Assimilation of lake water surface temperature observations using an extended Kalman filter

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Name (longitude, latitude)D (m)I (%)Name (longitude, latitude)D (m)I (%)

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Kuivajärvi (23.9, 60.8)2.294.8Rehja-Nuasjärvi (28.0, 64.2)8.595.5Tuusulanjärvi (25.1, 60.4)3.294.3Vaskivesi (23.8, 62.1)7.097.1Pääjärvi 1 (24.5, 62.9)3.896.6Haukivesi (28.4, 62.1)9.194.9Pesiöjärvi (28.7, 64.9)3.995.4Kallavesi (27.7, 62.8)9.796.3Kyyvesi (27.1, 62.0)4.496.5Pielinen (29.6, 63.3)10.194.6Jääsjärvi (26.1, 61.6)4.696.2Konnevesi (26.6, 62.6)10.695.4Nilakka (26.5, 63.1)4.996.6Saimaa (28.1, 61.3)10.894.5Pyhäjärvi (22.3, 61.0)5.596.4Ala-Rieveli (26.2, 61.3)11.292.4Längelmävesi (24.4, 61.5)6.894.4Päijänne (25.5, 61.6)14.193.7Ounasjärvi (23.6, 68.4)6.697.3Inarijärvi (27.9, 69.1)14.397.1Lappajärvi (23.7, 63.1)6.993.4Näsijärvi (23.8, 61.6)14.794.0Oulujärvi (27.0, 64.5)7.095.0Pääjärvi 2 (25.1, 61.1)14.896.7Unari (25.7, 67.1)7.094.0Kilpisjärvi (20.8, 69.0)19.596.8Kevojärvi (27.0, 69.8)7.098.0

Experiment nameData assimilation ObservationsCycling periodNumber of lakes

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FR (free run)NoNoNo27EKF_SYesSYKE24 h27EKF_MYesMerged24 h (48 h for cross-validation)4

InarijärviSaimaaLappajärviTuusunlajärvi

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FRDate06/0605/1005/0905/07EKF-SDate05/2305/1105/1305/06LWST7.6°C12.6°C12.0°C8.9°CEKF-MDate05/1605/0505/0404/26LWST4.6°C7.8°C2.7°C2.8°C

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FREKF-M

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LakeI (%)BiasRMSEESTDBiasRMSEESTD

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Inarijärvi97.9−2.035.024.590.120.960.96Saimaa96.8−1.073.673.52−0.041.111.11Lappajärvi97.50.192.872.870.461.511.44Tuusulanjärvi95.90.832.922.800.851.411.13

ComponentValueRemarks

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Mixed regime

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Matrix M

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$\frac{\partial {\overline{T}}^t}{\partial {\overline{T}}^0}$0.8–0.9 K/KAlmost constant value, less for shallow lakesVector H

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$\frac{\partial \overline{T}}{\partial \overline{T}}$1.0 K/KConstant valueMatrix B

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$var\left(e_{\overline{T}}\right)$4.5–4.8 K²Almost constant valueVector K (weights)

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for $\overline{T}$0.82–0.83Almost constant valueStratified regime

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Matrix M

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$\frac{\partial {\overline{T}}^t}{\partial {\overline{T}}^0}$0.5–1.0 K/KLarger values for deep lakes, smaller for shallow lakes; for shallow lakes the annual cycle has a minimum in midsummer$\frac{\partial {\overline{T}}^t}{\partial {\eta }^0}$(−1.2)–(−0.1) KLarger absolute values for shallow lakes, smaller for deep lakes; the annual cycle has a maximum in midsummer$\frac{\partial {\overline{T}}^t}{\partial T_b}$0.01–0.40 K/KLarger values for shallow lakes, smaller for deep lakes; the annual cycle has a maximum in midsummer$\frac{\partial {\overline{T}}^t}{\partial C_T}$(−4.5)–(−0.1) KLarger absolute values for shallow lakes, smaller for deep lakes; the annual cycle has a minimum in midsummer$\frac{\partial {\eta }^t}{\partial {\overline{T}}^0}$(−0.2)–0.2 K⁻¹Almost zero values in midsummer$\frac{\partial {\eta }^t}{\partial {\eta }^0}$0.1–1.4Larger values for deep lakes, smaller for shallow lakes; the annual cycle has a maximum in autumn$\frac{\partial {\eta }^t}{\partial T_b}$(−0.3)–0.5 K⁻¹Larger absolute values for shallow lakes, smaller for deep lakes; the annual cycle has a minimum in autumn$\frac{\partial {\eta }^t}{\partial C_T}$(−1.0)–1.0Almost zero values in midsummer$\frac{\partial T_b^t}{\partial {\overline{T}}^0}$(−0.6)–0.6 K/KThe annual cycle with a minimum in spring$\frac{\partial T_b^t}{\partial {\eta }^0}$(−2.0)–0.4 KClose to zero in midsummer$\frac{\partial T_b^t}{\partial T_b^0}$0.2–1.2 K/KFor shallow lakes an almost constant value of 1.0 K/K, for deep lakes small values in midsummer$\frac{\partial T_b^t}{\partial C_T^0}$(−6.0)–3.0 KFor shallow lakes low values in midsummer, for deep lakes high values in spring$\frac{\partial C_T^t}{\partial {\overline{T}}^0}$(−0.03)–0.60 K⁻¹Almost zero values in midsummer$\frac{\partial C_T^t}{\partial {\eta }^0}$(−0.01)–0.80Almost zero values in midsummer$\frac{\partial C_T^t}{\partial T_b}$(−1.4)–0.03 K⁻¹Almost zero values in midsummer$\frac{\partial C_T^t}{\partial C_T}$(−0.1)–1.2For shallow lakes almost zero values in midsummer, for deep lakes an almost constant value of 1.0Vector H

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$\frac{\partial T_{ML}}{\partial \overline{T}}$1.0–2.6 K/KThe annual cycle has a maximum in midsummer$\frac{\partial T_{ML}}{\partial \eta }$0.0–20.0 KThe annual cycle has a maximum in midsummer$\frac{\partial T_{ML}}{\partial T_b}$(−1.6)–(−0.1) K/KThe annual cycle has a minimum in midsummer$\frac{\partial T_{ML}}{\partial C_T}$0.0–30.0 KThe annual cycle has a maximum in midsummer, the amplitude is stronger for deep lakesMatrix B

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$var\left(e_{\overline{T}}\right)$4.0–16.0 K²The annual cycle has a maximum in midsummer, the amplitude is stronger for shallow lakes$cov\left(e_{\overline{T}},e_{\eta }\right)$(−2.0)–0.2 KThe annual cycle has a minimum in late summer, the amplitude is stronger for shallow lakes$cov\left(e_{\overline{T}},e_{T_b}\right)$(−2.0)–16.0 K²The annual cycle has a maximum in late summer, the amplitude is stronger for shallow lakes$cov\left(e_{\overline{T}},e_{C_T}\right)$(−0.80)–(−0.01) KNo annual cycle, oscillations$var\left(e_{\eta }\right)$0.01–0.90Almost constant small values, but outbreaks in autumn$cov\left(e_{\eta },e_{T_b}\right)$(−5.0)–1.0 KThe annual cycle has a minimum in late summer, the amplitude is stronger for shallow lakes$cov\left(e_{\eta },e_{C_T}\right)$(−0.05)–0.15Almost zero values in midsummer$var\left(e_{T_b}\right)$1.0–45.0 K²The annual cycle has a maximum in late summer$cov\left(e_{T_b},e_{C_T}\right)$(−2.5)–2.0 KNo annual cycle, oscillations$var\left(e_{C_T}\right)$0.01–0.18No annual cycle, oscillationsVector K (weighs)

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for $\overline{T}$0.20–0.83The annual cycle has a minimum in midsummerfor η(−0.06)–0.60 K⁻¹No annual cycle, oscillationsfor T _b (−0.6)–0.4No annual cycle, oscillationsfor C _T (−0.05)–0.02 K⁻¹The annual cycle has a maximum in midsummer