⚡ electrical engineering

Analysis of tropospheric noise at small spatial scales based on GNSS ZTD and Numerical Weather Prediction models

This study quantifies small-scale tropospheric noise in InSAR observations by comparing GNSS and Numerical Weather Prediction (NWP) ZTD data, revealing that residual errors depend primarily on latitude and elevation rather than NWP model resolution, and proposing a functional model to incorporate these sub-grid uncertainties into InSAR error estimates.

Original authors: Adeyinka Olaseinde, Jeremy Maurer

Published 2026-07-06

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Adeyinka Olaseinde, Jeremy Maurer

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Measuring the Earth's "Breath"

Imagine you are trying to take a super-precise photo of the ground to see if it is moving (like a volcano swelling or a mine sinking). Scientists use a technique called InSAR, which is like taking a satellite photo of the Earth's surface and comparing it to a photo taken months later. They can detect movements as small as a few millimeters.

However, there is a problem: the air between the satellite and the ground isn't empty. It's filled with the troposphere (the lowest layer of our atmosphere). This layer acts like a foggy window. When radar signals pass through it, the "fog" (specifically water vapor) slows the signal down, making the ground look like it moved when it actually didn't. This is called tropospheric noise.

The Current Solution: The "Weather Forecast" Map

To fix this, scientists use Numerical Weather Prediction (NWP) models. Think of these models as giant, digital weather maps that predict temperature, pressure, and humidity everywhere on Earth. They try to calculate how much the "fog" slowed down the signal so they can subtract that error from their photos.

The Big Assumption:
Most scientists have assumed that if you look at a very small area (like a neighborhood), the "fog" is pretty much the same everywhere. They thought, "If I correct for the weather using a big map, the tiny errors left over in a small neighborhood will be so small we can ignore them."

What This Study Did: The "Twin Test"

The authors of this paper wanted to test that assumption. They asked: "Is the weather really the same in a small neighborhood, or is there still 'noise' left over after we use the weather maps?"

To find out, they set up a clever experiment:

The Twins: They found groups of GPS stations (which measure the atmosphere very accurately) that were packed very close together—some within just a few kilometers of each other.
The Comparison: They compared the "real" atmosphere measured by these GPS stations against the "predicted" atmosphere from two different weather models:
- ERA5: A global model with a "grid" size of about 31 km (like looking at a map with large, blocky squares).
- HRRR: A regional model with a "grid" size of about 3 km (like looking at a map with much smaller, finer squares).

If the assumption were true, the GPS stations in the same small neighborhood should all agree with each other, and the weather models should match them perfectly.

The Surprising Findings

The results were a bit of a shock to the scientific community:

1. The "Fog" is still there, even in small neighborhoods.
Even though the GPS stations were close together, they didn't agree perfectly. There was still a significant amount of "noise" (variability) left over after using the weather models. It's like looking at a group of twins in the same room; you'd expect them to be dressed identically, but they were actually wearing slightly different outfits. The "fog" in the air was actually changing over very short distances.

2. Bigger maps aren't always better.
The researchers thought the HRRR model (the one with the tiny 3km squares) would do a much better job than the ERA5 model (the 31km squares).

The Result: They were wrong. The high-resolution model did not significantly improve the accuracy at these small scales.
The Analogy: Imagine trying to paint a detailed picture of a forest. You might think using a fine-tipped brush (HRRR) would be better than a thick marker (ERA5). But if the leaves are constantly moving and shifting in the wind (water vapor), even the finest brush can't capture the exact movement perfectly. The problem isn't the size of the brush; it's that the "leaves" (water vapor) are too chaotic to be predicted perfectly by the model.

3. Location matters more than height.
The study found that where you are on the map matters more than how high up you are.

Latitude: The "noise" was much higher near the equator and lower near the poles. This makes sense because the air near the equator is hotter and holds more water vapor, which is chaotic and hard to predict. The air near the poles is drier and more stable.
Elevation: Surprisingly, how high up the station was (mountain vs. valley) didn't have a consistent pattern. Sometimes high places had more noise, sometimes low places did.

The Takeaway: A New Rule of Thumb

The authors concluded that we cannot simply assume the air is "calm" in small areas. Even after using the best weather models, there is still a "point of uncertainty" left over.

They created a simple math formula (a recipe) that scientists can use. If you tell the formula your location (latitude) and your height (elevation), it will give you a number representing how much "noise" to expect.

Why this matters:
When scientists measure the ground moving, they need to know how much they can trust their numbers. This study gives them a way to add a "safety margin" to their calculations. Instead of saying, "We are 100% sure the ground moved 5mm," they can now say, "We are sure it moved 5mm, plus or minus this specific amount of weather noise based on where we are."

Summary in One Sentence

This paper proves that even with our best weather maps, the air is still too "wobbly" and unpredictable over tiny distances to be ignored, and we need a new way to account for this leftover messiness when measuring the Earth's surface.

Technical Summary: Analysis of Tropospheric Noise at Small Spatial Scales Based on GNSS ZTD and Numerical Weather Prediction Models

Problem Statement
Interferometric Synthetic Aperture Radar (InSAR) is a primary tool for monitoring ground surface displacements with millimeter-level precision. However, tropospheric delays represent a significant source of error, often reaching magnitudes of 15–20 cm, which can exceed the signals of interest. While Numerical Weather Prediction (NWP) models are commonly used to correct these delays, most studies assume that tropospheric variability is negligible at small spatial scales (e.g., within the "reference point" area of an InSAR time-series). There is a lack of systematic quantification regarding the magnitude of residual tropospheric noise at these small length scales after NWP-based correction. Furthermore, it remains unclear whether higher-resolution NWP models improve the accuracy of corrections at these sub-grid scales.

Methodology
The study quantifies small-scale tropospheric errors by comparing Zenith Total Delay (ZTD) derived from NWP models against GNSS-derived ZTD at clusters of closely spaced stations (maximum separation ≲13 km, with most <7 km). The analysis focuses on the spatial standard deviation of the residuals ( $\sigma_{sp}$ ) across stations within a group at a given time, rather than just temporal variability at a single station.

Two NWP models representing end-member spatial resolutions were evaluated:

ERA5: A global reanalysis model from ECMWF with ~31 km grid spacing.
HRRR: A regional reanalysis model from NOAA with ~3 km grid spacing.

Data sources included:

GNSS ZTD: Sourced from the Nevada Geodetic Laboratory (NGL) for 11 global sites (including Australia, Japan, Germany, Italy, Chile, Alaska, and California) and manually collected static survey data from four stations in Rolla, Missouri.
Processing: The RAiDER software package was used to download NWP data, reformat it to a regular grid, and perform 3D interpolation to GNSS station locations to calculate ZTD.
Statistical Analysis: The study calculated the mean spatial standard deviation of residuals ( $\sigma_{sp}$ ) for each site and fitted a functional model relating $\sigma_{sp}$ to latitude and elevation.

Key Results

Persistence of Small-Scale Noise: Significant residual variability persists at spatial scales of ≤13 km even after NWP correction. The spatial standard deviation ( $\sigma_{sp}$ ) is comparable in magnitude to global per-station temporal standard deviations reported in prior studies.
Latitude Dependence: Residual noise varies primarily with latitude. Variability is generally higher at low and mid-latitudes (e.g., Italy, California, Missouri) and lower at high latitudes (e.g., Alaska, Germany). A Gaussian-shaped model of latitude effectively captures this trend.
Elevation Dependence: The dependence on elevation is weaker and less consistent. While some trends were observed, they varied between models and datasets, suggesting elevation is not a dominant control compared to latitude.
Model Resolution Impact: Contrary to the hypothesis that higher resolution improves small-scale matching, the finer-resolution HRRR model (~~3 km) did not improve the match to GNSS ZTD compared to the coarser ERA5 model (~~31 km).
- Bias: Both models exhibited systematic biases (ERA5 ~50.1 mm; HRRR ~49.5 mm).
- Variability: HRRR actually exhibited slightly higher temporal variability ( $\sigma_t \approx 1.9$ cm) compared to ERA5 ( $\sigma_t \approx 1.4$ cm).
Nature of Residuals: The observed inter-station spread (1–2 cm) significantly exceeds typical GNSS ZTD uncertainties (2–4 mm), indicating that the residuals represent real, small-scale tropospheric variability (likely driven by water vapor) that NWP models fail to resolve, rather than GNSS measurement error.

Key Contributions

Quantification of Sub-Grid Error: The study provides the first systematic quantification of tropospheric noise at small spatial scales (≤13 km) after NWP correction, demonstrating that this noise is non-negligible.
Resolution Independence: It challenges the assumption that higher-resolution NWP models inherently yield better small-scale corrections, showing that ERA5 and HRRR perform similarly regarding residual spatial variability.
Functional Uncertainty Model: The authors propose a simple functional model (Eq. 6) to estimate small-scale uncertainty ( $\hat{\sigma}_{sp}$ ) as a function of latitude ( $\phi$ ) and elevation ( $z$ ):
$\hat{\sigma}_{sp} \approx (az + b)e^{-\frac{\phi^2}{2c^2}}$
This model allows practitioners to pre-compute a point-wise uncertainty estimate for their study region.

Significance and Claims
The paper claims that the identified small-scale tropospheric variability is a persistent global feature that persists even within closely spaced GNSS networks and at the 3 km resolution of HRRR. The authors assert that this variability must be accounted for in InSAR uncertainty budgets, particularly for tectonic applications where pixels are aggregated or where "noise-free" assumptions are made for reference points.

The proposed functional model offers a quantitative basis for incorporating sub-grid tropospheric error into InSAR uncertainty estimates. The authors suggest that because this residual structure is not resolved by current NWP physics or resolution, future improvements may require combining NWP corrections with complementary methods, such as machine learning approaches, to better resolve fine-scale atmospheric variability. The study concludes that while NWP models are effective for large-scale corrections, a "point-wise" error term derived from latitude and elevation is necessary to accurately represent the remaining uncertainty at the pixel scale.