Measuring mass redistribution from satellite gravity measurements
2026-02-27
The Central Valley of California overlies one of the largest and most heavily exploited aquifer systems in the world. Groundwater has been withdrawn for irrigation since the early twentieth century, but the rate accelerated dramatically during droughts — including those of 2007–2009, 2012–2016, and 2020–2022 — when surface water allocations were cut and farmers turned to pumping to survive. The loss of groundwater mass shows up in a remarkable measurement: the slight decrease in gravitational attraction that the two GRACE satellites sense as they fly over the region. The lead satellite accelerates slightly as it approaches an area of excess mass and decelerates as it leaves; the trailing satellite does the same, with a time delay. The changing separation between them — measured to micrometre precision by a microwave ranging system — is the raw data from which gravity anomalies are recovered.
GRACE (Gravity Recovery and Climate Experiment), launched in 2002 and succeeded by GRACE-FO in 2018, is the only instrument that measures large-scale changes in the distribution of mass within and at the surface of the Earth. Every month it produces a global map of gravitational anomalies that, when processed to remove known signals from ocean tides, atmospheric loading, and glacial isostatic adjustment, yields changes in terrestrial water storage — the sum of soil moisture, surface water, snow, ice, and groundwater. The spatial resolution is coarse (roughly 300 km) and the processing is complex, but the measurement is unique: no other technique can quantify groundwater depletion over a multi-state region or ice sheet mass balance over an entire continent. This model derives the gravitational potential equations, introduces spherical harmonic decomposition, and explains how mass anomalies are recovered from satellite inter-ranging data.
How much water is California losing during drought?
GRACE mission (2002-2017, GRACE-FO 2018-present):
Twin satellites measure gravity variations.
Principle:
Mass changes → gravity changes → satellite separation changes
Measurement:
Inter-satellite distance via microwave ranging.
Precision: ~10 μm range change → ~1-2 cm equivalent water thickness
Temporal resolution: Monthly
Spatial resolution: ~300-400 km
Applications: - Ice sheet mass balance (Greenland, Antarctica) - Groundwater storage changes - Drought monitoring - Flood assessment - Ocean mass change (sea level) - Terrestrial water storage - Earthquake mass redistribution
Newton’s law:
g = \frac{GM}{r^2}
Where: - g = gravitational acceleration (m/s²) - G = gravitational constant (6.674 × 10⁻¹¹ m³/kg/s²) - M = mass (kg) - r = distance (m)
Earth surface: g \approx 9.81 m/s²
Variations:
Latitude (centrifugal force): ±0.03 m/s²
Altitude (1 km): -0.003 m/s²
Mass anomalies: ±10⁻⁶ m/s² (1 μGal)
GRACE detects: ~10⁻⁸ m/s² (0.01 μGal)
Gravity anomaly → orbital velocity change
Two satellites in tandem:
Leading satellite over mass anomaly: - Accelerates (stronger pull) - Separation increases
Trailing satellite reaches anomaly: - Accelerates (catches up) - Separation decreases
Range rate:
\dot{\rho} = \frac{d\rho}{dt}
Where \rho = inter-satellite distance (~220 km)
Measured: Range rate via K-band ranging (24 GHz)
Precision: 0.1 μm/s
Gravitational potential:
V = \frac{GM}{r} + \text{anomalies}
Spherical harmonic expansion:
V = \frac{GM}{r} \sum_{l=0}^{\infty} \sum_{m=0}^{l} \left(\frac{a}{r}\right)^l P_{lm}(\sin\phi) (C_{lm}\cos m\lambda + S_{lm}\sin m\lambda)
Where: - l = degree (spatial scale) - m = order - P_{lm} = associated Legendre polynomials - C_{lm}, S_{lm} = Stokes coefficients - a = Earth radius - \phi = latitude - \lambda = longitude
Truncation: l_{\max} \approx 60 (GRACE resolution limit)
Degree 60: ~300 km wavelength
Mass anomaly to water depth:
\Delta h = \frac{\Delta \sigma}{\rho_w}
Where: - \Delta h = equivalent water thickness (m) - \Delta \sigma = surface density anomaly (kg/m²) - \rho_w = 1000 kg/m³
From gravity:
\Delta \sigma = \frac{a}{3} \sum_{l,m} \frac{2l+1}{1+k_l} \Delta C_{lm} Y_{lm}
Where k_l = load Love number (elastic deformation).
Gravity gradient along track:
\frac{\partial g}{\partial x} = -\frac{2GM}{r^3} + \text{anomaly gradient}
Differential acceleration:
\Delta a = \frac{\partial g}{\partial x} \times \Delta x
Where \Delta x = satellite separation (~220 km)
Range rate change:
\ddot{\rho} = \Delta a
Integrate:
\Delta \rho(t) = \int_0^t \int_0^{t'} \Delta a \, dt' \, dt
Observed: Range vs predicted (from baseline gravity model)
Residual: Indicates mass change
Power at each degree:
\sigma_l^2 = \sum_{m=0}^{l} (C_{lm}^2 + S_{lm}^2)
Kaula’s rule:
Expected variance:
\sigma_l \propto l^{-2}
GRACE measurement error:
Increases rapidly with degree:
\epsilon_l \propto l^2
Filtering required:
Low-pass (smooth) to reduce noise.
Gaussian filter:
W_l = e^{-l(l+1) b^2 / 2}
Where b = smoothing radius (typically 300-500 km)
Trade-off: Noise reduction vs spatial resolution
Time series at location:
\Delta h(t) = a + b \times t + \sum A_i \cos(\omega_i t + \phi_i) + \varepsilon
Where: - a = offset - b = linear trend (mass change rate) - A_i = seasonal amplitudes - \omega_i = annual, semi-annual frequencies - \varepsilon = noise
Least squares fit:
Solve for parameters.
Uncertainty:
Accounts for temporal correlation in residuals.
Problem: Calculate water storage change from GRACE.
Observations:
Region: California Central Valley (120°W, 37°N, radius 200 km)
GRACE data (simplified):
Month 1 (Jan 2022): Gravity anomaly = +50 μGal
Month 13 (Jan 2023): Gravity anomaly = -30 μGal
Change: -80 μGal
Calculate equivalent water thickness change.
Step 1: Convert gravity to mass
\Delta g = 2\pi G \Delta \sigma
Where \Delta \sigma = surface density change (kg/m²)
\Delta \sigma = \frac{\Delta g}{2\pi G}
Units:
1 μGal = 10⁻⁸ m/s²
\Delta \sigma = \frac{-80 \times 10^{-8}}{2\pi \times 6.674 \times 10^{-11}}
= \frac{-8 \times 10^{-7}}{4.19 \times 10^{-10}} = -1910 \text{ kg/m}^2
Step 2: Convert to water depth
\Delta h = \frac{-1910}{1000} = -1.91 \text{ m}
1.9 meters of water loss!
Step 3: Total volume
Area of region (circle, r = 200 km):
A = \pi r^2 = \pi \times (200000)^2 = 1.26 \times 10^{11} \text{ m}^2
Volume change:
V = A \times \Delta h = 1.26 \times 10^{11} \times (-1.91) = -2.4 \times 10^{11} \text{ m}^3
= 240 km³ loss
Step 4: Interpretation
240 cubic kilometers of water lost in one year.
Causes: - Groundwater extraction - Below-average precipitation - Snow deficit - Soil moisture depletion
This is severe drought (typical seasonal variation: ±50 km³)
Below is an interactive GRACE data simulator.
<label>
Region:
<select id="region">
<option value="greenland" selected>Greenland Ice Sheet</option>
<option value="amazon">Amazon Basin</option>
<option value="california">California</option>
<option value="ganges">Ganges Basin (India)</option>
</select>
</label>
<label>
Time range (years):
<input type="range" id="time-range" min="1" max="20" step="1" value="10">
<span id="time-val">10</span>
</label>
<div class="grace-info">
<p><strong>Linear trend:</strong> <span id="trend">--</span> cm/year</p>
<p><strong>Seasonal amplitude:</strong> <span id="seasonal">--</span> cm</p>
<p><strong>Total change:</strong> <span id="total-change">--</span> m</p>
<p><strong>Mass change rate:</strong> <span id="mass-rate">--</span> Gt/year</p>
</div><canvas id="grace-canvas" width="700" height="400" style="border: 1px solid #ddd;"></canvas>Observations: - Greenland shows strong negative trend (ice loss: -280 Gt/year) - Amazon shows large seasonal variations (wet/dry seasons) - California shows drought signal with seasonal variation - Ganges shows groundwater depletion trend - Blue line: actual GRACE signal (trend + seasonal + noise) - Red dashed: linear trend component - Seasonal variations evident as annual oscillations
Key findings: - GRACE detects both long-term trends and seasonal cycles - Ice sheet mass loss clearly visible as negative trend - Groundwater depletion measurable in major aquifers - Seasonal water storage changes reach 10-25 cm equivalent
Greenland (2002-2023):
GRACE trend: -280 Gt/year average
Acceleration: -20 Gt/year² (increasing loss)
Contributions to sea level:
280 Gt/year ÷ (ocean area × density) = 0.8 mm/year
Regional patterns: - Southeast: Largest losses - Northwest: Moderate losses - Interior: Slight gains (snow accumulation)
Antarctica:
GRACE trend: -150 Gt/year
Variations: - West Antarctica: -160 Gt/year (marine ice sheet collapse) - East Antarctica: +10 Gt/year (slight snow increase) - Antarctic Peninsula: -20 Gt/year
Combined: ~1 mm/year sea level rise from ice sheets
North India (Ganges-Brahmaputra):
GRACE: -4 cm/year water storage loss
Cause: Irrigation extraction > recharge
Volume: -54 km³/year
Unsustainable: Fossil aquifer depletion
California Central Valley:
2011-2015 drought: - GRACE: -15 cm/year peak loss - Groundwater contributed 60% of deficit - 50+ km³ cumulative loss
Recovery: 2017-2019 wet years partially replenished
2010-2011 Amazon drought:
GRACE detected: - -15 cm water storage anomaly - Preceded vegetation stress (optical NDVI) - Early warning capability
Operational use: - USDA drought monitor integrates GRACE - NASA FLDAS (Famine Early Warning System) - Water resource planning
Smoothing spreads signal:
300 km Gaussian filter → adjacent regions contaminate
Example:
Greenland ice loss “leaks” to ocean, land nearby.
Correction:
Forward modelling: - Assume spatial pattern (coast concentration) - Apply GRACE processing - Compute gain factors - Amplify observed signal
Typical: 10-30% underestimate without correction
Ice age deglaciation:
Mantle still rebounding.
GIA vertical motion:
Up to 10 mm/year (Scandinavia, Canada)
GRACE sees mass change:
Cannot distinguish GIA from contemporary changes.
Correction:
GIA models (ICE-6G, etc.) based on ice history.
Subtract from GRACE signal.
Uncertainty: ±20-30% in some regions
Earth’s center of mass moves:
Relative to crust surface.
Degree 1 (l=1) coefficients:
Not measured by GRACE (both satellites affected equally).
Estimate from: - Ocean models - Station position networks - Combination solutions
Impact: Small global (few mm), important for sea level budget
Large earthquakes:
Redistribute mass (coseismic + postseismic).
2004 Sumatra M9.1:
GRACE detected: - Coseismic: -5 cm water equivalent (localized) - Postseismic: Years of relaxation
Challenge:
Separate earthquake from hydrologic signals.
Solution:
Model earthquake, subtract before hydrology analysis.
GRACE-FO (launched 2018):
Improvements: - Laser ranging (addition to microwave) - Better accelerometers - Continuous from GRACE
Laser ranging:
10-100× better precision than microwave.
But: Atmospheric scattering limits (clouds block laser).
Combined system:
Microwave for continuous tracking, laser for highest precision.
Future missions:
Mass Change mission (2028+): - Lower orbit (improve resolution) - Better instruments - Goal: 150 km resolution
Applications expansion: - Smaller aquifers - Individual drainage basins - Urban water use - Irrigation monitoring
On sphere:
Y_{lm}(\theta, \phi) = P_{lm}(\cos\theta) e^{im\phi}
Where: - \theta = colatitude - \phi = longitude - P_{lm} = associated Legendre polynomial
Properties:
Orthogonal:
\int Y_{lm} Y_{l'm'}^* \, d\Omega = \delta_{ll'} \delta_{mm'}
Complete: Any function on sphere can be expanded.
Degree l corresponds to wavelength:
\lambda = \frac{2\pi a}{l}
Where a = Earth radius (6371 km)
Examples: - l = 2: ~20,000 km (Earth’s flattening) - l = 10: ~4,000 km (continents) - l = 60: ~670 km (GRACE resolution) - l = 360: ~110 km (EIGEN-6C4 model)