Phase 2: ground gravity and the equivalence principle (#3)

crates/geopotential/src/coefficients.rs

@@ -9,16 +9,21 @@ pub enum TideSystem {
     TideFree,
     /// Includes the permanent deformation (indirect) part, not the direct tidal potential.
     ZeroTide,
-    /// Includes both the direct and the indirect permanent parts.
+    /// Includes both the direct and the indirect permanent parts. The direct part, the tide-generating
+    /// potential itself, then sits in the Earth's own field, falling as `r⁻³` where it grows as `r²`:
+    /// right on the sphere `r = R` only ("there is no mean-tide C̄20", Mäkinen 2021, J. Geod. 95,
+    /// 106). Its gravity is +45.7 µGal at the equator against the true −30.5; take mean-tide gravity
+    /// from `gravimetry::tides::permanent`.
     MeanTide,
     /// Not stated; cannot be converted.
     Unknown,
 }
 
 impl TideSystem {
     /// `C̄20` in this system minus `C̄20` tide-free. Uses the EGM96/EGM2008 convention
-    /// `ΔC̄20 = −3.11080e-8·k/√5` with `k = 0.3` (Rapp 1989), which gives the IERS 2010 value
-    /// −4.1736e-9 for zero-tide minus tide-free.
+    /// `ΔC̄20 = −3.11080e-8·k/√5` with `k = 0.3` (Rapp 1989), which gives −4.1736e-9 for zero-tide
+    /// minus tide-free, the value the IERS Conventions 2010 quote for EGM2008 (their own
+    /// `A₀H₀k₂₀`, Eq. 6.14 with `k₂₀ = 0.30190`, gives −4.2007e-9).
     fn c20_offset(self) -> Option<f64> {
         let direct = -3.11080e-8 / 5f64.sqrt();
         let k = 0.3;

crates/gravimetry/README.md

@@ -133,6 +133,7 @@ maximum likelihood over many series at once.
 | `atom` | Phase, shot noise, and accelerometer and gradiometer noise of a cold-atom sensor, in orbit (double diffraction) or on the ground (single), its phase sensitivity and the gravitational Aharonov–Bohm rate ([Lévèque et al. 2021](https://arxiv.org/abs/2011.03382); [Stray et al. 2022](https://doi.org/10.1038/s41586-021-04315-3); [CARIOQA](https://arxiv.org/abs/2211.01215); [Trimeche et al. 2019](https://doi.org/10.1088/1361-6382/ab4548)) |
 | `gradiometry` | The error of each coefficient a gradiometer mission like GOCE achieves |
 | `local` | Gravity and gravity gradient of a buried sphere and horizontal cylinder, the shapes a ground survey fits, and the averaging time an instrument needs to see them |
+| `tides` | The lunar and solar tide at a gravimeter, the gravimetric and diminishing factors of the Love numbers, and the permanent tide that separates mean-tide, zero-tide and tide-free gravity and geoids ([Longman 1959](https://doi.org/10.1029/JZ064i012p02351), as [LongmanTide](https://github.com/jrleeman/LongmanTide) codes it; [Mäkinen 2021](https://doi.org/10.1007/s00190-021-01541-5)) |
 | `sim` | A closed-loop simulation of a GRACE-type pair: simulate, estimate the field back, and compare its errors with the predicted ones |
 | `noise` | Noise with any spectrum, its whitening, and the FFT |
 | `technotes` | NASA's TN-14 (C̄20 and C̄30 from laser ranging, [Loomis et al. 2020](https://doi.org/10.1029/2019GL085488)) and TN-13 (degree 1, [Sun et al. 2016](https://doi.org/10.1002/2016JB013073)), from [PO.DAAC](https://archive.podaac.earthdata.nasa.gov/podaac-ops-cumulus-docs/gracefo/open/docs/) |
@@ -161,6 +162,8 @@ errors of a real month.*
   files; `prompt_lead` gives how far ahead of the P-wave a light-speed gravity signal arrives.
 - **Site surveys and monitoring.** `local` gives the gravity and gradient of a buried void or
   dense body, and the averaging time a quantum gravimeter needs to resolve it.
+- **Absolute gravimetry.** `tides` corrects a reading for the lunar and solar tide and moves it
+  between the zero-tide convention of gravity and the mean-tide convention of the IHRS.
 - **Mission design.** `missions`, `atom` and `sim` size the sensors and simulate what field a
   satellite pair recovers, including cold-atom instruments.
 
@@ -179,6 +182,9 @@ errors of a real month.*
   equivalent mass sums to zero); on the 2025 Kamchatka fault model, its uplift matches the
   USGS's own surface displacement with r = 0.89.
 - **Point masses:** their potential matches Newton's law.
+- **Tides:** LongmanTide's test vector to 3×10⁻¹¹ mGal; averaged over the 18.61-year nodal cycle,
+  Longman's tide is δ₂ times the permanent tide to 0.2 µGal; the permanent tide matches Mäkinen's
+  (2021) Eqs. 19–21 to their printed digits.
 - **Kalman smoother:** it matches the exact Gaussian-process solution to 1e-10, its fit
   recovers the settings of simulated data, and on the real record withheld months are predicted
   with calibrated errors.

crates/gravimetry/examples/buried_tunnel.rs

@@ -1,30 +1,51 @@
 //! Finding a void from the surface: the gravity and gradient of a buried tunnel, and how long a
-//! quantum gravimeter must average to see it. The geometry is the Birmingham survey's
-//! ([Stray et al. 2022](https://doi.org/10.1038/s41586-021-04315-3)), which found a tunnel at
-//! about 1.9 m depth.
+//! quantum gravimeter or gradiometer must average to see it. The geometry and the gradiometer are
+//! the Birmingham survey's ([Stray et al. 2022](https://doi.org/10.1038/s41586-021-04315-3)),
+//! which found a tunnel at about 1.9 m depth.
 //!
 //! ```text
 //! cargo run -p gravimetry --example buried_tunnel
 //! ```
 
+use gravimetry::atom::AtomInterferometer;
 use gravimetry::local::{EOTVOS, MICROGAL, averaging_time, cylinder};
 
 fn main() {
     let (radius, depth, contrast) = (1.0, 1.9, -2000.0); // an air-filled tunnel in soil
-    println!("{:>8} {:>12} {:>12}", "offset/m", "Δg/µGal", "Γuu/E");
+
+    // Stray et al.'s gradiometer: two single-diffraction interferometers 1 m apart vertically,
+    // T = 85 ms, 10⁵ atoms at 0.7 Hz. It reads the difference in gravity between its sensors.
+    let stray = AtomInterferometer { t: 0.085, contrast: 1.0, atoms: 1e5, cycle: 1.0 / 0.7, diffraction: 1 };
+    let baseline = 1.0;
+    let difference = |offset: f64| (cylinder(radius, depth, contrast, offset).0 - cylinder(radius, depth + baseline, contrast, offset).0) / baseline;
+
+    println!("{:>8} {:>10} {:>10} {:>12} {:>12}", "offset/m", "Δg/µGal", "Γuu/E", "Δg/1 m in E", "phase/mrad");
     for offset in [0.0, 1.0, 2.0, 3.0, 5.0] {
         let (gz, gradient) = cylinder(radius, depth, contrast, offset);
-        println!("{offset:>8.1} {:>12.1} {:>12.1}", gz / MICROGAL, gradient / EOTVOS);
+        let d = difference(offset);
+        println!("{offset:>8.1} {:>10.1} {:>10.1} {:>12.1} {:>12.2}", gz / MICROGAL, gradient / EOTVOS, d / EOTVOS, stray.phase(d * baseline) * 1e3);
     }
     let (peak, _) = cylinder(radius, depth, contrast, 0.0);
     println!("peak anomaly {:.0} µGal", peak / MICROGAL);
 
     // An Exail-class gravimeter at 500 nm/s²/√Hz needs this long per station for a 10:1 signal.
     let target = peak.abs() / 10.0;
-    println!("to resolve it 10:1 ({:.0} nm/s²): {:.0} s per station", target * 1e9, averaging_time(500e-9, target));
+    println!("gravimeter, to resolve it 10:1 ({:.0} nm/s²): {:.0} s per station", target * 1e9, averaging_time(500e-9, target));
+
+    // The gradiometer at its projection limit, and at the 466 E/√Hz it reached outdoors.
+    let target = difference(0.0).abs() / 10.0;
+    for (what, asd) in [("projection limit", stray.gradient_asd(baseline)), ("outdoors", 466.0 * EOTVOS)] {
+        println!("gradiometer, {what} {:.0} E/√Hz: {:.0} s per station", asd / EOTVOS, averaging_time(asd, target));
+    }
+
     assert!((peak / MICROGAL + 44.1).abs() < 0.1);
     // The gradient changes sign away from the tunnel, which is what localises it.
     assert!(cylinder(radius, depth, contrast, 0.0).1 < 0.0 && cylinder(radius, depth, contrast, 3.0).1 > 0.0);
+    // Over a line mass the difference across the baseline is the point gradient at the geometric
+    // mean of the two depths, 2Gλ/(z₁z₂): about Stray et al.'s 150 E and 17.5 mrad.
+    let mid = cylinder(radius, (depth * (depth + baseline)).sqrt(), contrast, 0.0).1;
+    assert!((difference(0.0) / mid - 1.0).abs() < 1e-12);
+    assert!((difference(0.0) / EOTVOS + 152.2).abs() < 0.1 && (stray.phase(difference(0.0)) * 1e3 + 17.7).abs() < 0.1);
 }
 
 #[test]

crates/gravimetry/src/lib.rs

@@ -12,6 +12,7 @@ pub mod missions;
 pub mod noise;
 pub mod sim;
 pub mod technotes;
+pub mod tides;
 
 /// Area of the world ocean, m²: 3.618×10⁸ km², so 361.8 Gt of water raise sea level 1 mm.
 pub const OCEAN_AREA: f64 = 3.618e14;