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    Schematic of the relationship between the total OIOSE (line EF) and forecast errors (or improvements, dashed and solid lines) verified against the referenced analysis at time t xta (point G). The forecast errors et|6TCet|6 (dashed line) and et|0TCet|0 (solid line) are obtained from two forecasts xt|6f and xt|0f started from the analysis at t = −6 h, xt=6a, and from the analysis at t = 0 h, xt=0a (point B), respectively. The total OIOSE at t = te hour (evaluation lead time), ΔE2¯(t=te) (line CD), is quantified as the difference between the forecast errors and is approximated by the sum of the OIEFSO values Δε2. Here line AB corresponds to the analysis increment. The gray colors indicate when an OSE forecast is additionally performed. The OSE forecast xOSEf(t) started from the OSE analysis at t = 0, xOSEa (point H), leads to the forecast error of the OSE eOSET(t)CeOSE(t) (gray line). In the OSE a small subset of observations is excluded. Here, xt|0f are equivalent to the control forecast xCTLf(t), which is compared with xOSEf(t). Both xCTLf(t) and xOSEf(t) are verified against xta. The OIOSE of the subset at time t, ΔEexp2¯(t), is quantified by the difference between the forecast errors eCTLT(t)CeCTL(t) and eOSET(t)CeOSE(t) (line JF). The OIOSE at t = te hour, ΔEexp2¯(t=te) (line ID), is approximated by the OIEFSO value of the subset of the observations Δεexp2¯(t=te).

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    Time series of the sum of the 6-h total OIEFSO values Δε2 (J kg−1; blue dashed) and the 6-h total OIOSE ΔE2¯(t=6) (black) for every 0000 UTC during the period of December 2015–February 2016. It is important to note that the 6-h total OIOSE ΔE2¯(t=6) is calculated by Eq. (4) as the difference between the 12-h forecast and the first guess (6-h forecast) fields (Fig. A2).

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    The winter-averaged OIEFSO values Δε2¯(t=6) per cycle (J kg−1) for global radiosonde observations and the analyzed vertically averaged wind field (m s−1) in ALERA2. Note that only radiosonde observations for locations that reported more than 80 times at 0000 UTC during the cycling period (winter) are shown. (Total number of the radiosonde observations at each site is 91.)

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    Schematic of the experimental designs. The solid red arrow denotes the forecast–analysis (data assimilation) cycle of ALERA2, and the dashed blue arrows indicate the 7-day ensemble forecast experiments initialized with ALERA2 (red circles, CTL experiments) and the data-denied analyses by each OSE (blue circles). The number of OSEs is 12, that is, the Mid, Mid2, Mid3, Mid4, Pol, Pol2, Pol3, Pol4, Tro, Tro2, Tro3, and Tro4 experiments. The 7-day forecast experiments started every day at 0000 UTC during winter; and in total, the experiments were conducted 91 times. Each ensemble forecast (each dashed blue arrow) includes two ensemble forecasts, initialized from the 63-member CTL and OSE analyses.

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    Twelve observation locations (×, □, ○, and △ symbols) consisting of three routine radiosonde sites at the midlatitude (black), the Arctic (blue), and the tropical (red) bands of the OSE experiments. For each band, four locations denoted by the ×, □, ○, and △ symbols were selected. Gray dots denote all the radiosonde observation points in the Northern Hemisphere and tropics displayed in Fig. 3.

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    Scatter diagram of Δεexp2¯(t=6) (horizontal axis) and ΔEexp2¯(t=6) (vertical axis) of each experiment averaged for the 91-time forecasts (over the winter period). Colors and symbols are the same as Fig. 5 (black for the midlatitude, blue for the Arctic, and red for the tropical OSEs). The black line indicates the y = x function, and the gray solid and dashed lines indicate 10% and 40% deviations from y = x.

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    Ratios of OIOSE ΔEexp2¯(t) to OIEFSO Δεexp2¯(t) of each experiment at different evaluation times (t = 6–48 h). Colors and symbols are the same as Fig. 5. The gray line indicates the y = 1 function.

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    Time sequences of OIOSE ΔEexp2¯(t) (J kg−1) from 6-h to 7-day forecast periods (h) for the (a) Arctic, (b) midlatitude, and (c) tropical OSEs. Colors and symbols are the same as in Fig. 5. The OIOSE at the 6-h forecasts (symbols at the leftmost fringe) correspond to ΔEexp2¯(t=6) in Fig. 6. Deficits indicate negative values. Thick solid and dashed lines indicate statistical confidence exceeding 99% and 95% significant levels against the 91-time forecasts, respectively.

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    Horizontal maps of OIOSE ΔEexp2(t) (J kg−1) at 1–3-day forecast times. The observation impacts of the three OSEs—(a)–(c) the Pol (blue × in Fig. 6), (d)–(f) the Mid4 (black △), and (g)–(i) the Tro (red ×) experiments—are displayed. The hatching indicates statistical confidence exceeding 95%.

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    As in Fig. 9, but for OIEFSO Δεexp2(t) (J kg−1).

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    (a)–(c) As in Fig. 9 but for the averaged OIOSE during days 4–7 (medium ranges) for (a) ΔEPol2(t), (b) ΔEMid42(t), and (c) ΔETro2(t). The hatching indicates statistical confidence exceeding 90%. (d)–(f) Standard deviations of the averaged ΔEexp2(t) (J kg−1) for the (d) Pol, (e) Mid4, and (f) Tro experiments.

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    Composites of the averaged OIOSE ΔEexp2(t) (J kg−1) during (a) days 0.25–3 and (b) days 4–7 (medium ranges) for the four Arctic (Pol, Pol2, Pol3, and Pol4) experiments. The arrows in (a) indicate the flux of the OIOSE vΔEexp2(t) (J kg−1 m s−1) (see text for more details). (c),(d) As in (b), but the composites for the four midlatitude (Mid, Mid2, Mid3, and Mid4) and the four tropical (Tro, Tro2, Tro3, and Tro4) experiments, respectively.

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    Time sequences of the area-averaged OIOSE ΔEexp2(t) within (top) the Arctic (0°–360°E, 60°–90°N) and (bottom) the midlatitude North American (240°–300°E, 20°–60°N) domains. Thick solid and dashed lines indicate statistical confidence exceeding 95% and 90% levels, respectively.

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    As in Fig. 8a, but for the repeated (purple lines) and the non-repeated (gray lines) Arctic OSEs. Note that the displayed forecast periods are from 6 h to 5 days.

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    As in Figs. 12a and 12b, but for the repeated Arctic OSEs.

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    Latitude–height cross section of (a) observation numbers per cycle and (b) its density distribution used in ALERA2 in free atmosphere (above 1000 hPa) during winter. In the density distribution in (b), the numbers normalized by areas (weighted by cos−1θ, where θ is latitude) are displayed. (c) Δε2¯ and area-separated Δε2¯ per observation (J kg−1) of the global, the tropical (10°S–20°N), the Northern Hemisphere midlatitude (20°–60°N), and the Arctic (60°–90°N) latitudinal bands.

  • View in gallery

    Snapshots of (a) 500-hPa geopotential height (Z500) and (b) sea level pressure (SLP) fields in ALERA2 (blue or red contours) and ERA-Interim (black contours; Dee et al. 2011) at 0000 UTC 1 Jan 2016. Contour intervals in (a) and (b) are 100 m and 4 hPa, respectively. (c) Root-mean-square differences of Z500 [right axis (m), blue] and SLP [left (hPa), red] for the period of December–February in 2015/16. Black vertical line indicates the date shown in (a) and (b).

  • View in gallery

    Flowchart of forecast–analysis cycles in ALEDAS from the analyzed times t = 0 and t = +6 h. Squared and rounded-corner boxes indicate inputs and outputs (data) and the executed processes, respectively. Details of each process and data are shown in bottom. In this figure, Xt=6|0f in Eq. (2) is obtained from “guess,” et=6|0 is from “guess” and “analysis,” et=6|6 is from “12-hr fcst” and “analysis,” δy0ob and R are from “O-B,” and Y0a is from “O-A,” respectively. The process “obsope” corresponds to the observation operator H to obtain δy0ob. Parallel process and ensemble data are shown with additional three boxes. Red indicates newly implemented processes and added outputs for the EFSO diagnosis.

  • View in gallery

    As in Fig. 3 but for all the seasons during (top left) winter 2015/16, (top right) spring 2016, (bottom left) summer 2016, and (bottom right) fall 2016. Seasonal averages of Δε2¯(t=6) (J kg−1) are displayed. Only radiosonde observations for locations that reported more than 80 times at 0000 UTC during each season are shown. The wind fields are not displayed here.

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    As in Fig. B1, but for special observing periods (SOPs) of YOPP when frequencies of radiosonde observations in (a),(b) the Arctic and (c) the Antarctic regions were increased at many routine stations. The SOPs are from 1 Feb 2018 to 31 Mar 2018 in (a), from 1 Jul 2018 to 30 Sep 2018 in (b), and from 16 Nov 2018 to 15 Feb 2019 in (c). Averaged values of Δε2¯(t=6) during the SOPs (J kg−1) are displayed. Note that only radiosonde observations for locations that reported more than 50 [in (a)] and 80 times [in (b) and (c)] at 0000 UTC during each SOP are shown.

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EFSO at Different Geographical Locations Verified with Observing System Experiments

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  • 1 a Application Laboratory, JAMSTEC, Yokohama, Japan
  • | 2 b RIKEN Center for Computational Science, Kobe, Japan
  • | 3 c National Institute of Polar Research, Tachikawa, Japan
  • | 4 d Disaster Prevention Research Institute, Kyoto University, Uji, Japan
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Abstract

An ensemble-based forecast sensitivity to observations (EFSO) diagnosis has been implemented in an atmospheric general circulation model–ensemble Kalman filter data assimilation system to estimate the impacts of specific observations from the quasi-operational global observing system on weekly short-range forecasts. It was examined whether EFSO reasonably approximates the impacts of a subset of observations from specific geographical locations for 6-h forecasts, and how long the 6-h observation impacts can be retained during the 7-day forecast period. The reference for these forecasts was obtained from 12 data-denial experiments in each of which a subset of three radiosonde observations launched from a geographical location was excluded. The 12 locations were selected from three latitudinal bands comprising (i) four Arctic regions, (ii) four midlatitude regions in the Northern Hemisphere, and (iii) four tropical regions during the Northern Hemisphere winter of 2015/16. The estimated winter-averaged EFSO-derived observation impacts well corresponded to the 6-h observation impacts obtained by the data denials and EFSO could reasonably estimate the observation impacts by the data denials on short-range (from 6 h to 2 day) forecasts. Furthermore, during the medium-range (4–7 day) forecasts, it was found that the Arctic observations tend to seed the broadest impacts and their short-range observation impacts could be projected to beneficial impacts in Arctic and midlatitude North American areas. The midlatitude area was located just downstream of dynamical propagation from the Arctic toward the midlatitudes. Results obtained by repeated Arctic data-denial experiments were found to be generally common to those from the non-repeated experiments.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Akira Yamazaki, yzaki@jamstec.go.jp

Abstract

An ensemble-based forecast sensitivity to observations (EFSO) diagnosis has been implemented in an atmospheric general circulation model–ensemble Kalman filter data assimilation system to estimate the impacts of specific observations from the quasi-operational global observing system on weekly short-range forecasts. It was examined whether EFSO reasonably approximates the impacts of a subset of observations from specific geographical locations for 6-h forecasts, and how long the 6-h observation impacts can be retained during the 7-day forecast period. The reference for these forecasts was obtained from 12 data-denial experiments in each of which a subset of three radiosonde observations launched from a geographical location was excluded. The 12 locations were selected from three latitudinal bands comprising (i) four Arctic regions, (ii) four midlatitude regions in the Northern Hemisphere, and (iii) four tropical regions during the Northern Hemisphere winter of 2015/16. The estimated winter-averaged EFSO-derived observation impacts well corresponded to the 6-h observation impacts obtained by the data denials and EFSO could reasonably estimate the observation impacts by the data denials on short-range (from 6 h to 2 day) forecasts. Furthermore, during the medium-range (4–7 day) forecasts, it was found that the Arctic observations tend to seed the broadest impacts and their short-range observation impacts could be projected to beneficial impacts in Arctic and midlatitude North American areas. The midlatitude area was located just downstream of dynamical propagation from the Arctic toward the midlatitudes. Results obtained by repeated Arctic data-denial experiments were found to be generally common to those from the non-repeated experiments.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Akira Yamazaki, yzaki@jamstec.go.jp

1. Introduction

Observing system experiments (OSEs) are commonly used for evaluating the impact of observations on data assimilation systems. OSEs enable us to quantify the impact of observations in specific regions or of type, by excluding them from or adding them to the total observations assimilated into a data assimilation system. OSEs are usually used to evaluate the direct influence of observations on weather phenomena occurring over regions at all distances, from the local to the global domains (e.g., Weissmann et al. 2011; Ito et al. 2018; Sato et al. 2017; Schäfler et al. 2018); they are also used to estimate the total impact of some observation types or observations in specific latitudinal bands on the global observing system (e.g., Bormann et al. 2019; Day et al. 2019; Lawrence et al. 2019). In previous work, we have conducted many OSEs using in-house developed data assimilation system known as the Atmospheric General Circulation Model for the Earth Simulator—a local ensemble transform Kalman filter (AFES-LETKF) ensemble data assimilation system (ALEDAS; see appendix A). This system has been used to generate an experimental atmospheric global ensemble reanalysis (ALERA2). So far, these studies have focused on the remote influences of small subsets of observations obtained during field campaigns (e.g., Inoue et al. 2013; Yamazaki et al. 2015). To clarify how observation impacts propagate and where their impacts accumulate in quasi-operational data assimilation and forecast systems can provide useful information for the design of observational campaigns. However, conducting OSEs during a campaign is not very practical because they are too expensive in computational resources and time, as additional data assimilation and forecast cycles must be performed to evaluate specific observations.

Alternatively, the forecast sensitivity to observation (FSO) technique pioneered by Langland and Baker (2004) allows us to diagnose the impacts of all observations by quantifying the extent to which each observation improves or degrades subsequent forecasts. FSO estimates (diagnoses) individual observation impacts at early short-range forecast times, which are typically 6–24 h. In this study we adopt the acronym FSO instead of FSOI (forecast sensitivity observation impact) also used in many previous studies (e.g., Sommer and Weissmann 2016). The FSO formulation has been originally developed for variational data assimilation systems as the adjoint-based FSO. Recent studies by Liu and Kalnay (2008) and Kalnay et al. (2012) have applied the formulation to the ensemble Kalman filter (EnKF) system, known as the ensemble-based FSO (called EFSO). The EFSO diagnosis can be used in current data assimilation schemes. Recently, Kotsuki et al. (2019) showed the usefulness of EFSO in a nonhydrostatic model. Alongside the global NWP model, Sommer and Weissmann (2016) and Necker et al. (2018) successfully adopted the EFSO diagnosis for a convection-permitting regional model. The adjoint-based FSO and EFSO diagnoses can be used to quantify the impact of each observation individually, without the need to conduct OSEs. The adjoint-based FSO and EFSO techniques have been successfully used to estimate the impacts of specific observation types (e.g., satellite observations) against conventional ones (e.g., Gelaro and Zhu 2009). Recently, Lien et al. (2018) proposed the use of the EFSO diagnosis in an offline approach to develop new data selection strategies. Elsewhere, Ota et al. (2013) and Hotta et al. (2017) proved that the EFSO technique can be used for the proactive quality control of observations.

Previous studies used the adjoint-based FSO and EFSO techniques for targeted observations. In targeted observations (Majumdar 2016), one deploys and assimilates additional observations through an observational campaign to improve the numerical forecast of a weather event. Since targeted observations are conducted to better forecast weather events that occur near campaign sites, an FSO diagnosis that estimates a forecast impact in less than a day would be most useful. However, it is not known yet whether the adjoint-based FSO or EFSO is useful for estimating the remote impacts of observations or impacts on time scales longer than a day. This may be desirable for field campaigns conducted from ships or near-stationary mobile platforms.

In the adjoint-based FSO and EFSO context the observation impact represents the difference between a selected measure of error on the analysis and background model state trajectories (see section 2a and Fig. 1). In this study, we define observation impacts obtained by actual data assimilation cycles (OSEs) as OIOSE and those estimated by the EFSO technique which has been implemented as OIEFSO.

Fig. 1.
Fig. 1.

Schematic of the relationship between the total OIOSE (line EF) and forecast errors (or improvements, dashed and solid lines) verified against the referenced analysis at time t xta (point G). The forecast errors et|6TCet|6 (dashed line) and et|0TCet|0 (solid line) are obtained from two forecasts xt|6f and xt|0f started from the analysis at t = −6 h, xt=6a, and from the analysis at t = 0 h, xt=0a (point B), respectively. The total OIOSE at t = te hour (evaluation lead time), ΔE2¯(t=te) (line CD), is quantified as the difference between the forecast errors and is approximated by the sum of the OIEFSO values Δε2. Here line AB corresponds to the analysis increment. The gray colors indicate when an OSE forecast is additionally performed. The OSE forecast xOSEf(t) started from the OSE analysis at t = 0, xOSEa (point H), leads to the forecast error of the OSE eOSET(t)CeOSE(t) (gray line). In the OSE a small subset of observations is excluded. Here, xt|0f are equivalent to the control forecast xCTLf(t), which is compared with xOSEf(t). Both xCTLf(t) and xOSEf(t) are verified against xta. The OIOSE of the subset at time t, ΔEexp2¯(t), is quantified by the difference between the forecast errors eCTLT(t)CeCTL(t) and eOSET(t)CeOSE(t) (line JF). The OIOSE at t = te hour, ΔEexp2¯(t=te) (line ID), is approximated by the OIEFSO value of the subset of the observations Δεexp2¯(t=te).

Citation: Weather and Forecasting 36, 4; 10.1175/WAF-D-20-0152.1

Our purpose is to understand how well EFSO can estimate OIOSE of individual observations and their downstream influence. We also aim to understand which points are most influential on short- and/or medium-range weekly forecasts; in other words, we seek to discover the “optimal spot” of observational locations to enhance global NWP forecasts. To the best of our knowledge, only a few recent studies directly compared the adjoint-based FSO and EFSO diagnoses and OIOSE obtained by data-denial experiments in the global observing framework. Gelaro and Zhu (2009) compared the values obtained from the adjoint-based FSO with those obtained from data-denial experiments. For satellite observations they used the adjoint-based FSO technique to estimate the relative contributions by observation type, with a focus on conventional and satellite observations. Ota et al. (2013) conducted a data-denial experiment for a case where several satellite-wind observations that were estimated as candidates to degrade a forecast by EFSO. They showed that the estimation could capture the actual 24-h forecast change. Hotta et al. (2017) further developed the study of Ota et al. (2013) and proposed the concept of “proactive QC (quality control)” in which observations with the potential to degrade a forecast could be detected by EFSO and rejected during the data assimilation process. Very recently, Lawrence et al. (2019) compared observation impacts estimated by the adjoint-based FSO with OIOSE from data-denial experiments to study how Arctic observations influence global weather forecasts. They found that the relative contributions of observation types obtained by the adjoint-based FSO technique were consistent with the results of the data-denial experiments for short and even medium forecast ranges. The aforementioned studies illustrated that adjoint-based FSO and EFSO techniques are useful for estimating OIOSE on short- and medium-range forecasts. Their focus, however, was on large numbers of globally distributed observations rather than on small subsets obtained through observational campaigns.

To demonstrate the usefulness of EFSO for evaluating OIOSE on short- and medium-range forecasts and for estimating the remote impacts of localized observations obtained in field campaigns, we need to address the following question:

  • Can the EFSO technique reasonably approximate the short-range OIOSE obtained by data-denial experiments?
Since the adjoint-based FSO and EFSO are formulated to estimate OIOSE at an early short-range forecast, it is not obvious whether the early short-range (initial) OIOSE can be retained in longer forecast ranges. Privé et al. (2021) discussed time evolution of observation impacts estimated by the adjoint-based FSO, and showed that their behaviors are different between extratropical and tropical observations. Thus, discussion on mechanisms of how the initial OIOSE amplify and propagate starting from observed locations at different latitudinal bands is helpful to suggest whether EFSO can be useful to short- to medium-range forecasts. The following questions are also need to be addressed:
  • Are initial (at an early short-range forecast) OIOSE estimated by the EFSO diagnosis are dynamically propagated in the forecast to retain their impacts on longer forecasts?
  • Which is the most important for improving medium-range weather forecasts the Arctic, the midlatitudes, or the tropics?
To clarify those points, we conduct multiple data-denial experiments in which several radiosonde observations launched from various geographical locations are removed from the real global observing network. Medium-range forecast experiments (up to 7 days) are then initialized by the data-denied analysis fields to measure OIOSE and subsequently compared with OIEFSO. We focus on the impacts of radiosonde observations because they are the most reliable observation type and include both the lower and upper atmospheric state; they are also more robust than satellite observations, in the sense that they do not strongly depend on seasons and weather conditions. Nonetheless, the knowledge and methods obtained in this study should be applicable to other observation types. Our results suggest that the planning of observational campaigns could benefit from the use of the EFSO technique. To the best of our knowledge, this is the first study to investigate how well EFSO can estimate OIOSE derived from multiple data-denial experiments for various locations selected wholly from a global domain. The paper is organized as follows. Section 2 introduces the formulation of the ensemble-based FSO (EFSO), and the implementation of this technique in ALEDAS. Section 3 introduces experimental designs of data-denial experiments (OSEs). In section 4, first, OIEFSO are compared with OIOSE in the short-range forecast. Second, we investigate how to propagate OIOSE in each OSE during the medium-range forecast. Last, it is discussed whether EFSO can be useful to estimate OIOSE by repeated OSEs that mimics long-lasting near-stationary observational campaigns. Section 5 discusses the summary and our speculations.

2. Implementation of EFSO in ALEDAS

ALEDAS is a global atmospheric data assimilation system that combines AFES (an AGCM) with the LETKF (an EnKF). See appendix A for detailed configurations.

a. Physical interpretation of the observation impact and its approximation by EFSO

In this subsection, we define the observation impact, give a physical interpretation, and discuss how it can be approximated by the EFSO technique.The details of the EFSO formulation are available in Kalnay et al. (2012), Ota et al. (2013), and Hotta et al. (2017).

1) The total observation impact (total OIOSE)

First, we define the total OIOSE at time t, which is the area average of OIOSE for a target domain, ΔE2¯(t), quantified by the difference between the forecast errors initialized 6 h before the analysis time t = 0, et|−6, and initialized at t = 0, et|0:
ΔE2¯(t)et|6TCet|6et|0TCet|0=(xt|0fxt|6f)TC(et|0+et|6),
where et|0xt|0fxtυ, et|6xt|6fxtυ, and C is a matrix that defines an energy norm and an area-averaged operation. The terms xt|0f and xt|6f represent the ensemble-mean forecast from the analyzed time (t = 0) and 6 h before (t = −6) to the verifying time t; xtυ is the reference (truth) verifying forecast errors. The terms x and e are the state vectors for the target domain. In this study, our reference is the ensemble mean of the analysis xtυxta. The meaning of ΔE2¯(t) (total OIOSE) is conceptualized in Fig. 1. Total OIOSE quantifies how close (similar) xt|0f and xt|6f are to xta and how large the difference between xt|0f and xt|6f is. If xt|6f is closer to xta than xt|0f over the target domain, ΔE2¯(t) becomes negative. At t = 0, the total OIOSE ΔE2¯(t=0) is equivalent to the analysis increment (line AB in Fig. 1). Whereas at t → ∞, the total OIOSE approaches to zero (Privé et al. 2021, their Fig. 5).

The total OIOSE at the evaluation lead time te obtained by the difference between xt|0f and xt|6f, ΔE2¯(t=te), is expressed as the “te-hour” total OIOSE.

2) The EFSO approximation

Second, we explain how OIEFSO can be estimated through the use of the EFSO technique. ΔE2¯(t=te) is diagnosed by the sum of the OIEFSO values Δε2¯(t=te) in the following way. In Eq. (1), by assuming xtυ=xta, we can obtain (et|0+et|6). Normally, one would now use the adjoint model to solve (xt|0fxt|6f)T. The EFSO formulation, however, enables the calculation of this term without the expensive calculation of the adjoint model. This is done by alternatively using the ensemble forecast at t, Xt|0f, and the ensemble perturbation at t = 0 in observation space Y0a, which approximates the Kalman gain (Kalnay et al. 2012):
Δε2¯(t)=(δy0ob)T1K1ρ[R1Y0a(Xt|0f)T]C(et|0+et|6),
where δy0ob is the observation-minus-background (O-B) innovation of the ensemble mean (δy0ob=yHx0|6f, where y and Hx0|6f are observation and the ensemble-mean forecast from t = −6 to t = 0 in observation space, respectively), K is the ensemble size, R is the observation error matrix, Xt|0f is the ensemble perturbation matrix of xt|0f; here, and ρ are the Schur product and localization matrix, which define the localization function same as that used in the LETKF of ALEDAS. Note that observations with positive or beneficial (negative or nonbeneficial) ΔE2¯(t=te) or Δε2¯(t=te) in Eqs. (1) and (2) are expected to improve (degrade) a subsequent forecast.1

Because δy0ob is a vector composed of individual observations, we can decompose Δε2¯(t) into each observation’s OIEFSO value. Here, Δε2¯(t=te) is sum of Δε2¯(t=te) and approximates ΔE2¯(t=te). A decomposed te-hour OIEFSO value for an observation estimates a te-hour OIOSE, which is obtained by conducting a data-denial (or an OSE) experiment of the observation (Fig. 1).

3) The matrix C and the difference between the forecast errors

Last, we define OIOSE at a grid point, ΔE2(t). As the matrix C, we adopt the vertically summed moist total energy error norm (Ota et al. 2013; Hotta et al. 2017; Kotsuki et al. 2019), which includes an area weighting function of latitude. The area average of ΔE2(t) over the target domain is equivalent to ΔE2¯(t) because of the weighting function. From the definition of Eq. (1):
ΔE2(t)121psps0(u2+υ2)+cpTrT2+L2cpTrq2dp+RdTrprps2,
where p is the vertical pressure coordinate; ps is the surface pressure; u, υ, T, q, and ps are the zonal wind, meridional wind, temperature, specific humidity, and surface pressure, respectively; cp is the specific heat capacity of the air; Rd is the gas constant of dry air; L is the latent heat of condensation per unit mass; and Tr and pr are the reference temperature and surface pressure, respectively. Here, the “difference” (or sensitivity) is defined as
x2=(xt|0fxt|6f)(et|0+et|6),
where x (e) is an element of x(e) and is identical to u, υ, T, q, or ps. If xt|6f is closer to xta than xt|0f at the grid point, ΔE2(t) becomes negative.

b. Implementation

To calculate OIEFSO, we need to select several parameters. The target domain of EFSO is set to the global domain. The evaluation lead time te is mainly set to 6 h. The moving localization scheme (Ota et al. 2013; Hotta et al. 2017) with coefficient 1.0 is adopted, in which the localization function is advected by the wind fields of x0a and xteυ (i.e., xtea) multiplied by a weighting coefficient between 0 and 1. Those EFSO parameters are selected based on practical considerations that we explain below.

In this paper, we use as our reference xtυ the ensemble mean of the analysis, although some recent studies suggested that the observation should be used (Cardinali 2018; Necker et al. 2018; Kotsuki et al. 2019; Privé et al. 2021). The rationale is that we would like to use the EFSO diagnosis as an alternative to OSE, mainly because OSEs are normally compared against their own analysis. In addition, as shown in Fig. A1, ALERA2 (analysis) reproduces reasonably well the synoptic and general circulations that we are interested in. EFSO is also sensitive to the error metric (Necker et al. 2018; Kotsuki et al. 2019) and the target domain used. Our choice of using the error metric [Eq. (3)] and the target area (global domain) is typical for EFSO and the adjoint-based FSO studies and serves to emphasize the forecast errors in the densely populated midlatitudes.

The reasons for evaluating at lead time 6 h rather than at longer ones are as follows.

  • The 6-h EFSO has a much lower computational cost. An additional process that one substantially uses to calculate is extended forecasts for 3 h (See appendix A).
  • The 6-h EFSO suffers less from the impact of the moving localization scheme than EFSO with longer evaluation (lead) times (Hotta et al. 2017; Kotsuki et al. 2019). It would be difficult to determine the optimal weighting coefficient of the moving localization, since OIEFSO can be propagated in the Lagrangian way and by the Rossby wave transport owing to phase and group velocity (Kalnay et al. 2012). For example, Ota et al. (2013) chose the coefficient 0.75 for 24-h EFSO in an operational global data-assimilation system.

c. Evaluation

We first examine whether the implemented EFSO technique works correctly in ALEDAS. ALEDAS forecast–analysis cycles with EFSO diagnoses are examined for the period of December–February 2015/16 (Fig. 2).

Fig. 2.
Fig. 2.

Time series of the sum of the 6-h total OIEFSO values Δε2 (J kg−1; blue dashed) and the 6-h total OIOSE ΔE2¯(t=6) (black) for every 0000 UTC during the period of December 2015–February 2016. It is important to note that the 6-h total OIOSE ΔE2¯(t=6) is calculated by Eq. (4) as the difference between the 12-h forecast and the first guess (6-h forecast) fields (Fig. A2).

Citation: Weather and Forecasting 36, 4; 10.1175/WAF-D-20-0152.1

Total OIEFSO Δε2¯(t=6) can estimate ΔE2¯(t=6) similar to the previous adjoint-based FSO and EFSO studies, as evidenced by the winter average of Δε2¯(t=6) (2.42 J kg−1), which is mostly the same as the winter average of ΔE2¯(t=6) (2.82); the daily variation of Δε2¯(t=6) is also highly correlated with that of ΔE2¯(t=6) (0.95). Both the average difference and the correlation are consistent with OIEFSO in Ota et al. (2013), Hotta et al. (2017), and Kotsuki et al. (2019), and the total OIEFSO slightly underestimates the total OIOSE in the same sense as Kotsuki et al. (2019) (EFSO) and Privé et al. (2021) (the adjoint-based FSO), implying that the EFSO diagnosis works accurately when implemented in ALEDAS.

The distribution map of the winter-averaged OIEFSO values Δε2¯(t=6) for radiosonde observations in the Northern Hemisphere and the tropics is shown in Fig. 3. OIEFSO is greater where the radiosonde observation density is lower. When comparing the OIEFSO values in the latitudinal direction, generally the values become larger at the higher latitudes (see Fig. 16c and appendix B). In addition, Fig. 3 shows that the time-averaged impacts are distributed smoothly (i.e., not too localized). Therefore, the global distribution of Δε2¯(t=6) of global routine observations provides information on regions where field campaigns might have the largest impact. We note that observational campaigns may temporally modify the global distribution. However, those OIEFSO features are kept during such observational campaigns when the extra radiosonde observations in the Arctic or the Antarctic regions are performed (see appendix B).

Fig. 3.
Fig. 3.

The winter-averaged OIEFSO values Δε2¯(t=6) per cycle (J kg−1) for global radiosonde observations and the analyzed vertically averaged wind field (m s−1) in ALERA2. Note that only radiosonde observations for locations that reported more than 80 times at 0000 UTC during the cycling period (winter) are shown. (Total number of the radiosonde observations at each site is 91.)

Citation: Weather and Forecasting 36, 4; 10.1175/WAF-D-20-0152.1

3. Experimental designs

To use the EFSO diagnosis as an alternative to an OSE, the extent to which a 6-h EFSO can estimate the individual OIOSE for short- and medium-range forecasts must be examined. In this study, we conduct multiple data-denial experiments for various regions in the Northern Hemisphere and in the tropics, where radiosondes are routinely launched.

The experiments are conducted from 1 December 2015 to 29 February 2016. During this period, 7-day ensemble forecasts are initialized every day from 0000 UTC using ALERA2 (Fig. 4); that is, 91 times of 7-day forecast experiments (Fig. 4). This is our control experiment (CTL hereafter).

Fig. 4.
Fig. 4.

Schematic of the experimental designs. The solid red arrow denotes the forecast–analysis (data assimilation) cycle of ALERA2, and the dashed blue arrows indicate the 7-day ensemble forecast experiments initialized with ALERA2 (red circles, CTL experiments) and the data-denied analyses by each OSE (blue circles). The number of OSEs is 12, that is, the Mid, Mid2, Mid3, Mid4, Pol, Pol2, Pol3, Pol4, Tro, Tro2, Tro3, and Tro4 experiments. The 7-day forecast experiments started every day at 0000 UTC during winter; and in total, the experiments were conducted 91 times. Each ensemble forecast (each dashed blue arrow) includes two ensemble forecasts, initialized from the 63-member CTL and OSE analyses.

Citation: Weather and Forecasting 36, 4; 10.1175/WAF-D-20-0152.1

We compare CTL with 12 data-denial experiments (OSEs). In each of these experiments, a subset of radiosonde observations from three adjacent sites is excluded. Four locations are in the midlatitudes (20°–60°N), four in the Arctic (60°–90°N), and four in the tropics (10°S–10°N), as shown in Fig. 5. The midlatitude locations are hereafter denoted as the Mid, Mid2, Mid3, and Mid4 experiments (black symbols in Fig. 5); the Arctic locations as Pol, Pol2, Pol3, and Pol4 experiments (blue symbols); and the tropical locations as Tro, Tro2, Tro3, and Tro4 experiments (red symbols). After the data denial, 7-day ensemble forecasts are initialized every day from 0000 UTC using OSE analyses (Fig. 4); that is, 91 times of 7-day forecast experiments for each OSE. Note that the data denial in the data assimilation process is not repeated to avoid the temporal accumulation of OIOSE; i.e., the data denial is only performed once before each forecast. These non-repeated data-denial experiments will help us to understand how OIOSE propagate dynamically. Each CTL and OSE ensemble forecast is initialized from the 63 members of the CTL (ALERA2) and OSE analyses. To evaluate the OIOSE in each OSE, we compare the ensemble forecast mean of CTL xCTLf(t) with that of the OSE xOSEf(t). This is because for the definition of EFSO the ensemble mean is used (for et|0 and et|6).

Fig. 5.
Fig. 5.

Twelve observation locations (×, □, ○, and △ symbols) consisting of three routine radiosonde sites at the midlatitude (black), the Arctic (blue), and the tropical (red) bands of the OSE experiments. For each band, four locations denoted by the ×, □, ○, and △ symbols were selected. Gray dots denote all the radiosonde observation points in the Northern Hemisphere and tropics displayed in Fig. 3.

Citation: Weather and Forecasting 36, 4; 10.1175/WAF-D-20-0152.1

We define the OIOSE of the excluded (denied) radiosondes for the global domain, ΔEexp2¯(t), as the differences between xCTLf(t) and xOSEf(t):
ΔEexp2¯(t)=eOSET(t)CeOSE(t)eCTLT(t)CeCTL(t)=[xCTLf(t)xOSEf(t)]TC[eCTL(t)+eOSE(t)],
where eOSE(t)xOSEf(t)xta and eCTL(t)xCTLf(t)xta (Fig. 1) are calculated for lead times from 6 h to 7 days. Also, ΔEexp2¯(t) at each grid point, ΔEexp2(t) is calculated in the same way as Eqs. (3) and (4). The relationship between ΔEexp2¯(t) and ΔE2¯(t) is schematically shown in Fig. 1: ΔEexp2¯(t) is an component of ΔE2¯(t) (i.e., point J is located on line EF). Also, point H xOSEa (the initial value of the OSE) is near point B xCTLa (the initial value of the CTL or x0a) and on the line connecting points A and B.

The OIEFSO of a specific OSE Δεexp2¯(t) and Δεexp2(t) are derived from Eq. (2) by replacing δy0ob with δy0ob,deny whose elements are set to 0 except those of the observations denied in the OSE (Hotta 2014). Thus, Δεexp2¯(t) approximates ΔEexp2¯(t) and also composes Δε2¯(t), and the area average of Δεexp2(t) over the target domain is equivalent to Δεexp2¯(t).

We first compare Δεexp2¯(t=6) of the subsets of the radiosondes with ΔEexp2¯(t=6). Subsequently, we show how long and to what extent the 6-h EFSO diagnosis is useful for the estimation of ΔEexp2¯(t) for short- and/or medium-range forecasts.

4. Results

a. Comparison of OIOSE and OIEFSO

We compare the 6-h OIEFSO Δεexp2¯(t=6) with the 6-h OIOSE ΔEexp2¯(t=6) (Fig. 6). In this study, we focus on comparison of the winter-averaged observation impacts rather than their daily variations, since we are interested in observation impacts which can remain in short- to medium-range forecasts; such impacts must last long enough to be reflected in the seasonal averages. These averages can be useful for deciding the location of medium-term field campaigns, such as those in the Year of Polar Prediction (YOPP, see appendix B). Hereafter we merely refer to the winter-averaged OIEFSO (winter-averaged OIOSE) as OIEFSO (OIOSE). The OIEFSO values show a linear relation with ΔEexp2¯(t=6) (12 plots), with the 12 points close to the y = x line. All of the points are within 40% deviation from the line except Mid2 and most (8 of the 12) points are within 10%, indicating that the EFSO technique can estimate ΔEexp2¯(t=6) except Mid2. We note that all OIOSE are beneficial, which suggests that the radiosondes can always improve the subsequent forecasts. Moreover, the magnitudes of ΔEexp2¯(t=6) are not sensitive to the latitudinal band. This is because we selected the denied observations for the OSEs with various OIEFSO values at each latitudinal band (Fig. 3), though 6-h OIEFSO values are in general larger in the polar latitudinal bands than those in the midlatitude and tropical bands (see Figs. 16c and B1).

Fig. 6.
Fig. 6.

Scatter diagram of Δεexp2¯(t=6) (horizontal axis) and ΔEexp2¯(t=6) (vertical axis) of each experiment averaged for the 91-time forecasts (over the winter period). Colors and symbols are the same as Fig. 5 (black for the midlatitude, blue for the Arctic, and red for the tropical OSEs). The black line indicates the y = x function, and the gray solid and dashed lines indicate 10% and 40% deviations from y = x.

Citation: Weather and Forecasting 36, 4; 10.1175/WAF-D-20-0152.1

To elucidate to what extent EFSO has potential to estimate OIOSE within short-range forecasts, we compare Δεexp2¯(t) at the evaluation time t = te with ΔEexp2¯(t) from 6 to 48 h. Figure 7 shows the ratios of ΔEexp2¯(t) to Δεexp2¯(t) at different evaluation times. As plotted point of an experiment is closer to 1 at t hours, t-hour EFSO better estimates OIOSE. Among the different evaluation times, EFSO at t = 6 and 12 h can best estimate ΔEexp2¯(t). This result suggests that 6-h EFSO can be more useful than 24 h or longer-hour EFSO and as useful as 12-h EFSO to estimate OIOSE in ALEDAS. At the evaluation times increase, the ratios of all the experiments increase. After t = 24 h, Δεexp2¯(t) tends to underestimate ΔEexp2¯(t), indicating that EFSO becomes meaningless to directly approximate OIOSE.

Fig. 7.
Fig. 7.

Ratios of OIOSE ΔEexp2¯(t) to OIEFSO Δεexp2¯(t) of each experiment at different evaluation times (t = 6–48 h). Colors and symbols are the same as Fig. 5. The gray line indicates the y = 1 function.

Citation: Weather and Forecasting 36, 4; 10.1175/WAF-D-20-0152.1

In the next step, we examine whether the 6-h EFSO-derived Δεexp2¯(t=6) can be useful to estimate ΔEexp2¯(t) at longer lead times (i.e., on the 12-h to 7-day forecasts). Since Δεexp2¯(t=6) specifically estimates ΔEexp2¯(t=6), the examination is reworded as whether ΔEexp2¯(t=6) can be projected to ΔEexp2¯(t) at different lead times, or how long initial OIOSE can remain against the forecast error (Privé et al. 2021). For this data analysis we extend the lead time time t for ΔEexp2¯(t) from 6 h to 7 days.

Figure 8 shows the time sequences of ΔEexp2¯(t) (OIOSE) for lead times up to 7 days at each latitudinal band (Arctic, midlatitude, or tropical). During short-range forecasts, the Arctic and midlatitude OIOSE keep amplifying, while the tropical OIOSE earlier amplify and keep their values after lead time of 1 day: These are consistent with different behaviors of increasing observation impacts between extratropical and tropical observations found in Privé et al. (2021). In our experiments, the Arctic OIOSE more strongly amplify than the midlatitude OIOSE. Note that ΔEMid22¯(t) most strongly amplifies among the midlatitude OIOSE: the observed location of Mid2 is nearest to the Arctic (Fig. 5).

Fig. 8.
Fig. 8.

Time sequences of OIOSE ΔEexp2¯(t) (J kg−1) from 6-h to 7-day forecast periods (h) for the (a) Arctic, (b) midlatitude, and (c) tropical OSEs. Colors and symbols are the same as in Fig. 5. The OIOSE at the 6-h forecasts (symbols at the leftmost fringe) correspond to ΔEexp2¯(t=6) in Fig. 6. Deficits indicate negative values. Thick solid and dashed lines indicate statistical confidence exceeding 99% and 95% significant levels against the 91-time forecasts, respectively.

Citation: Weather and Forecasting 36, 4; 10.1175/WAF-D-20-0152.1

The figure reveals that among each latitudinal band (each color in Fig. 8) the relative differences (rankings) of the 6-h (initial) OIOSE remain unchanged up to lead times of 1 to 2 days except for Mid2. Only the relative ranking of ΔEMid22¯(t) is not retained and cannot be estimated by 6-h EFSO. This could be because ΔEMid22¯(t) amplifies too slowly at early short-range forecast times among the 12 OSEs (Fig. 8), and only ΔεMid22¯(t=6) largely departs from ΔEMid22¯(t=6), as indicated by Fig. 6. Overall, this result and Fig. 6 suggest that the 6-h EFSO can be useful for the estimation of initial OIOSE and amplification of OIOSE within a short-range forecast.

Starting from lead times of 3 days, however, the relative rankings do not remain intact as the time series of ΔEexp2¯(t) vary drastically for almost all the experiments, with many of them dipping to negative values. Thus, EFSO cannot directly estimate ΔEexp2¯(t) for medium-range forecast in the sense of the globally averaged OIOSE. It should again be noted that ΔEexp2¯(t) denotes the global average of ΔEexp2(t).

It is worth mentioning that ΔEexp2¯(t) is gradually increasing in whole until 7 days for almost all OSEs (see Fig. 1). It may reflect that OIOSE in some areas of the global domain amplify and survive against the forecast error growth until the 7-day forecast period (Privé et al. 2021, their Fig. 5). Therefore, OIOSE without the global averaging can be useful for short- to medium-range forecasts. We note that the ensemble spread of the CTL forecast also does not reach saturation over the 7-day forecast period (not shown). For operational forecasts, Zhang et al. (2019) have reported error saturation at around 10 days. For longer lead times, however, it should approach zero as the forecast error saturates and OIOSE become meaningless.

b. Propagation of OIOSE

The results so far indicate that, for short-range forecasts, EFSO is useful to estimate the globally averaged OIOSE, but not for medium-range forecasts. However, when we focus on the distributions of OIOSE in medium ranges, that is, ΔEexp2(t) before spatial averaging to obtain ΔEexp2¯(t), the OIOSE in some areas can be projected from the magnitudes of the initial (6-h) OIOSE. In this subsection, we examine horizontal maps to investigate how the observation impact of each experiment spreads out in the Northern Hemisphere. Our aim is to find out why OIEFSO or the initial OIOSE cannot predict ΔEexp2¯(t) for medium-range forecasts (Fig. 8), and whether they can predict ΔEexp2(t) in this range.

First, we examine the temporal evolution of OIOSE [ΔEexp2(t)]. Initially, we focus on the three experiments Pol, Mid4, and Tro, because they show the largest 6-h OIEFSO values or 6-h OIOSE within their latitudinal band (symbols closest to the upper rightmost fringes in Fig. 6). Figures 9 and 11 show the evolution for 1-day (Day 1) to 7-day (Day 7) forecasts. We discover that on Day 1, OIEFSO extends near the observed location in each OSE.

Fig. 9.
Fig. 9.

Horizontal maps of OIOSE ΔEexp2(t) (J kg−1) at 1–3-day forecast times. The observation impacts of the three OSEs—(a)–(c) the Pol (blue × in Fig. 6), (d)–(f) the Mid4 (black △), and (g)–(i) the Tro (red ×) experiments—are displayed. The hatching indicates statistical confidence exceeding 95%.

Citation: Weather and Forecasting 36, 4; 10.1175/WAF-D-20-0152.1

During the short-range and the early medium-range forecasts (Days 1–3), beneficial OIOSE mainly propagates dynamically; in other words, they are propagated in the Lagrangian way (by wind advection) and by the Rossby wave transport owing to phase and group velocity (Moteki et al. 2011; Inoue et al. 2013; Yamazaki et al. 2015; Hattori et al. 2017; Sato et al. 2020b). We, hereafter, refer to this mechanism as dynamical propagation. In the Pol experiment, initial OIOSE in northern East Siberia propagates southward and eastward and extends over the Arctic Ocean. In the Mid4 experiment, initial OIOSE propagates eastward (downstream) with the westerly jet. In the Tro experiment, initial OIOSE over the western central Pacific shifts westward with the prevailing trade winds. All these patterns are consistent with dynamical propagation.

Again to justify OIEFSO, we compare the distributions of Δεexp2(t) and ΔEexp2(t) for Day 1 to Day 3 (Fig. 10). In the three experiments, EFSO can approximate the distributions at Day 1 and to some extent at Day 2, though Δεexp2(t) underestimate the amplitudes of OIOSE as implied by Fig. 7. Thus, EFSO can estimate the dynamical propagation of OIOSE within 24 forecast hours. This supports that EFSO is useful to estimate the initial amplitudes and subsequent dynamical propagation of OIOSE during the short-range forecast.

Fig. 10.
Fig. 10.

As in Fig. 9, but for OIEFSO Δεexp2(t) (J kg−1).

Citation: Weather and Forecasting 36, 4; 10.1175/WAF-D-20-0152.1

In the medium range (Figs. 11a–c), however, many spots of nonbeneficial OIOSE appear and both beneficial and nonbeneficial OIOSE grow in the midlatitudes after Day 4, regardless of the latitudinal band of the data denial. This explains the sign changes of the globally averaged ΔEexp2(t), i.e., ΔEexp2¯(t) during the medium-range forecast (Fig. 8).

Fig. 11.
Fig. 11.

(a)–(c) As in Fig. 9 but for the averaged OIOSE during days 4–7 (medium ranges) for (a) ΔEPol2(t), (b) ΔEMid42(t), and (c) ΔETro2(t). The hatching indicates statistical confidence exceeding 90%. (d)–(f) Standard deviations of the averaged ΔEexp2(t) (J kg−1) for the (d) Pol, (e) Mid4, and (f) Tro experiments.

Citation: Weather and Forecasting 36, 4; 10.1175/WAF-D-20-0152.1

During the medium range, the regions of amplified OIOSE irrespective of beneficial or nonbeneficial ones seem to be concentrated in the midlatitudes, because standard deviations of ΔEexp2(t) against the 91-time forecasts have peaks around the midlatitudes, regardless of the latitudinal bands of data denial (Figs. 11d–f). When comparing the standard-deviation maps of the Mid4 and Tro experiments, the amplified regions seem to concentrate over the westerlies, including the Pacific, North America, and the Atlantic, where the Northern Hemisphere storm tracks are active (e.g., Hoskins and Hodges 2002). The storm-track activity can enhance a chaotic nature of the general atmospheric circulation, which causes the rapid growth of errors and the contamination of OIOSE (Hodyss and Majumdar 2007; Sellwood et al. 2008; Ancell et al. 2018).

The beneficial and nonbeneficial OIOSE in the Pol experiment in medium range broadly extend throughout the Arctic and toward the midlatitudes (Figs. 11a,d), and beneficial OIOSE are distributed in the midlatitudes more broadly than those in the Mid4 and Tro experiments (Figs. 11a–c). Therefore, Arctic observations can be the most influential and can have broadest beneficial OIOSE regions even for medium-range forecasts. If the amplified initial OIOSE by the Arctic observations systematically propagates via dynamical propagation toward the midlatitudes, beneficial OIOSE areas can appear downstream of the propagation in the midlatitudes.

Dynamical propagation of the OIOSE of the Arctic observations for medium range forecasts is examined by composite maps of OIOSE during Days 0.25–3 and Days 4–7 (medium ranges) for the 4 Arctic OSEs (Figs. 12a,b). During the short range, beneficial Arctic OIOSE originating from the initial observation points amplifies within the Arctic regions (Fig. 12a). In the medium range, areas of beneficial OIOSE are found at the east coast of North America and over the northeastern Pacific in the midlatitudes (Fig. 12a). The former area is possibly due to dynamical propagation of the initial Arcitic OIOSE. The dynamical propagation is partly illustrated by vertically averaged flux of OIOSE for the Arctic OSEs as composite of vΔEexp2(t) where v is wind vectors of the CTL. We can find that the flux is from the eastern Hemisphere of the Arctic Circle toward the eastern coastal region of North America during short- to medium ranges (Figs. 12a,b). The flux pattern supports the existence of the dynamical propagation toward the midlatitudes. It should be noted the flux just visualizes one aspect of the propagation, i.e., propagation in the Lagrangian way.

Fig. 12.
Fig. 12.

Composites of the averaged OIOSE ΔEexp2(t) (J kg−1) during (a) days 0.25–3 and (b) days 4–7 (medium ranges) for the four Arctic (Pol, Pol2, Pol3, and Pol4) experiments. The arrows in (a) indicate the flux of the OIOSE vΔEexp2(t) (J kg−1 m s−1) (see text for more details). (c),(d) As in (b), but the composites for the four midlatitude (Mid, Mid2, Mid3, and Mid4) and the four tropical (Tro, Tro2, Tro3, and Tro4) experiments, respectively.

Citation: Weather and Forecasting 36, 4; 10.1175/WAF-D-20-0152.1

To further examine the dynamical propagation, we plot the time evolution of the ΔEexp2(t) box-budgets over the Arctic and over the midlatitude north American domains for each OSE (Fig. 13). We find amplifications of the initial beneficial OIOSE for the Pol, Pol2, and Pol4 experiments in the Arctic domain during Days 2–4. After the amplifications, beneficial OIOSE for the same experiments appear in the midlatitude (North American) domain during short and early medium ranges (Days 1–4) and during the late-medium range (Days 6–7). On the other hand, for the Pol3 experiment in which initial OIOSE and OIEFSO are smallest (most non-beneficial) among the Arctic OSEs, the amplification in the Arctic domain is much weaker and no beneficial OIOSE appear in the midlatitude domain. These results also support the existence of the dynamical propagation of the Arctic OIOSE in the medium range, and show that the Arctic observations with more beneficial initial OIOSE can have more beneficial OIOSE regions within the Arctic Circle and in middle North America in the medium range. Since the initial Arctic OIOSE can be estimated by OIEFSO, EFSO for the Arctic observations are useful even for the medium-range forecasts.

Fig. 13.
Fig. 13.

Time sequences of the area-averaged OIOSE ΔEexp2(t) within (top) the Arctic (0°–360°E, 60°–90°N) and (bottom) the midlatitude North American (240°–300°E, 20°–60°N) domains. Thick solid and dashed lines indicate statistical confidence exceeding 95% and 90% levels, respectively.

Citation: Weather and Forecasting 36, 4; 10.1175/WAF-D-20-0152.1

Note that there can be two possible major propagation routes for the Arctic OIOSE over North America. One is along the surface. This propagation can be caused by meridional direct circulations in the extratropics; this is suggested by the pattern of vΔEexp2(t) being similar to the North American stream of cold airmass flux (Iwasaki et al. 2014, their Fig. 5). This flux is calculated based mass-weighted isentropic means (MIM, Iwasaki 1989; Iwasaki and Mochizuki 2012), and has been shown to describe the Lagrangian mean meridional circulation and Northern Hemisphere cold air outbreaks (Kanno et al. 2015). The other major route is along the upper troposphere. This upper-level propagation can be via both phase and/or group velocity of Rossby waves (Takaya and Nakamura 2001) and Lagrangian potential-vorticity advection (Yamazaki and Itoh 2013). Such propagation can be enhanced along climatologically strong potential-vorticity gradient zones (i.e., waveguides) in the upper troposphere. Although the Rossby-wave propagation cannot be expressed by the vΔEexp2(t) flux, a strong gradient zone exists just over North America starting from east of Alaska (northern North America) to east of middle North America (e.g., Yamazaki et al. 2019, their Fig. 5d). Sato et al. (2020b) reported that importance of the upper-level propagation of Arctic observation impacts. Their relative contributions and quantification of OIOSE propagation should be investigated in future studies.

In addition, the composite map in the medium range for the Arctic OSEs evidently shows different features than those in the midlatitude and tropical OSEs (Figs. 12b–d); areas of beneficial OIOSE are found within the Arctic Circle and at the east coast of middle North America. For the midlatitude and tropical OSEs, an area of the beneficial OIOSE is found over the northeastern Pacific; however, such beneficial area is also found for the Arctic composite map. Therefore, the beneficial OIOSE area over the northeastern Pacific is not related to dynamical propagation.

In summary, the following points are found.

  • OIOSE are able to dynamically propagate during short-range forecasts, irrespective of latitude.
  • For the medium-range forecasts, the Arctic OIOSE most effectively amplify and dynamically propagate.
  • Areas of systematic beneficial OIOSE during medium-range forecasts over the Arctic circle and the east coast of middle North America are associated with Arctic observations.
For the dynamical propagation, the Arctic OIOSE can be mediated by the extratropical direct meridional circulation (Iwasaki et al. 2014) and Rossby waves in the upper troposphere: These OIOSE are amplified before reaching the midlatitudes.

c. Discussion: Comparison with repeated OSEs for the Arctic experiments

Finally we discuss how the results above can provide useful information when observations over a limited geographical region are repeatedly collected, which is common in field campaigns. In this case, OIOSE accumulate in the analysis field (e.g., Yamazaki et al. 2015; Sato et al. 2020b). Here we test the Arctic observations because the Arctic OIOSE can most effectively amplify and propagate OIOSE in the short- and medium-range forecasts.

In section 3, the data denial was not repeated for any OSE (non-repeated OSEs hereafter). Here, we conduct two additional data assimilation and forecast experiments for each Arctic OSE, namely the repeated OSE (rep-OSE) and the semi CTL (semi-CTL) forecast experiments. Here, the rep-OSEs are conducted for the Pol, Pol2, Pol3, and Pol4 observations. For example, the rep-Pol analysis is the same analysis of ALERA2 (red arrow in Fig. 4) except the Pol observations for every 0000 UTC (rep-Pol analysis). In the rep-OSE analysis, data denials are repeated during the winter. The semi-CTL analysis is generated from the rep-OSE analysis but with one data-assimilation process for adding the OSE (e.g., Pol) observations every day at 0000 UTC (semi-CTL analysis). This is similar to generating a non-repeated OSE analysis, as described in section 3 (blue circles in Fig. 4). As in the non-repeated OSE, 7-day forecast experiments are conducted initialized with the rep-OSE and the semi-CTL analyses at every 0000 UTC during winter (91 times for each forecast experiment). The OIrep-OSE [ΔErepexp2¯(t) or ΔErepexp2(t)] is calculated as the forecast difference between semi-CTL xsemiCTLf(t) and rep-OSE xrepOSEf(t) using Eqs. (5) and (3), where xsemiCTLf(t) and xrepOSEf(t) correspond to xCTLf(t) and xOSEf(t) in the (non-repeated) OSE experiment, respectively. By comparing OIrep-OSE with OIOSE, we can assess whether the results obtained by a non-repeated OSE allows estimating the impact obtained by a repeated OSE.

Time sequences of ΔErepexp2¯(t) and ΔEexp2¯(t) are plotted in Fig. 14. During the short-range forecast (6–36 h), ΔErepexp2¯(t) and ΔEexp2¯(t) are similar, though ΔErepexp2¯(t=6) are slightly smaller than ΔEexp2¯(t=6). In particular, this similarity remains true up to 72 h (early medium range) for the rep-Pol and rep-Pol2 experiments. The relative differences (rankings) of 6-h (initial) OIrep-OSE almost retains unchanged up to Day 2. In particular, the difference between ΔErepPol32¯(t) and the other ΔErepexp2¯(t) is kept until Day 5. These are similar with the non-repeated Arctic OSEs, suggesting that EFSO for the (non-repeated) Arctic OSEs can also be useful to estimate the Arctic OIrep-OSE in short-range forecasts.

Fig. 14.
Fig. 14.

As in Fig. 8a, but for the repeated (purple lines) and the non-repeated (gray lines) Arctic OSEs. Note that the displayed forecast periods are from 6 h to 5 days.

Citation: Weather and Forecasting 36, 4; 10.1175/WAF-D-20-0152.1

Amplification and dynamical propagation of ΔErepexp2(t) including the medium-range period are evaluated by examining the composite maps averaged over Days 0.25–3 and Days 4–7 (Fig. 15). The maps are compared with those of the Arctic ΔEexp2(t) maps (Figs. 12a,b). The OIrep-OSE map during the short range shows very similar patterns with the OIOSE map (Fig. 12a). Furthermore, the OIrep-OSE distribution in the medium range shows the similar beneficial areas in middle North America and within the Arctic Circle to those in the Arctic OIOSE map (Fig. 12b). Therefore, the same mechanism (amplification and dynamical propagation) in the non-repeated Arctic OSEs can be adopted to the repeated Arctic OSEs.

Fig. 15.
Fig. 15.

As in Figs. 12a and 12b, but for the repeated Arctic OSEs.

Citation: Weather and Forecasting 36, 4; 10.1175/WAF-D-20-0152.1

Interestingly, the beneficial regions of OIrep-OSE and OIOSE are consistent with Jung et al. (2014), in which the influence of Arctic atmospheric reproducibility (observation impact) improved the medium-range forecast in the midlatitudes, especially in North America (their Fig. 2).

Therefore, non-repeated OSEs and their OIEFSO can provide useful information on the outcome of repeated OSEs, such as obtained during long-term observational campaigns. It would be interesting topic for future studies to compare repeated and non-repeated OSEs in other latitudinal bands.

5. Conclusions

We have successfully implemented the EFSO technique in the AFES-LETKF data assimilation system (ALEDAS). Based on this system, we have investigated whether the 6-h EFSO diagnosis can approximate the results of OSEs.

We conducted 12 OSEs in each of which a subset of radiosondes launched at three neighboring sites was excluded from the data assimilation. To obtain the observation impacts (OIOSE), we compared these experiments with the control experiment, in which all radiosondes were included. The 12 observation sites were selected from the Northern Hemisphere (4 in the Arctic, 4 in the midlatitudes, and 4 in the tropics). Through non-repeated data-denial experiments for the winter (December–February) of 2015/16, we have examined to what extent the 6-h OIOSE obtained by the OSEs are estimated by the EFSO technique. Our results suggest that the winter-averaged EFSO technique can accurately estimate 6-h OIOSE in all three latitude bands (Arctic, midlatitudes, tropics).

We have also tested whether the observation impacts estimated by EFSO (OIEFSO) obtain at lead time 6 h can provide information on OIOSE at short- (1–2 day) and medium-range (3–7 day) forecasts. For short-range forecasts, EFSO well estimated OIOSE at all latitudes. Furthermore, by focusing on spatial distribution of OIOSE, Arctic OIOSE in the short range tended to amplify within the Arctic. During the medium range, these OIOSE dynamically propagated and remained beneficial over several regions: the Arctic Circle and the eastern coastal region of middle North America. Therefore, we conclude that the EFSO technique can be useful to estimate OIOSE in short-range forecasts at all latitudes, and in medium-range forecasts for the Arctic. We have also performed four repeated Arctic OSEs that mimicked Arctic field campaigns covering the winter period. The results showed consistent evolution and dynamical propagation of Arctic observation impacts obtained from the repeated OSEs that are similar to OIOSE from the non-repeated OSEs. This implies that the results from the non-repeated Arctic OSEs and estimations by EFSO can provide valuable information for repeated Arctic observation campaigns of YOPP.

In other words, our results show that, for medium-range forecasts, the Arctic observations can be the most influential. They can have therefore the highest potential for improving medium-range forecasts in the Northern Hemisphere. Based on the discussion in sections 4b and 4c, we summarize the factors why the Arctic observations can be the most influential.

  1. The small number of Arctic observations. The Arctic regions are still relatively poorly observed (e.g., Jung et al. 2016, and Fig. 16a). Thus, Arctic observations tend to have a relatively large OIEFSO (Fig. 16c). We note, however, that the tropospheric observation density (observations per km2), is not very different between the Arctic and tropics (Fig. 16b).
  2. Due to the existence of the extratropical direct meridional circulation toward the midlatitudes and due to the ubiquitous Rossby waves in the extratropical upper troposphere, the Arctic OIOSE propagate toward the midlatitudes.
  3. Choice of error norm (metric) for quantifying OIEFSO and OIOSE. We chose the globally averaged moist total energy, which may emphasize errors on midlatitude storm-track regions (Figs. 11d–f). This choice is motivated by our interest in the midlatitude storm tracks, which are one of the most important elements of the general circulation.
In this paper, AFES, a typical atmospheric general circulation model with moderate horizontal and vertical resolution and a hydrostatic dynamical core, is used. The model is sufficient for our purposes, since we have used AFES and ALEDAS to conduct weather predictability studies for synoptic–large-scale atmospheric circulations (Yamazaki et al. 2015; Inoue et al. 2015; Sato et al. 2017; Kawai et al. 2017; Sato et al. 2018a,b, 2020a,b). In addition, high-resolution and convection-permitting forecast models can better represent the dynamical propagation, compared with trivial numerical errors; this is because the high-resolution models are more realistic. In short, while the former has a realistic process, the latter is due to unrealistic numerical noises. Convection-permitting models can reproduce more realistic tropical OIOSE and dynamical propagation; these may be underestimated in this study. Ensemble forecasting with more members can also reduce the trivial errors. Note that recent studies using the ECMWF forecast system have highlighted the importance of the Arctic observation impacts on the midlatitude weather systems in medium-range forecasts (Jung et al. 2014; Lawrence et al. 2019; Day et al. 2019). In addition, our results pertain to weekly weather (from 6 h to 7 days) and its impacts on the global domain. The relative importance of observations in the Arctic, midlatitudes, and tropics may vary for longer forecast periods and also vary by region (cf. Zhang et al. 2019).
Fig. 16.
Fig. 16.

Latitude–height cross section of (a) observation numbers per cycle and (b) its density distribution used in ALERA2 in free atmosphere (above 1000 hPa) during winter. In the density distribution in (b), the numbers normalized by areas (weighted by cos−1θ, where θ is latitude) are displayed. (c) Δε2¯ and area-separated Δε2¯ per observation (J kg−1) of the global, the tropical (10°S–20°N), the Northern Hemisphere midlatitude (20°–60°N), and the Arctic (60°–90°N) latitudinal bands.

Citation: Weather and Forecasting 36, 4; 10.1175/WAF-D-20-0152.1

Acknowledgments

The authors thank the organizer and the participants of the workshop “Observational Campaigns for Better Weather Forecasts” held at ECMWF in June 2019 (Magnusson and Sandu 2019), which inspired and helped us summarize the paper. We are grateful to Dr. Ingo Richter for his thoughtful comments to improve the paper and English language assistance. We also would like to thank three anonymous reviewers, and Drs. Daisuke Hotta and Yoichiro Ota for their helpful comments. The ALERA2 dataset is available from http://www.jamstec.go.jp/alera/alera2.html. This work was supported by JSPS KAKENHI (26282111, 15H02129, 17K05663, 18K13617, 18H03745, 18KK0292, and 19H05702), and the Arctic Challenge for Sustainability Research Project (ArCS) by MEXT. Numerical simulations were conducted on the Earth Simulator with the support of JAMSTEC.

APPENDIX A

ALEDAS

The ALEDAS configuration used in the present study is described in Yamazaki et al. (2015): The horizontal resolution is T119, which roughly corresponds to a 1° × 1° latitudinal and longitudinal horizontal resolution. There are 48 vertical levels, with the top at 3 hPa. The ensemble size is 63. The uniform multiplicative 10% inflation method is used. The horizontal and vertical localization scales of the LETKF are set to 400 km and 0.4lnp, respectively (Miyoshi et al. 2007; Enomoto et al. 2013; Yamazaki et al. 2017). Observations include conventional types and satellite winds, which are prepared from the PREPBUFR datasets, compiled by the National Centers for Environmental Prediction (NCEP) and archived at University Corporation for Atmospheric Research (UCAR). ALERA2 can reasonably well reproduce the observed synoptic–large-scale atmospheric circulations (Fig. A1). Further details of ALEDAS and ALERA2 can be found in Miyoshi and Yamane (2007), Enomoto et al. (2013), and Yamazaki et al. (2017).

Fig. A1.
Fig. A1.

Snapshots of (a) 500-hPa geopotential height (Z500) and (b) sea level pressure (SLP) fields in ALERA2 (blue or red contours) and ERA-Interim (black contours; Dee et al. 2011) at 0000 UTC 1 Jan 2016. Contour intervals in (a) and (b) are 100 m and 4 hPa, respectively. (c) Root-mean-square differences of Z500 [right axis (m), blue] and SLP [left (hPa), red] for the period of December–February in 2015/16. Black vertical line indicates the date shown in (a) and (b).

Citation: Weather and Forecasting 36, 4; 10.1175/WAF-D-20-0152.1

ALEDAS has been used to evaluate impacts of observations obtained in observational campaigns in various regions for various seasons; the Arctic Ocean (Inoue et al. 2013; Yamazaki et al. 2015; Sato et al. 2017, 2018b, 2020b), midlatitudes (Kawai et al. 2017; Sato et al. 2020b), the Philippine Sea (Hattori et al. 2017), the Southern Ocean (Sato et al. 2018a), and the Antarctic (Sato et al. 2020a) during summer (Yamazaki et al. 2015; Kawai et al. 2017; Sato et al. 2018a,b, 2020a), fall (Inoue et al. 2013; Hattori et al. 2017; Sato et al. 2020b), and winter (Sato et al. 2017).

Flowchart of forecast–analysis cycles in ALEDAS are shown in Fig. A2. In terms of code modifications to calculate the 6-h EFSO diagnosis, we must additionally calculate Y0a and et|−6 in the ALEDAS flowchart, to output OIEFSO. Additional processes that one uses to calculate the OIEFSO values are (i) extended forecasts for 3 h in addition to the 9-h forecast of the 4-D LETKF process (i.e., the “12-hour forecast” process in Fig. A2), (ii) calculation of Ya, and (iii) calculation of OIEFSO. We consider that these additional processes are minimal and optimal for our system because the computational cost of 9-h ensemble forecasting (forecast) is more than twice that for the LETKF (analysis) in the original ALEDAS cycle (denoted by black in Fig. A2) and the EFSO calculation requires as much time as the LETKF. Those choices make the technique more suitable for the use in observational campaigns with limited resources.

Fig. A2.
Fig. A2.

Flowchart of forecast–analysis cycles in ALEDAS from the analyzed times t = 0 and t = +6 h. Squared and rounded-corner boxes indicate inputs and outputs (data) and the executed processes, respectively. Details of each process and data are shown in bottom. In this figure, Xt=6|0f in Eq. (2) is obtained from “guess,” et=6|0 is from “guess” and “analysis,” et=6|6 is from “12-hr fcst” and “analysis,” δy0ob and R are from “O-B,” and Y0a is from “O-A,” respectively. The process “obsope” corresponds to the observation operator H to obtain δy0ob. Parallel process and ensemble data are shown with additional three boxes. Red indicates newly implemented processes and added outputs for the EFSO diagnosis.

Citation: Weather and Forecasting 36, 4; 10.1175/WAF-D-20-0152.1

APPENDIX B

Seasonal OIEFSO Values and the Values during the Special Observing Periods

ALEDAS and the EFSO diagnosis continued until winter 2019. We show the seasonality of OIEFSO for the whole globe. Figure B1 shows the seasonal averages of the OIEFSO values of the radiosondes. We can find two features common to all seasons; (i) the OIEFSO values are smoothly distributed, and (ii) the OIEFSO values are larger in the higher latitudes.

Fig. B1.
Fig. B1.

As in Fig. 3 but for all the seasons during (top left) winter 2015/16, (top right) spring 2016, (bottom left) summer 2016, and (bottom right) fall 2016. Seasonal averages of Δε2¯(t=6) (J kg−1) are displayed. Only radiosonde observations for locations that reported more than 80 times at 0000 UTC during each season are shown. The wind fields are not displayed here.

Citation: Weather and Forecasting 36, 4; 10.1175/WAF-D-20-0152.1

The same features are found in the maps during special observing periods (SOPs) of the Year of Polar Prediction (YOPP, Jung et al. 2016; Bromwich et al. 2020) when the extra radiosonde observations were performed (i.e., the polar observations were extremely increased) at many routine stations and mobile stations by field campaigns in the Arctic or Antarctic regions (Fig. B2).

Fig. B2.
Fig. B2.

As in Fig. B1, but for special observing periods (SOPs) of YOPP when frequencies of radiosonde observations in (a),(b) the Arctic and (c) the Antarctic regions were increased at many routine stations. The SOPs are from 1 Feb 2018 to 31 Mar 2018 in (a), from 1 Jul 2018 to 30 Sep 2018 in (b), and from 16 Nov 2018 to 15 Feb 2019 in (c). Averaged values of Δε2¯(t=6) during the SOPs (J kg−1) are displayed. Note that only radiosonde observations for locations that reported more than 50 [in (a)] and 80 times [in (b) and (c)] at 0000 UTC during each SOP are shown.

Citation: Weather and Forecasting 36, 4; 10.1175/WAF-D-20-0152.1

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1

Note that the sign of our definition of Δε2¯(t) is opposite to that in some previous studies.

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