By applying an annual impulse sample-and-hold on a common-mode basis set of tidal factors, a wide range of climate indices can be modeled and cross-validated. Whether it is a biennial impulse or annual impulse, the slowly modulating envelope is roughly the same, thus models of multidecadal indices such as AMO and PDO show similar skill — with cross validation results evaluated here for a biennial impulse. Now we will evaluate for annual impulse.

PDO

This if for PDO using 1955-1960 as a test interval:

The reference annual impulsed tidal forcing is the same as the model foring — keep that in mind for the other indices to be modeled. The basis set is

  PAR file =>

  -0.13692705024 :offset:
  -0.04314519998 :bg:
   0.13229139920 :impA:
  -4.03799882549 :impB:
  -2.39218908851 :impC:
   0.16854494793:delA:
  -0.73331170792:delB:
   0.01109793846:asym:
   0.00839251691:ann1:
  -1.03073472279:ann2:
   0.00184817010:sem1:
   4.16936186945:sem2:
   0.00007130008:year:
  -0.10325499530:IR:
   0.00602250837 :mA:
   0.00064550631 :mP:
  -0.42821620213 :shiftT:
   0.20548474515 :init:
---- Tidal ----
  27.32166155400,   -2.19095043415,  -63.63291308143,  1, -1084177,  2.02102769668380E-04
  27.21222081500,    0.09832619924,   -4.25045172775,  2,  7252,  1.33739927797268E-03
1095.17502233364,    0.33172573682,   -6.04902659488,  3,  324604,  1.02162405535151E-04
 346.61626917641,   -0.06539705871,    1.18567582246,  4, -9890,  6.68020756457130E-04
 173.30813458820,    0.16806370542,   10.34466371459,  5,  15179,  1.09998339712384E-03
  13.63341568476,   -0.42340566621,  -19.39406777984,  6, -432,  1.27536054918252E-01
  13.66083077700,   -1.94091032595,    1.51731358004,  7, -714,  3.15864689649859E-01
  13.60611040750,    0.30948930091,  -50.06363844404,  8,  3132,  9.57478917081662E-03
  27.55454988600,   -1.62732619750,   -9.51057122480,  9, -2000,  8.56364901331059E-02
  13.77727494300,   -0.28591561635,    6.47278750491,  10, -2238,  1.33743330759702E-02
  27.10365333842,    0.11139947467,   -0.73586456453,  11,  5012,  2.17928150447984E-03
-411.77905984329,   -0.05396567287,   -0.15841271984,  12, -25819,  2.09824536021795E-04
  27.09267692660,   -0.14832487224,    5.52646418425,  13, -4872,  3.10843568311545E-03
-205.88952992164,   -0.58753042983,    0.99856517446,  14, -84237,  6.98301918830809E-04
  13.69115772864,   -0.12782330144,    0.84501837112,  15, -4343,  3.01288414361810E-03
   9.10846048884,   -0.14534552103,   -5.47264666329,  16, -2768,  5.44762246983538E-03
  27.44323926226,   -0.27265601678,   -3.63353207379,  17, -6042,  4.58825378044224E-03
  27.66676714572,   -0.29015512107,   15.84274532526,  18, -4113,  7.23042231444426E-03
2190.35004466729,   -0.25654631516,    0.61781320509,  19, -46580,  5.51950580521193E-04
  26.87829334283,    0.11540722833,    0.17774456874,  20,  13808,  8.29786129130903E-04
   9.13295078376,    0.36241572937,    1.36195556850,  21,  323,  8.55829600504843E-02
   9.12068919638,   -0.07419122439,    3.46624876399,  22, -315,  3.44509453473477E-02
   6.83041538850,    0.11325181464,   -0.93801006957,  23,  11774,  9.53751477200680E-04
   9.10722051800,   -0.20414818695,    6.14936970634,  24, -5089,  4.09226346158531E-03
---- LTE ----
   0.00330968229 :trend:
   0.00007993470 :accel:
  -0.35902200391 :K0:
   5.46337907325 :level:
  -6.60953539988,    1.13153845912,   -0.19334495098-1
   1.71629419521,    0.31536146157,   -1.39268402544 0
  -0.65865167169,    0.37586081951,    1.06598619069 0
   3.66315312511,    0.39441710722,    1.64464117876 0
  10.44533042746,    0.76306819301,   -0.29448810594 1

Note that the strongest tidal factors are the 27.32/13.66d Mf combination (reaching a relative amplitude of ~4 when reinforcing) followed by the 27.55d Mm perigean factor (with amplitude 1.6). The 206d factor (with amplitude of 0.6) is the perigee-syzygy cycle exerting its maximum effect when the sun, moon and earth are aligned at a full moon. The 13.66d factor (at amplitude 0.42) and 13.606d (at amplitude 0.31) is due to the modulation of the 18.6 year nodal cycle. All other factors are less than 0.4, including the 9.13d Mt cycle due to Mf + Mm interaction, responsible for the multidecadal variation.

AMO

Used the same basis factors as a starting point for AMO, with different LTE modulation factors

IOD East

For the eastern portion of the Indian Ocean Dipole

NINO4

Again, little variation from the basis function. What happens when the cross-validation training region is reduced and the testing region is widened?

Restarting from the PDO fit, to avoid memory bias from the previous:

NINO34

Darwin

NINO3

EMI

NAO

The raw NAO time series appears noisy, often attributed to it being more sensitive to the atmosphere rather than the ocean. I applied a 1-year averaging window filter to remove faster than annual variations, and then applied a 2-year differencing for reasons as follows:

Say one has an unsolved time-series f(t), which is derived from a composition of f(t)=g(t)-g(t-T), where T is a fixed lag. Fitting f(t) by using a model h(t)-h(t-T) — a la a differenced model approach — is commonly used in time-series analysis to remove trends or non-stationarity. Specifically, by modeling using a function of the form h(t)-h(t-T), one is directly fitting the differenced structure rather than the raw time-series itself. This approach is variously referred to as differential modeling, lagged differencing, or in some contexts, state-space differencing. The key idea is that instead of modeling the absolute behavior of g(t), you model its changes over a lag T, which can be useful for detecting cyclic patterns, filtering out low-frequency trends, or emphasizing high-frequency variations. In some ways this provides stronger discriminating power:

1. Stationarity Enhancement: Differencing often helps make a time series more stationary by removing trends and long-term correlations, which improves the efficacy of statistical models. (Will return to this with an AMO comparison later in the post)
2. Structural Matching: If the underlying process generating follows a differenced form (e.g., cumulative flows, impulse responses, or autoregressive processes), then modeling it with a similar structure ensures that the model captures the fundamental behavior. See the post on Sydney SLH readings to see how this enhanced the match to the ENSO behavior.
3. Error Minimization in Relative Terms: Since both the observed series and the model are expressed in terms of changes rather than absolute values, potential biases or scale mismatches (e.g., due to drift) may be reduced.

This method is widely used in fields such as: time series forecasting (e.g., ARIMA models where differencing is used for stationarity), system identification in controls engineering (e.g., estimating impulse responses in dynamical systems), and geophysical and climate analysis (e.g., analyzing anomalies rather than absolute values).

In the following, we start from the basic PDO fit and look at various 20-year test windows, to evaluate the skill of the cross-validation.

1940-1960 test interval

1910-1930 test interval

1965-1985 test interval

1985-2005 test interval

1880-1905 test interval

The 2-year differencing wasn’t strictly required for NAO as it doesn’t show nearly the multidecadal variations as AMO, yet the cross-validation results are striking — as are the results for the other indices without the delay differencing. Yet, it’s instructive to show how it works for AMO — note below how the differencing removes the multidecadal variations and thus focuses primarily on the shorter interannual variations.

1880-1945 test interval

The implication here is that the short-term variations may actually show more feasibility for prediction, than the long-term, which requires a lengthy training interval.

Conclusions

1. Note how the Annual Impulse Tidal Forcing modulation (center panel in each chart) showed strong stability across the evaluated climate indices, indicating a fixed common-mode basis for the forcing tidal factors.
2. Note the excellent cross-validation results across the climate indices. Of course, these are not pure, untainted cross-validation results, yet they appear to be resistant to overfitting, which is a good sign.
3. Note how the nonlinear LTE Modulation (lower panel in each chart) is the primary discriminator amongst the indices. Many of the modulation factors are shared among the indices, but the amplitudes vary, thus generating the unique characteristics of each index.

Once again, cross-validation in the temporal domain combined with cross-validation of tidal invariants across the indices is a powerful tool to substantiate the general unified model.

As a final note consider this just published paper “Topology shapes dynamics of higher-order networks”,