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Earth, Planets and Space

Table 1 Summary of the numerical experiments

From: A candidate secular variation model for IGRF-13 based on MHD dynamo simulation and 4DEnVar data assimilation

Case code

Window length

\(\varvec{\alpha}_{S}\)

\(\varvec{\alpha}_{UW}\)

[\(\times 30\)]

T (trend), B (bias)

Forecast type

\(\sqrt {\varvec{dP}} \varvec{ }\) 4.5 years after release [nT]

\(\sqrt { \varvec{dP}_{\varvec{wo}}}\) 4.5 years after release [nT]

(O1)

10 years

\(\alpha '\)

2

no T, B

MHD (\(k^{\prime} = 0\))

228.5

168.2

(O2)

10 years

\(\alpha '\)

2

only B

MHD (\(k^{\prime} = 0\))

212.9

164.0

(A1)

10 years

\(\alpha '\)

2

T and B

Ens. wei. sum

153.8

129.9

(A2)

10 years

\(\alpha '\)

2

T and B

MHD (\(k^{\prime} = 0\))

168.2

136.2

(A3)

10 years

\(\alpha '\)

2

T and B

MHD (\(k^{\prime} = 0\))

155.7

131.5

(A4)

10 years

\(0.1\alpha '\)

2

T and B

Ens. wei. sum

100.9

95.1

(A5)

10 years

0.1 \(\alpha '\)

2

T and B

MHD (\(k^{\prime} = 0\))

107.8

102.5

(A6)

10 years

0.1 \(\alpha '\)

2

T and B

KD (\(k^{\prime} = 0\))

106.3

99.3

(B1)

5 years

\(\alpha ''\)

2

T and B

Ens. wei. sum

146.8

137.3

(B2)

5 years

\(\alpha ''\)

2

T and B

MHD (\(k^{\prime} = 0\))

147.6

136.4

(B3)

5 years

\(\alpha ''\)

2

T and B

MHD (\(k^{\prime} = 0\))

145.7

136.2

(B4)

5 years

\(0.1\alpha ''\)

2

T and B

Ens. wei. sum

115.0

111.4

(B5)

5 years

0.1 \(\alpha ''\)

2

T and B

MHD (\(k^{\prime} = 0\))

118.7

114.5

(B6)

5 years

0.1 \(\alpha ''\)

2

T and B

KD (\(k^{\prime} = 0\))

137.6

134.1

(C1)

 

IGRF-12

96.9

94.7

(C2)

 

Extrapolation using SV of MCM at 2014.25

72.4

73.4

(C3)

 

No SV from 2014.25

442.7

398.9

  1. In calculations of \(\sqrt {{\text{d}}P}\) (Eq. 40) and \(\sqrt {{\text{d}}P_{wo} }\) (Eq. 41), the MCM6 model, which covers up to 2019.50, is used as \({g_{l}^{m}}_{\text{data}}\) and \({h_{l}^{m}}_{\text{data}}\). Release times for all cases are assumed to be 2014.75, except 2015.00 for IGRF-12. In the “Forecast type” column, \(k^{\prime} = 0\) and \(k^{\prime} = K\) mean the MHD dynamo or KD simulations running from the optimized state vector at 2004.25 and 2014.25, respectively. See text for details of the three forecast types. In the \(\alpha_{S}\) and \(\alpha_{UW}\) columns, we specify those for the final 5th step in Eq. 39, where \(\alpha^{\prime} = 14\) and \(\alpha^{\prime\prime} = 7\)
  2. Italic—Case A4 was found to be the best setting in our numerical experiments
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