quantmet

Documentation: scorvec.com/quantmet-tools

Quantitative meteorology tools, extracted from the diagnostics pipelines behind scorvec.com, whose daily products import this package. dct_spectra and tc_detect come from products since retired and conservation from a one-off study; verify, thermo and degdays (0.4) are general-purpose implementations written for the package. Each module is self-contained and documented with the numerical subtleties that matter in practice, and each has a test file in tests/, against analytic cases wherever one exists.

Module	What it does	Key references
quantmet.tem	Transformed-Eulerian-mean diagnostics: eddy covariances by zonal wavenumber (Nyquist-exact), residual circulation v*, ω*, the full primitive-equation E–P flux and its divergence by wave group, residual streamfunction, downward control, tropical upwelling w*, eddy heat flux [v′T′] by wavenumber	Andrews, Holton & Leovy (1987); Haynes et al. (1991, JAS)
quantmet.epflux	Quasi-geostrophic Eliassen–Palm flux + divergence as a zonal force, with a per-member-flux ensemble path; zonal-mean static stability, ln-p derivatives and below-ground masking; Charney–Drazin propagation ceiling for stationary planetary waves	Eliassen & Palm (1961); Edmon, Hoskins & McIntyre (1980, JAS); Charney & Drazin (1961, JGR)
quantmet.waf	Takaya–Nakamura wave-activity flux, per ensemble member, with divergence, sign agreement and a low-passed basic state	Takaya & Nakamura (2001, JAS)
quantmet.pv	Ertel PV on pressure levels, the dynamic tropopause (θ, p, wind on 2 PVU by an upward search interpolated in PV), PV on isentropes	Hoskins, McIntyre & Robertson (1985, QJRMS); Morgan & Nielsen-Gammon (1998, MWR)
quantmet.packets	Rossby wave-packet envelope (Hilbert transform along latitude circles) and latitude-band means for packet Hovmöllers	Zimin et al. (2003, MWR)
quantmet.tc	ECMWF tropical-cyclone track BUFR decoding (every ensemble member), recurvature, the outflow–jet interaction metric (−v_χ·∇PV), Hart cyclone phase space and extratropical-transition timing	Archambault et al. (2013, 2015, MWR); Hart (2003, MWR); Evans & Hart (2003, MWR)
quantmet.ensemble	Ensemble sensitivity (regression across members) and member-group composites with FDR masking, and a no-lookahead estimator of systematic 00Z/12Z cycle offsets	Ancell & Hakim (2007, MWR); Torn & Hakim (2008, MWR)
quantmet.conservation	Global dry-air mass, the column water budget (implied evaporation), column total energy on pressure levels, atmospheric angular momentum and its mountain and friction torques	Trenberth & Smith (2005, J. Climate); Peixoto & Oort (1992); Egger et al. (2007, Rev. Geophys.)
quantmet.gill	Gill model: steady damped shallow-water response to tropical heating by one sparse direct solve; heating from SST anomalies (total-SST convective threshold) or from rainfall	Gill (1980, QJRMS)
quantmet.qbo	QBO zero-wind line: the latitude where the zonal-mean wind turns easterly equatorward of the winter westerlies, with the dip rule and censoring	Holton & Tan (1980, JAS)
quantmet.stats	Benjamini–Hochberg FDR, Prais–Winsten regression with AR(1) errors and segment breaks, Bartlett variance inflation, event-block bootstrap composites, Freedman–Lane partial-correlation test, bootstrap r comparison, AUC	Wilks (2016, BAMS); Freedman & Lane (1983)
quantmet.helmholtz	Global Helmholtz decomposition (velocity potential / streamfunction, divergent / rotational wind) by spherical-harmonic Poisson inversion on a pole-free Gauss–Legendre grid	—
quantmet.wk_filter	Wheeler–Kiladis space-time filtering of equatorial waves (Kelvin / ER / MJO or any custom band) with per-wavenumber dispersion bounds from the full cubic; Hermitian-safe FFT masking; any number of middle dimensions; soft-landing end padding; Lanczos low-pass	Wheeler & Kiladis (1999, JAS)
quantmet.dct_spectra	Kinetic-energy power spectra on limited-area grids via 2-D DCT + elliptic-ring binning; spectral slopes and effective resolution	Denis, Côté & Laprise (2002, MWR); Skamarock (2004)
quantmet.tc_detect	Closed-circulation tropical-cyclone candidate detection with an O(1)-per-gridpoint summed-area-table ring test (C kernel, compiled on first use)	—
quantmet.verify	Forecast verification: ensemble, fair (unbiased for finite and weighted multi-model ensembles) and Gaussian CRPS; tie-randomised rank histograms and PIT; the Brier score with its exact decomposition and reliability curves; the Diebold–Mariano test with a prewhitened Newey–West variance and the Harvey–Leybourne–Newbold correction; strict pairing of dated series	Gneiting & Raftery (2007, JASA); Ferro (2014, QJRMS); Hamill (2001, MWR); Stephenson et al. (2008, WAF); Diebold & Mariano (1995); Harvey et al. (1997, IJF)
quantmet.thermo	Moist thermodynamics on SI inputs: saturation vapour pressure (Magnus/Alduchov–Eskridge, Bolton, Murphy–Koop; water and ice), pressure-aware humidity conversions, θ (dry and moist κ), T_v, θ_v, density temperature, θ_e (Bolton pseudo-adiabatic and reversible), θ_es, θ_l, θ_il, θ_w, Stull and psychrometric wet bulb, exact LCL, static energies, enthalpy, the pseudo-adiabat, column water / MSE / enthalpy 0 → p_s, hypsometric thickness; NWS heat index and wind chill; unit helpers	Bolton (1980, MWR); Alduchov & Eskridge (1996, JAM); Murphy & Koop (2005, QJRMS); Emanuel (1994); Davies-Jones (2008, MWR); Romps (2017, JAS); Stull (2011, JAMC); Rothfusz (1990); Osczevski & Bluestein (2005, BAMS)
quantmet.degdays	Degree days: midpoint and integral conventions, degree hours by time-of-day block, degree days from daily extremes and a diurnal shape, expected degree days of a Gaussian temperature, kink-first weighted aggregation, DST-safe local days from UTC data, a 366-slot day of year, two-hinge balance-point fit	Thom (1954, MWR); Kissock, Haberl & Claridge (2003, ASHRAE RP-1050)
quantmet.constants	One SI constant set (R_d, R_v, c_pd, c_pv, c_l, c_i, L_v0, L_f0, g, …) used by thermo	—

Quick starts

Stratospheric momentum budget and the Brewer–Dobson circulation (zonal means and eddy covariances from (plev, lat, lon) snapshots, levels 0.1–200 hPa):

from quantmet.tem import eddy_covariances, tem_terms, downward_control, fill_tropics, upwelling_w

r = eddy_covariances(u, v, t, omega)  # [.] and [a'b'], total and by k1 / k2 / k3+
tt = tem_terms(lat, plev_hpa, r)  # fhat, v*, omega*, EP flux, div F (m/s^2) by wave group
psi = downward_control(lat, plev_hpa, tt["fhat"], tt["epd"] + gwd)  # kg/s; parts add linearly
psi = fill_tropics(psi, lat, edge=15)  # fhat -> 0 near the equator
up = upwelling_w(psi, lat, plev_hpa, lat_s=-25, lat_n=25)  # tropical w* (m/s) by level

Is the planetary wave going up? (100 hPa eddy heat flux, poleward positive):

from quantmet.tem import eddy_heat_flux

hf = eddy_heat_flux(v100, t100, lat, kmax=72, lat_band=(45, 75))
hf.total, hf.by_k[..., 0], hf.by_k[..., 1]  # all waves, wave-1, wave-2

Stratospheric wave driving (QG E–P flux, per-member ensemble path) and the Charney–Drazin ceiling:

from quantmet.epflux import ep_flux, ensemble_ep_flux, charney_drazin_excess

r = ep_flux(u, v, t, lat, plev_hpa, kmax=3)  # (plev, lat, lon) snapshots in
r.force  # ∇·F as m/s per day, (plev, lat)
ens = ensemble_ep_flux(((u_m, v_m, t_m) for u_m, v_m, t_m in members), lat, plev_hpa)  # per-member fluxes averaged
lid = charney_drazin_excess(ubar, lat, plev_hpa, theta_profile, k=1)  # u - U_c; 0 < u < U_c propagates

Rossby wave packets (250 hPa wave-activity flux from an ensemble):

from quantmet.waf import streamfunction_anomaly, ensemble_tn01_flux, lead_smooth

psi_a, lat, lon = streamfunction_anomaly(u250, v250, psi_clim)  # needs quantmet[waf]
w = ensemble_tn01_flux(member_psi_anomalies, U, V, lat, lon, p_hpa=250)
w.wx, w.wy, w.div, w.agree  # mean of per-member fluxes; sign agreement 0.5-1

Gill response to a heating pattern:

from quantmet.gill import gill_response, heating_from_ssta, heating_from_precip, to_dimensional

Q = heating_from_ssta(ssta, sst_clim, lat)  # shape only; or, in physical units:
Q = heating_from_precip(precip_anom_mm_day)  # 1 mm/day = 28.9 W/m^2 of column heating
u, v, p = gill_response(Q, lat, lon, eps=0.1)  # low-level fields, nondimensional
u_ms, v_ms, phi = to_dimensional(u, v, p)

QBO zero-wind line (per day, level or member):

from quantmet.qbo import zero_line

lat0, flag = zero_line(ubar_50hpa, lat, hemi="nh")  # flag 0 found, 1 undefined, 2 censored

Significance:

from quantmet.stats import benjamini_hochberg, prais_winsten, acf_inflation, event_bootstrap

sig = benjamini_hochberg(pvals, alpha=0.10)  # field significance, NaN-aware
fit = prais_winsten(y, X, start=starts)  # trend with AR(1) errors, segment breaks
infl = acf_inflation(daily_residuals)  # variance inflation of a mean (Bartlett)
mean, se = event_bootstrap(composite_days, event_ids)

Equatorial wave decomposition (time × ... × longitude anomalies):

from quantmet.wk_filter import wk_filter

kelvin = wk_filter(olr_anom, "Kelvin")  # same shape, real
kelvin = wk_filter(chi_anom, dict(k=(1, 14), p=(2.5, 20), h=(8, 90), n=None), pad_end=60)  # custom band, live end

Walker circulation strength (velocity potential of the divergent wind):

from quantmet.helmholtz import velocity_potential, irrotational_wind

chi, lats, lons = velocity_potential(u200, v200)  # xarray DataArrays in
uchi, vchi = irrotational_wind(chi, lats, lons)

Model effective resolution (used to compare HRRR vs RRFS):

from quantmet.dct_spectra import ke_spectrum, spectral_slope, effective_resolution

wl, p = ke_spectrum(u10, v10, res_km=3.0)
print(spectral_slope(wl, p, 30, 300))  # ≈ -5/3 in the resolved mesoscale
print(effective_resolution(wl, p))  # where numerics start dissipating

Dynamic tropopause (θ on 2 PVU; one member — PV does not survive averaging):

from quantmet.pv import ertel_pv, dynamic_tropopause, on_isentrope

pv, theta = ertel_pv(t, u, v, p_pa, lat, lon)  # (lev, lat, lon) in, PVU out
dt = dynamic_tropopause(pv, theta, u, v, p_pa)  # dt["theta"], dt["p"] (hPa), dt["u"], dt["v"]
pv330 = on_isentrope(pv, theta, u, v, 330.0)["pv"]

Rossby wave packets (a packet Hovmöller):

from quantmet.packets import packet_envelope, band_mean

env = band_mean(packet_envelope(v250, kmin=4, kmax=15), lat, 35, 60)  # v250 (time, lat, lon) -> (time, lon)

Tropical cyclones meeting the jet:

from quantmet.tc import decode_tracks, recurvature, pv_advection, outflow_index, phase_series, et_times

storms = decode_tracks("20261002000000-360h-enfo-tf.bufr")  # ECMWF open data, every member
ctl = storms[0]["tracks"][0]  # member 0 = control
recurvature(ctl)  # lead (h) of the westward-to-eastward turn
adv = pv_advection(u_chi, v_chi, pv_300_200, lat, lon)  # PVU/day
outflow_index(adv, lat, lon, clat, clon)  # Archambault et al. interaction metric
hart = phase_series(z_steps, levs, lat, lon, ctl, steps)  # B, -VT_L, -VT_U along the track
et_times(hart)  # (onset, completion) leads

Ensemble sensitivity with field significance:

from quantmet.ensemble import sensitivity

r = sensitivity(day3_storm_latitude, z500_day7)  # (members,) and (members, lat, lon)
r["slope"], r["sig"]  # m per degree; Benjamini-Hochberg mask at FDR 10 %

Do AI weather models conserve mass, water, energy and angular momentum?

from quantmet import conservation as C

C.dry_air_mass(ps, tcw, lat, lon)  # kg, ~5.135e18
C.column_energy(t, phi, q, p_pa, ps, u, v, phis=g * orography_m)  # J/m2: total = cpT + Lq + KE + Phi_s p_s/g
C.relative_aam(u, p_pa, ps, lat, lon, hemispheres=True)
C.mountain_torque(ps, orography_m, lat, lon)

Ensemble TC tracking building block:

from quantmet.tc_detect import detect_candidates

clat, clon, cp = detect_candidates(mslp_hpa, lat, lon, ring_deg=2.5, depth_hpa=2.0)

Is the new ensemble better, or just bigger? (fair CRPS, a significance test, honest pairing):

from quantmet import verify as V

crps_a = V.crps_fair(ens_a, obs)  # (..., m) members last; unbiased whatever m
crps_b = V.crps_fair(ens_b, obs)
blend = V.crps_fair(
    np.concatenate([ens_a, ens_b], -1), obs, weights=w_members, groups=model_of_member
)  # a weighted multi-model mixture
V.crps_gaussian(mu, sigma, obs)  # EMOS / any Gaussian forecast
dm = V.diebold_mariano(daily_crps_a, daily_crps_b)  # dm.statistic, dm.pvalue; mean_diff < 0: a better
pr = V.paired(score_a_by_date, score_b_by_date)  # dates both legs have; report pr.n / pr.window
V.rank_histogram(ens, obs, rng=0)
V.pit(ens, obs, rng=0)  # ties randomised, seedable
d = V.brier_decomposition(p, o, bins=10)  # bs = reliability - resolution + uncertainty + wbv - wbc

Moist thermodynamics (K, Pa, kg/kg, RH in %):

from quantmet import thermo as T

q = T.specific_humidity_from_dewpoint(td, p)
the = T.equivalent_potential_temperature(t, td, p)  # Bolton (1980) eq. 39, pseudo-adiabatic
thw = T.wet_bulb_potential_temperature(t, td, p)  # Davies-Jones (2008)
t_lcl, p_lcl, z_lcl = T.lcl_romps(t, p, rh)  # exact LCL (Romps 2017)
tw = T.wet_bulb_temperature(t, p, rh=rh)  # psychrometric, at the real pressure
T.wet_bulb_stull(t, rh)  # Stull (2011): sea-level fit, NaN outside its domain
pw = T.precipitable_water(q3d, p_levels, ps)  # kg/m2, integrated 0 -> p_s
dz = T.thickness(t3d, p_levels, 100000.0, 50000.0, q=q3d)  # 1000-500 hPa thickness, m
T.heat_index(T.k_to_f(t2m), rh2m)
T.wind_chill(T.k_to_f(t2m), T.ms_to_mph(wind10))

Degree days (any units; base 65 by default, pass 18 for °C):

from quantmet import degdays as D

ld = D.local_days(times_utc, "America/Chicago")  # 23/24/25-hour local days, partial days dropped
tmax, tmin = ld.reduce(t_hourly, np.max), ld.reduce(t_hourly, np.min)
hdd_mid, cdd_mid = D.midpoint_dd(tmax, tmin)  # the published convention
hdd_int = ld.reduce(t_hourly, lambda x, axis: D.integral_dd(x, axis=axis)[0])  # time mean of the hinge
hdh, cdh = D.degree_hours(t_day, local_hours, blocks=((7, 19), (19, 24), (0, 7)))
hdd, cdd = D.weighted_dd(t_by_location, population)  # kink per location, THEN weight
hdd, cdd = D.expected_dd(ens_mean, ens_sd)  # E[max(65 - T, 0)] for T ~ N(mu, sigma^2)
fit = D.fit_balance_points(daily_temp, daily_load, X=dow_dummies)  # fit.heat_base, fit.cool_base, slopes

Case studies

• AIFS single vs AIFS-ENS control: spectral fidelity by lead time — the deterministic AIFS blurs progressively (MSE objective) while the CRPS-trained ENS control holds a realistic spectrum ten days in; measured with quantmet.dct_spectra on open data.

Numerical notes worth stealing

• Quadratics are formed per member, then averaged. E–P flux, [v′T′], the Takaya–Nakamura flux and every other eddy quadratic: the quadratic of the ensemble-MEAN fields fades with lead time as the members' phases decorrelate, which reads as waves dying when they are merely uncertain. The mean of the members' quadratics does not fade.
• Nyquist halving. Summing a covariance from the cross-spectrum, each wavenumber contributes 2 Re(A_k B_k*)/n², except the Nyquist term of an even grid, which has no conjugate partner and contributes once. Doubling it is invisible at 0.25° and wrong by exactly the Nyquist variance when [v′T′] is summed to k = 72 on a 2.5° grid. With the halving the wavenumber sum equals the direct covariance to rounding.
• Gill by direct solve, heated by total SST. The steady Gill problem is one sparse linear system; an explicitly time-stepped version was unstable (the Coriolis term grows with |y|) and needed spin-up anyway. Heating is switched on where the total SST (climatology + anomaly) supports deep convection, not the climatology — that is what lets a warm cold tongue convect, and it took the modelled El Niño response from ~50 % to ~70 % of the observed regression. The model is checked against the exact damped dispersion relation (Kelvin decay = ε; n = 1 Rossby decay without the long-wave approximation), and the docstring quantifies the periodic wrap of the Kelvin wave at small ε.
• The zero-line dip rule. A zero crossing only counts as a critical line if the easterlies beyond it reach 1 m/s; shallower dips are stepped over. Otherwise a −0.3 m/s wobble in a westerly-QBO autumn reads as a critical line and the index jumps 20° for a day.
• Bartlett, not AR(1), for the noise of a mean. An AR(1) model fills the variance-inflation sum from ρ₁ alone; for an oscillating daily index (ρ₁ = 0.59 with a ~6-day oscillation) that gives 3.9 where the Bartlett sum of the measured autocorrelations gives ~1.3 — a threefold overstatement of the noise of a 30-day mean.
• DCT over FFT for regional spectra: no periodicity assumption, so no detrending step and no leakage from the domain-scale gradient (the difference is not subtle — it's the difference between seeing a k⁻⁵/³ range and seeing an artifact).
• Gauss–Legendre sampling before spherical differentiation: the GLQ grid has no pole points, so divergence/vorticity never touch 1/cos φ = ∞.
• Fold-to-positive-frequency masking in space-time filtering keeps the spectral mask Hermitian, so the inverse FFT is exactly real — no imaginary residue to silently discard. A soft-landing pad at the end of the record keeps the taper off the most recent days.
• Summed-area tables turn a radius-r ring mean into four lookups, making closed-circulation tests O(1) per gridpoint.
• A profile, never a pointwise, static stability in the QG E–P heat-flux term: a local ∂θ/∂p crosses zero in the troposphere and blows the flux up by orders of magnitude. Since v0.3 the profile is the ZONAL mean [θ](φ, p): the global mean carries tropospheric stability above the polar tropopause, 3–4× too small. Take vertical derivatives in ln p on the actual levels, mask below ground before differentiating, and floor cos φ before smoothing — smoothing first lets the polar 1/cos φ blow-up bleed equatorward and bury the real signal.
• Smooth PV before ∇PV: the divergent wind from a spherical-harmonic inversion is smooth, the 0.25° PV gradient is not.
• Integrate pressure-level columns 0 → p_s exactly: layer edges midway, top at 0 hPa, bottom clipped to the surface pressure — stopping at the top level drops the mass above it, and extrapolated below-ground values must get zero weight.
• Column total energy is ∫(c_pT + Lq + k) dp/g + Φ_s p_s/g, not ∫(c_pT + Φ + Lq) dp/g. Internal plus potential energy is ∫(c_vT + Φ) dp/g; integrating p dΦ by parts turns it into ∫c_pT dp/g + Φ_s p_s/g. Adding Φ to the enthalpy counts the pressure work R T twice — +28 % of the total (quantmet ≤ 0.3.0 did exactly that; fixed in 0.3.1).
• Recurvature needs a westward phase: a storm that heads north or east from the start has no recurvature in the forecast; a westward wobble of a few tenths of a degree is not one either (thresholds −1 and +2 m/s).
• Filter along named axes, never positional ones. Longitude is the last axis everywhere; a band-pass along "axis 1" silently filters latitude (or members) the day a leading axis is added. The tests pin it.
• The plain ensemble CRPS rewards size, not skill. For members drawn from the true distribution its expectation is the true CRPS plus E|X − X′|/(2m): +50 % at m = 2, +10 % at m = 10, +2 % at m = 50. The fair CRPS (Ferro 2014) divides the spread term by m(m − 1) and is unbiased at every m (tested 2–50); for a weighted multi-model blend the correction is applied within each model only.
• Diebold–Mariano needs a HAC variance — and prewhitening. Daily loss differentials are persistent; on equal-skill AR(1) errors the naive test rejects 35 % of the time at a nominal 5 %, Bartlett/Newey–West alone 8–10 %, AR(1)-prewhitened Newey–West 5–6 %.
• Pair before you compare. Two series that stop on different days make "the last 7 days" mean two different windows; the error always makes the newer, less-published quantity look low. Compare only on dates both legs have, and print the count ("6/7 d"). Missing is NaN, never 0.
• Stull's wet bulb is a sea-level fit. At 840 hPa, 40 °C and 15 % the psychrometric wet bulb is 19.2 °C; Stull (2011) says 20.8 °C — 1.1 °C of the difference is pressure alone. Even at sea level it runs up to ~1 °C warm in hot, moderately dry air.
• Heat index: the simple formula first. The NWS computes Steadman's simple formula, averages it with T, and uses the Rothfusz regression (and its two adjustments) only when that average reaches 80 °F; Rothfusz everywhere is wrong below 80 °F. Tested against the NWS chart.
• Kink first, weight after. Degree days are convex in temperature, so the degree day of a weighted-mean temperature is never larger than the weighted mean of the degree days (Jensen) — and a mild national mean reads zero HDD while half the population heats. The same goes for ensemble members.
• Midpoint and integral degree days are different quantities. A day swinging 58–78 °F around a 65 °F base has a midpoint HDD of 0 and an integral HDD of 1.83; they agree only when the whole day sits on one side of the base, so shoulder seasons are where they part. Expected degree days of N(μ, σ²) exceed the degree days of μ by up to 0.4 σ.

Install

pip install quantmet-tools                 # core: numpy, scipy
pip install "quantmet-tools[helmholtz]"    # + pyshtools, xarray (helmholtz; the waf streamfunction path)
pip install "quantmet-tools[tc]"           # + eccodes for ECMWF track BUFR, pyshtools/xarray for the outflow wind

The package imports as quantmet (from quantmet.tem import tem_terms). Releases before 0.4.4 are not on PyPI and carry the old distribution name; install one from its GitHub tag:

pip install "quantmet @ git+https://github.com/scorvec/quantmet-tools@v0.4.2"

quantmet.degdays.local_days uses the standard-library zoneinfo, which needs the IANA time-zone database (pip install tzdata where the system has none).

The TC kernel compiles itself with cc on first use into a per-user cache directory ($QUANTMET_CACHE_DIR, else $XDG_CACHE_HOME/quantmet, else ~/.cache/quantmet).

Development

pip install -e ".[test]"   # editable install with pytest
pre-commit install         # ruff lint and format on every commit, as in CI
pytest

Documentation builds with pip install -e ".[docs]" and mkdocs serve.

Citing

If you use quantmet in published work, please cite it: CITATION.cff holds the metadata, and GitHub's "Cite this repository" button formats it.