LTE formulation (physics-first)

This page describes the Feb2026 LTE pipeline as a sequence of physically meaningful transforms. The implementation is deliberately split so that configuration/bookkeeping and generic numeric primitives do not obscure the physics.

Notation

Pipeline overview

1) Tidal synthesis → raw tide coordinate $T(t)$

2) Seasonal impulse comb → impulsed tide $I(t)$

3) Lagged accumulation (IIR-like) → forcing manifold $F(t)$

4) Bessel-like expansion of the forcing coordinate (nonlinear adjustment)

5) Regression fit to obtain LTE coefficients

6) LTE reconstruction $\hat d(t)$ and metrics (CC, dLOD reference)

1) Tidal input → raw tide coordinate

We synthesize a tide-like coordinate by summing constituents evaluated at the sample times.

A representative form is:

\[T(t) = \sum_k A_k \cos\big(2\pi f_k t + \phi_k\big) \; + \; \text{(optional integration/phase term)}\]

with (dimensionless) frequency:

\[f_k = \frac{Y}{P_k}\]

Code:

2) Seasonal impulse comb → impulsed tide

A discrete seasonal impulse comb selects months (or sampling slots) where impulses fire.

Define the within-year sample index:

\[n(t) = \operatorname{AdaInt}\Big( (t - \lfloor t \rfloor)\,S \Big)\]

where $S$ is samples-per-year (typically 12). The “comb position” is:

\[n_B = \operatorname{AdaInt}(|\Delta B|\,S)\]

The impulse magnitude is:

\[\delta(t) = \begin{cases} \Delta A, & n(t)=n_B\\ \text{asym}, & n(t)= (n_B + S/2)\bmod 12\\ 0, & \text{otherwise} \end{cases}\]

Then the impulsed tide is:

\[I(t) = T(t)\,\delta(t)\]

Important parity note: Ada’s Integer(real) conversion is round-to-nearest, ties away from zero. Python’s int() truncates; using Ada-compatible rounding is required for the impulse month to land correctly.

Code:

3) Lagged accumulation → forcing manifold

We convert impulsed inputs into a lagged forcing coordinate via a causal/anti-causal pass that behaves like an IIR with a sign-dependent ramp term.

Forward pass (conceptually):

\[F_{i} = I_i + m\,F_{i-1} - \operatorname{sign}(F_{i-1})\,c\]

Backward pass (conceptually):

\[F_{i-1} = -I_{i-1} + m\,F_i + \operatorname{sign}(F_i)\,c\]

with memory $m\in[0,1]$ derived from parameters.

Code:

4) Bessel-like expansion (forcing coordinate adjustment)

A compact nonlinear expansion is applied to the forcing coordinate (this is “Bessel-like” in the sense of a truncated harmonic/nonlinear expansion in $x$):

\[F'(t) = x + e_s\sin(2\pi k x) + e_c\cos(2\pi k x) + e_{s2} e_s\sin(2\pi k_2 x) + e_{c2} e_c\cos(2\pi k_2 x)\]

where $x=F(t)$ and $k,k_2$ come from the modulation periods list $M$.

Code:

5) Regression fit (normal equations)

We fit coefficients by solving the normal equations:

\[(X^T X)\,c = X^T y\]

where $y$ is the observed data and $X$ contains:

Code:

6) LTE reconstruction

Given fitted coefficients, we reconstruct the model time series:

\[\hat d(t) = \text{offset} + k_0 F(t) + \sum_j a_j\,g_j(F(t)) + \text{(trend/annual terms)}\]

where $g_j$ are the nonlinear sin/cos basis functions (with optional nonlinearity exponent) and annual terms are sin/cos in $t$ and $2t$.

Code:

Metrics and gates

Correlation coefficient (CC)

The gate uses the correlation coefficient computed on training and out-of-band segments.

Code:

dLOD reference comparison

A reference comparison against dlod3.dat provides an additional stability signature.

Code:

Regression test harness (must stay green)

Links:

Code map (where to read what)