), using the mean value theorem again, we have

where q_b is between q₀ and q(y). Note that q_b may not be known, but f^′(q_b) can be precisely evaluated from the radiative transfer calculation. Subtracting Eq. 19 from Eq. 17 yields

Since q₀ and $\hat{q}$ are sufficiently close (up to the calibration uncertainty) and if the brightness temperature function f(q) is approximately linear for a range of q that yields small variation of $T_b^1$ compared to $T_b^0$, we can safely assume that

without knowing $\hat{q}$ and q_a exactly. Denote the ammonia concentration anomalies as

the differential fitting formula reads as

and the minimization function is

If Eq. 21 is strictly valid, then the fitted ammonia anomaly δ^Mq converges to the true ammonia anomaly δ^Aq. Otherwise, the inference would be biased. The validity of Eq. 21 depends on how close q₀ is to $\hat{q}$ and how linear the brightness temperature function is with respect to the ammonia concentration. Figure S7 shows the optical depths across various VLA radio bands. It shows that different bands are sensitive to different pressure levels. At about 10 bars, S band’s optical depth is close to 1. At about 20 bars, S band’s optical depth increases to 4. Linearity is best observed when the optical depth is close to 1. We terminate our inversion algorithm at 25 bars, as beyond this point, the sensitivity is lost and the linearity assumption is no longer valid. Nevertheless, using the differential fitting formula, Eq. 25 bypasses the large calibration uncertainty ɛ and leverages the measurement precision δ.

The fitting and radiative transfer were performed using the Markov chain Monte Carlo (MCMC) method as described in (24, 41). First, we divide the atmosphere into five layers with pressure boundaries at 25, 12.2, 5.98, 2.93, 1.43, and 0.7 bars, respectively. These pressure levels are chosen on the basis of log spacing between 25 and 0.7 bars, which denote the lower and upper boundary of the spectral inversion domain. Second, we fix the ammonia concentration at 200 ppm at 25 bars (see the Supplementary Materials, section A, for the choice) and vary the ammonia concentration at the other pressure levels. The ammonia profiles in between these pressure levels are smoothly interpolated. In total, we have six sampling variables. Third, we calculate the brightness temperatures of this model atmosphere at the emission angle specified by the observing geometry for each frequency band. Last, we minimize Eq. 25 based on the MCMC algorithm. More discussions on the inversion parameters are provided in the Supplementary Materials, section C.

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