Open AccessArticle

Toward Utilizing Similarity in Hydrologic Data Assimilation

Haksu Lee

^1,*

Haojing Shen

and

Yuqiong Liu

LEN Technologies, Oak Hill, VA 20171, USA

Department of Civil Eng, The University of Texas at Arlington, Arlington, TX 76019, USA

Earth Resources Technology, Inc., Laurel, MD 20707, USA

Author to whom correspondence should be addressed.

Hydrology 2024, 11(11), 177; https://doi.org/10.3390/hydrology11110177

Submission received: 7 September 2024 / Revised: 22 October 2024 / Accepted: 23 October 2024 / Published: 24 October 2024

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Abstract

Similarity to reality is a necessary property of models in earth sciences. Similarity information can thus possess a large potential in advancing geophysical modeling and data assimilation. We present a formalism for utilizing similarity within the existing theoretical data assimilation framework. Two examples illustrate the usefulness of utilizing similarity in data assimilation. The first, theoretical example shows changes in the accuracy of the amplitude estimate in the presence of a phase error in a sine function, where correcting the phase error prior to the assimilation reduces the degree of ill-posedness of the assimilation problem. This signifies the importance of accounting for the phase error in order to reduce the error in the amplitude estimate of the sine function. The second, real-world example illustrates that timing errors in simulated flow degrade the data assimilation performance, and that the flow gradient-informed shifting of rainfall time series improved the assimilation results with less adjusting model states. This demonstrates the benefit of utilizing streamflow gradients in shifting rainfall time series in a way to improve streamflow timing—vital information for flood early warning and preparedness planning. Finally, we discuss the implications, potential issues, and future challenges associated with utilizing similarity in hydrologic data assimilation.

Keywords:

1. Introduction

Similarity to reality is a necessary attribute of a (hydrologic) model as a representation of an aspect of the world as scientists perceive it [1,2,3]. To avoid ambiguity, we use the notion of similarity originating from the principle of association by similarity used in studying the operation of mental processes by associating one mental state to its successor state. For example, “α is similar to β” is interpreted as “α reminds β because α resembles β”. Similarity and resemblance are interchangeable in this paper [4,5]. Similarity concepts and measures are often very useful in developing and verifying geophysical models. For instance, similarity in spatial patterns between distributed observations and distributed model simulations helps assess the fidelity of the model [6], improve our understanding of the processes involved [7], and enable physically more realistic modeling [8,9]. Recognizing position errors helps avoid over-adjusting states and amplitudes to improve forecasts [10,11]. Capturing spatially connected features, e.g., hydrologic connectivity, is useful in understanding the behavior of the modeled system and its underlying processes [12,13,14,15,16,17]. Whereas the notion of similarity has been used pervasively in almost all stages of geophysical modeling [18], only a handful of studies have applied similarity in geophysics to data assimilation [19,20]. These include wildfire modeling with a morphing ensemble Kalman filter (EnKF) [21], hurricane vortex modeling with EnKF [22] and variational assimilation (VAR) [11], marine tropical cyclones with VAR [23], storm scale weather prediction with phase-correcting data assimilation [24], and synthetic channelized aquifer modeling with ensemble pattern matching (EnPAT) [25]. To the best of the authors’ knowledge, the use of similarity in hydrologic data assimilation has not been reported in the literature, where the prevailing practice is to use only the paired observations and model simulations that are valid at the same times and locations, which limits the incorporation of similarity information that may exist across wider space–time domains.

Similarity may be found in gradient, shape, pattern, distribution, position, direction, connectivity, timing, etc. In geophysical modeling, similarity information may be contained in spatial patterns of soil moisture [26], temporal patterns of streamflow [27], snow cover [9], or water tables [8], phases of thunderstorms [24], the timing of streamflow [28,29,30], the spatial orientation of rainfall [31], the shape of rainfall areas [32] or drainage basins [33], and the positions of hurricanes [11,22], among others. Modelers often use the notion of similarity in order to relate physiographic properties to the variables modeled, such as relating the topographic index to water table depths [34]. To simplify spatial complexities in the model domain, the similarity concept has been used to divide an entire study domain into a number of similarly behaving areas based on, for example, the topographic wetness index [34], elevation [35], or vegetation types [36]. The above approach has played a key role in developing models such as TOPMODEL [34] and SWAT [37]. Vegetation patterns may arise due to ecological and ecohydrological processes or eco-evolutionary dynamics [38,39,40]. Representing interconnected saturated hydraulic conductivity values may improve modeling preferential flow paths [41]. Capturing the timing of short-fused flash floods helps reduce property and life losses [42]. Capturing the timing of peak river discharges may improve the discovery and understanding of changes relevant to climate [43]. Utilizing hurricane position observations helps improve hurricane track forecasts [22]. Direction information helps forecast the movement of squall lines, hurricanes, or tornadoes better. Studying the evolution of the shape drawn by rainfall areas may assist precipitation forecasting [32]. Capturing the degree of heterogeneous mass distribution over the spatial domain helps model the natural variabilities of the physical variables under consideration. Streamflow gradients, or fluctuations with time, helps estimate basin-averaged precipitation rates [44]. Table 1 summarizes similarity objects and examples in hydrology. Prioritizing a similarity object over another may depend on the type of application. Hydrologic application examples in Table 1 include the flow gradient-based estimation of basin-averaged precipitation rates, since a change in precipitation is the primary cause of fluctuations in streamflow. Hence, estimating the precipitation rate may suit a gradient-based method.

To utilize similarity in data assimilation, it is necessary to quantify it. Quantifying similarity is inherently fuzzy. For two disparate things, one may still find similarities because “anything is similar to anything else in some respect or other” [45]. To move similarity from the psychological space to a physical space, mathematical measures are used. The quantification of similarity may thus follow steps similar to model building [46]: (1) identify the similarity that exists in the psychological space; (2) conceptualize it by defining a particular similarity object; (3) mathematically formulate the conceptualized similarity; (4) develop the metrics, equations, or techniques to compute it; (5) develop a computer code using equations or numerical algorithms; and (6) extract and quantify the similarity information using the computer code. Table 2 summarizes a number of widely used similarity measures. For additional measures, the reader is referred to the literature on image matching, pattern recognition, or image registration, among others [47,48,49,50,51].

In this paper, we describe how similarity may be incorporated in hydrologic data assimilation with the goal of assimilating information that may not be captured effectively by simply adding more data. To that end, we offer a formalism for assimilating similarity information within the existing data assimilation framework, and present two illustrative examples to demonstrate the process and the impact. The first example shows analytically how reducing the phase error in the sine function prior to the assimilation can improve amplitude estimates of the sine function. The second example represents an important data assimilation problem in operational hydrology, in which the soil moisture states of a hydrologic model are updated in the presence of phase (or timing) errors [58,59]. The significant contributions of this paper include a formalism for assimilating similarity information within the existing theoretical framework of data assimilation, example applications of the proposed approach in the context of hydrology, and a discussion of the implications, potential issues, and future challenges associated with utilizing similarity in hydrologic data assimilation. This paper is organized as follows. Section 2 describes the formalism for utilizing similarity in the context of the existing data assimilation framework. Section 3 presents the illustrative example of the proposed approach. Section 4 offers discussions. Section 5 provides a concluding remark.

2. Formalism for Utilizing Similarity Within the Existing Theoretical Data Assimilation Framework

In this section, we present a formalism for utilizing similarity information within the existing theoretical framework of data assimilation. In general, a data assimilation technique solves via the Bayes theorem for the posterior probability density function (PDF),

f (X_{k}| Z_{k})

, of the true state of the system,

X_{k}

, given the available observations,

Z_{k}

[60,61]:

f (X_{k}| Z_{k}) \propto f (Z_{k} | X_{k}) f (X_{k})

(1)

In the above,

f (X_{k})

denotes the prior distribution of

X_{k}

, representing the a priori knowledge about

X_{k}

, and

f (Z_{k} | X_{k})

denotes the likelihood function of

Z_{k}

, representing the likelihood of

Z_{k}

being realized given the state of the system,

X_{k}

, or the information content in the newly available observation,

Z_{k}

. Depending on the choice of the data assimilation technique, one may estimate only the first two conditional moments of

X_{k}

instead of the full conditional distribution. If

X_{k}

and

Z_{k}

are jointly normal, evaluating

E [X_{k} | Z_{k}]

and

C o v [X_{k} | Z_{k}]

amounts to solving the following least-squares minimization problem [62]:

Minimize

J_{K} = \frac{1}{2} {[X_{k} - X_{B, k}]}^{T} Ψ_{k}^{- 1} [X_{k} - X_{B, k}] + \frac{1}{2} {[Z_{k} - H_{k} (X_{k})]}^{T} R_{k}^{- 1} [Z_{k} - H_{k} (X_{k})]

(2)

subject to

X_{k + 1} = M (X_{k}) + W_{k + 1}

(3)

where

X_{k}

denotes the model state at time

k

;

X_{B, k}

represents a priori (i.e., background) estimates of

X_{k}

at time

k

;

Ψ_{k}

denotes the covariance of the background error, i.e.,

X_{B, k} - X_{k}

;

M ()

denotes the dynamical model acting on the state vector

X_{k}

; and

W_{k + 1}

denotes the dynamical model error,

W_{k + 1} ~ N (0, Q_{k + 1})

. The second term in Equation (2) arises from the following observation equation:

Z_{k} = H_{k} (X_{k}) + V_{k}

(4)

where V_k denotes the measurement error vector with mean zero and covariance

R_{k}

, and

H_{k} (X_{k})

denotes the modeled observation based on

X_{k}

for the actual observation

Z_{k}

. The first and second terms in Equation (2) prescribe

C o v (X_{k}, X_{k}^{T})

in the prior distribution,

f (X_{k})

, and

C o v (V_{k}, V_{k}^{T})

in the likelihood function,

f (Z_{k} | X_{k})

in Equation (1), respectively.

In the presence of similarity information, we may rewrite

f (X_{k}| Z_{k})

in Equation (1) via the total probability law [63] as

f (X_{k}| Z_{k}) = \int f (X_{k}| Z_{k}, ∆_{k}) f (∆_{k}| Z_{k}) d ∆_{k}

(5)

In Equation (5),

∆_{k}

denotes the transform model inferred from the similarity information. The transform model is chosen such that, when applied to the state vector,

X_{k}

, it minimizes the dissimilarity between the modeled observation and the actual observation (see Equation (2)). The transform operation may comprise some forms of translation, rotation, and scaling, and is denoted as

∆_{k} \oplus X_{k}

. Because the above basic operations provide a wide range of morphing that may be necessary to achieve similarity, it is very likely that only a limited number of transform models are necessary in practice. For example, if only a single model is considered for each of the three operations, there will be up to 2³ transform models, including the null model with no transforms. We may thus write Equation (5) as

f (X_{k}| Z_{k}) = Σ_{i = 1}^{n} f (X_{k}| Z_{k}, ∆_{i, k}) P r (Δ_{i, k}| Z_{k})

(6a)

= Σ_{i = 1}^{n} f (Δ_{i, k} \oplus X_{k}| Z_{k}) P r (Δ_{i, k}| Z_{k})

(6b)

where

Δ_{i, k}

denotes the

i

-th transform model considered for time step k and n denotes the total number of transform models considered. In Equation (6a),

f (X_{k}| Z_{k}, ∆_{i, k})

or its first two moments may be estimated by solving the standard data assimilation problem of Equations (2) through (4), in which the initial conditions of the hydrologic model,

X_{k}

, are replaced by

Δ_{i, k} \oplus X_{k}

to reflect the similarity information. The above development is conceptually similar to using similarity information as a preconditioner to the a priori initial conditions (or, background fields) before applying other analysis schemes in order to correct, e.g., position or phase errors [10,24]. The posterior distribution

P r (Δ_{i, k}| Z_{k})

in Equation (6) may be estimated via the Bayes theorem:

P r (Δ_{i, k}| Z_{k}) \propto \Pr (Z_{k}| Δ_{i, k}) P r (Δ_{i, k})

(7)

In Equation (7), the prior,

P r (Δ_{i, k})

, represents the a priori knowledge about the i-th transform model,

Δ_{i, k}

, and the likelihood function

\Pr (Z_{k}| Δ_{i, k})

includes the dissimilarity measures between the actual observation and the modeled observation resulting from the transformed model initial conditions of

Δ_{i, k} \oplus X_{k}

. It is very likely that no a priori knowledge exists for the transform models, in which case

P r (Δ_{i, k})

may be modeled as diffuse [64,65]. The above development indicates that the incorporation of similarity in data assimilation may be considered as a Bayesian model averaging problem [66,67]. In this framework, the transform-dependent data assimilation solutions are weight-averaged according to the posterior inclusion probabilities of the participating transform models,

Δ_{i, k}, i = 1, \dots, n

. If there is only a single transform model considered, Equation (6) is reduced to

f (X_{k}| Z_{k}) = f (Δ_{k} \oplus X_{k}| Z_{k})

in which

Δ_{k}

is estimated by minimizing the dissimilarity between the model-simulated and actual observations. In such a case, a single analysis scheme is used to incorporate similarity information [23]. Once the transform

Δ_{k}

is fully prescribed, one may solve the regular data assimilation problem of Equations (2) through (4) in which

X_{k}

is replaced by

Δ_{k} \oplus X_{k}

as the initial condition.

3. Examples

This section presents two examples. Section 3.1 describes a simple theoretical example using a sine function to show the benefit of reducing the phase error prior to estimating the amplitude by analytically solving the minimization problem. This is similar to solving the ill-posed streamflow assimilation problem due to timing errors in a modeled hydrograph. Section 3.2 demonstrates a real-world example showing timing errors in a modeled streamflow for a rainfall event during the period of 18 October 1998 to 22 October 1998 at the United States Geological Survey (USGS) stream gauge, ID 08065800, located at the outlet of the 870 km² Madisonville basin in Texas. Prior to streamflow assimilation, the flow timing error is addressed by the transform model that shifts rainfall time series based on a streamflow gradient. It is worth noting that the examples in this section are not exhaustive. since the aim is to present easily understandable examples of how the formalism described in Section 2 can be applied in the context of hydrologic data assimilation.

3.1. Example 1: A Sine Function

This example analytically illustrates the negative effect of a phase error, or a phase dissimilarity, on the amplitude estimate of the sine function, and the benefit of reducing the phase error prior to the amplitude estimation. Reducing the phase error before estimating the amplitude corresponds to applying the transform model

∆_{k}

in Equations (5) and (6) prior to data assimilation. Note that the phase error in the sine function may be analogous to the flow timing error in rainfall–runoff and routing modeling. Estimating the amplitude of the sine function without addressing the phase error may be similar to the case of the conventional hydrologic data assimilation that tries to improve flow magnitudes in the presence of flow timing errors. This can lead to overly adjusted model states to compensate for the timing errors.

The data assimilation problem to be solved in this example is to estimate the amplitude in the presence of the phase error in the sine function where the control vector includes the amplitude only. Changes in the accuracy of the amplitude estimate can be quantified depending on the amount of the phase error reduced prior to the assimilation. Let the observation operator (

H_{s i n}

) and the truth (T) be defined in Equations (8) and (9), respectively.

H_{s i n} = X_{A} s i n (k + X_{P})

(8)

T = X_{A, T} s i n (k + X_{P, T})

(9)

where X_A and X_P are the amplitude and the phase of

H_{s i n}

; X_A,T and X_P,T are the truth corresponding to X_A and X_P, respectively; and k is the time index. Let

Z_{H_{s i n}}

be the noise (

ε_{k}

)-added truth so that the assimilation solution converges to the truth as

ε_{k}

dissipates.

Z_{H_{s i n}} = X_{A, T} s i n (k + X_{P, T}) + ε_{k}

(10)

where

ε_{k}

denotes the observational error at the k-th time step, which follows the normal distribution with a zero mean and a variance of

σ_{Z}^{2}

, or

ε_{k} ~ N (0, σ_{Z}^{2})

. To analyze the effect of the phase error (

X_{P}

) on the amplitude estimate

X_{A}^{+}

, the control vector

X

includes

X_{A}

only. The objective function

J

may be formulated as follows:

J (X_{A}) = {\frac{1}{2} R}_{X_{A}}^{- 1} {({Z_{X_{A}} - X}_{A})}^{2} + {\frac{1}{2} R}_{H_{s i n}}^{- 1} \sum_{k = K - L + 1}^{K} {(Z_{H_{s i n}} - H_{s i n})}^{2}

(11)

For brevity, the ratio of the

X_{A}^{+}

estimate to

X_{A, T}

is given in Equation (12) by analytically solving

\frac{d J}{d X_{A}} = 0,

for which derivation details are found in Appendix A. Equation (12) is valid under the following conditions: (a) the a priori values of

X_{A}

, or

X_{A}^{-}

, are the best estimate of

Z_{X_{A}}

in the absence of amplitude observations, i.e.,

Z_{X_{A}} = X_{A}^{-}

, (b)

σ_{X_{A}}^{2} ≫ σ_{Z}^{2}

and

σ_{Z}^{2} \to 0

, and (c)

X_{P, T} = 0

\frac{X_{A}^{+}}{X_{A, T}} = \frac{2 L \cos (X_{P}) - s i n (2 L + X_{P}) + s i n (X_{P})}{2 L - \sin 2 (L + X_{P}) + \sin ({2 X}_{P})}

(12)

Equation (12) shows that for a given

X_{A, T}

, the minimization solution

X_{A}^{+}

is a function of the assimilation window length

L

and the amount of the phase error

X_{P}

, or

\frac{X_{A}^{+}}{X_{A, T}} = f (L, X_{P})

Ideally, as

L

increases,

\frac{X_{A}^{+}}{X_{A, T}}

should approach 1.0, i.e.,

\lim_{L \to \infty} \frac{X_{A}^{+}}{X_{A, T}} = {\frac{X_{A}^{+}}{X_{A, T}}|}_{L \to \infty} = 1

, which may not happen with too large an

X_{P}

; For example, with

X_{P} = \pm π

(perfectly out of phase), Equation (12) converges to −1 regardless of

L

, i.e.,

X_{A}^{+} = - X_{A, T}

, to improve the curve fitting by compensating for the phase error. In this case, utilizing similarity prior to data assimilation helps reduce the degree of ill-posedness of the assimilation problem by reducing the phase error such that the data assimilation finds its ideal solution with an increase in

L

. Figure 1 shows how the solution of Equation (12), i.e.,

\frac{X_{A}^{+}}{X_{A, T}}

, changes as a function of

L

and

X_{P}

. When

|X_{P}| > 0.25 π

, the quality of the assimilation solution

X_{A}^{+}

is rapidly degraded in order to compensate for the phase error. When

|X_{P}| < 0.25 π

, the assimilation solution behaves as expected, i.e.,

X_{A}^{+}

approaches

X_{A, T}

with an increase in L and a decrease in

|X_{P}|

. This implies the importance of reducing the phase error prior to the assimilation in order to reduce the degree of ill-posedness of the assimilation problem. When

L = 0.5 π

X_{A}^{+}

tends to be overestimated in the case of

- 0.25 π < X_{P} < 0

, i.e., the model is leading the observations; the opposite is true in the case of 0

< X_{P} < 0.25 π

, i.e., the model is trailing the observations. When

L = π

, both cases generally converge to the same

X_{A}^{+}

value for a given

X_{P}

Figure 2 shows the percent reduction in the error in

X_{A}^{+}

as quantified by

(1 - \frac{|X_{A}^{+} - X_{A, T}|}{|X_{A, X_{P} = 0.25 π}^{+} - X_{A, T}|}) \times 100

when the phase error

X_{P}

is reduced from

0.25 π

prior to the assimilation in the case of

L = π

. Figure 2 suggests that reducing the a priori phase error

X_{P} = 0.25 π

by a half through

Δ_{k} \oplus X_{k}

in Equation (6) can reduce the error in

X_{A}^{+}

by 74%, where the transformation of

X_{k} = {[X_{A}^{-} X_{P}]}^{T}

Δ_{k} \oplus X_{k} = {[X_{A}^{-} {0.5 X}_{P}]}^{T}

may be achieved by adopting the Hausdorff distance [68], series distance [28], or the hydrograph matching algorithm [30]. Figure 1 and Figure 2 show the importance of reducing the phase (or timing) error prior to the assimilation in order to improve the accuracy of the amplitude estimate. The following section presents an assimilation problem for a flood event with the presence of timing errors in the simulated flow, describes an attempt at detecting and reducing the timing error prior to the assimilation, and analyzes the impact on the assimilation results.

3.2. Example 2: A Flood Event

For a real-world example of correcting streamflow timing errors prior to the assimilation, the selected study basin is Madisonville, which is a headwater basin with an area of 870 km² in Texas. The USGS stream gauge (USGS ID: 08065800) is located at the basin outlet [69]. Visual examination was carried out in order to isolate an event with conspicuous timing errors in simulated flow over the whole event period. The selected event presented in Figure 3 shows about a 4 to 9 h early rise and recess of simulated flow relative to observed flow exceeding 50 m³/s from the period of 18 October 1998 to 22 October 1998. For streamflow assimilation, a conditional bias (CB)-penalized ensemble Kalman filter (CBEnKF) [70] is used because of its improved estimation of extremes such as floods. The derivation of the CB-penalized KF and its ensemble extension, CBEnKF, are comprehensively described in [70,71]. For brevity, the CBEnKF equations used in this study is summarized in Appendix B. Section 3.2.1 describes the hydrologic model used. Section 3.2.2 presents the transform model

∆_{k}

that addresses flow timing errors based on the similarity between rainfall and flow gradients. Section 3.2.3 compares the CBEnKF results with or without a flow timing error correction scheme for a single flood event. Note that this example is not intended to provide a general solution for correcting flow timing errors, given the diverse sources of errors and their unknown effects on flow timings. Instead, the example illustrates how the formalism described in Section 2 can be applied to a real-world hydrologic data assimilation study.

3.2.1. Hydrologic Models

The hydrologic models used include the Sacramento (SAC) soil moisture accounting [72] and the unit hydrograph [73]. The SAC is forced with mean areal precipitation (MAP) and mean areal potential evapotranspiration (MAPE), and outputs Total Channel Inflow (TCI) and evapotranspiration. The SAC models soil moistures, using two conceptual soil storages, i.e., the upper zone (UZ) and the lower zone (LZ). The UZ contains the Upper Zone Tension Water Content (UZTWC) and Upper Zone Free Water Content (UZFWC). Precipitation feeds the UZTWC, and precipitation exceeding the tension water capacity feeds the UZFWC. The LZ is fed by percolation from the UZFWC. The LZ contains Lower Zone Tension Water Content (LZTWC), Lower Zone Free Primary Content (LZFPC), and Lower Zone Free Supplemental Content (LZFSC). The LZ is generally thicker than the UZ [74]. The TCI from the SAC consists of fast and slow runoff. Fast runoff is composed of the rainfall intensity-dependent surface runoff from the unsaturated area, direct runoff from the saturated area, and impervious runoff from the impervious area, where in the SAC, additional impervious area water content is denoted as ADIMC. Slow runoff processes produce interflow from the UZFWC, supplemental baseflow from the LZFSC, and primary baseflow from the LZFPC. The UH routs fast and slow runoff components and generates streamflow at the basin outlet. The Adjoint-Based OPTimizer (AB_OPT) [75] calibrates the SAC model parameters and estimates the UH ordinates on an hourly time step.

3.2.2. The Transform Model $∆_{k}$ That Addresses Streamflow Timing

The timing of streamflow, as well as peak discharge, is inherently related to the timing of precipitation through mass balance and storage–discharge equations, as one may estimate rainfall time series from streamflow fluctuations [44]. To improve the timing of simulated streamflow, the transform model

∆_{k}

utilizes the similarity between rainfall and flow gradient and shifts rainfall time series within the assimilation window based on the gradients of both the observed and modeled flows, as described in the following.

The gradient of the observed flow at time

k

S_{t}^{o b s}

, is calculated by Equation (13).

S_{k}^{o b s} = \frac{Q_{k}^{o b s} - Q_{k - 1}^{o b s}}{d k}

(13)

Similarly, Equation (14) computes the gradient of the forecasted flow at time

k

S_{k}^{f c t}

S_{k}^{f c t} = \frac{Q_{k}^{f c t} - Q_{k - 1}^{f c t}}{d k}

(14)

The start time of a flood event is denoted as

t_{i n}^{o b s}

and

t_{i n}^{f c t}

for observation and forecasts, respectively. Experience with the Madisonville streamflow data suggested the following three criteria to find

t_{i n}^{o b s}

and

t_{i n}^{f c t}

, including

S_{t - 1}^{o b s} > 0

and

S_{t}^{o b s} > 1

to detect a flow rise and

Q_{t}^{o b s} > 5 m^{3} / s

to draw the threshold for an event. The end time of the flood event is defined by a flow less than

5 m^{3} / s

. Once

t_{i n}^{o b s}

and

t_{i n}^{f c t}

are known, the timing error,

δ^{e r r}

, can be computed by Equation (15).

δ^{e r r} = t_{i n}^{f c t} - t_{i n}^{o b s}

(15)

The rainfall data are then shifted as much as the flow timing error

δ^{e r r}

, and this forces the model to drive new a priori model states for the assimilation. The above procedure of reflecting the flow timing error in shifting rainfall time series as a new forcing to the model acts as a transform model

Δ_{k}

that replaces the initial conditions

X_{k}

with

Δ_{k} \oplus X_{k}

, as described in Section 2. Figure 4 shows a schematic of implementing data assimilation with shifting rainfall time series based on flow gradients (left), without similarity utilization (middle) or an open loop (right).

3.2.3. Results

Figure 5 compares streamflow and SAC soil moisture states with or without utilizing similarity information in assimilating streamflow observations at the basin outlet of Madisonville for the period of 18 October 1998 to 22 October 1998. Without a separate accounting of flow timing errors, the CBEnKF (blue lines in Figure 5) tends to lower soil moisture states in rising limbs to compensate for the timing error. This helps to improve rising limb flow but at a cost of missing four consecutive streamflow observations around the peak flow by the 99% confidence interval (CI) of the CBEnKF analysis. The CI was calculated using 100 ensembles. Although the CBEnKF is designed to improve extremes by accounting for Type-II CB [70], the presence of the apparent flow timing error may elevate the degree of the ill-posedness of the assimilation problem for the CBEnKF, resulting in missing peak flow by the 99% confidence interval. On the other hand, utilizing a flow timing error correction scheme helped the CBEnKF capture peak flows within its 99% CI (red lines in Figure 5). Compared to the original CBEnKF results (blue lines in Figure 5), those with the timing error correction scheme (red lines in Figure 5) overall produced the delayed soil moisture time series with less attenuation, which may explain its improved estimation of peak flow. Table 3 summarizes the validation statistics for the ensemble mean streamflow time series presented in Figure 5, including the root mean squared error (RMSE), mean absolute error (MAE), peak flow magnitude error (E_P), and peak flow timing error (E_T). The statistics suggest the overall performance of the CBEnKF has been improved with the aid of a flow timing error correction scheme. Compared to the open loop, the CBEnKF decreased the RMSE and MAE at the cost of an increased E_P, which did not happen in the case of the CBEnKF with a flow timing error correction scheme.

Equation (16) shows the normalized absolute difference (D), which computes the absolute value of the difference between the ensemble mean of state updates (

X_{j, k}^{+}

) and base model states (

X_{j, k}^{-}

) normalized by the model parameter value (

{P A R}_{j}

). In Equation (16), j and k denote the model state index and the hourly time index, respectively;

{P A R}_{j}

represents the j-th SAC model parameter, i.e., UZTWM, UZFWM, LZTWM, LZFSM, and LZFPM for j = 1,…5, respectively.

D = \frac{1}{5} \sum_{j = 1}^{5} (\sum_{k = 1}^{K} \frac{|\bar{X_{j, k}^{+}} - X_{j, k}^{-}|}{{P A R}_{j}})

(16)

The D values computed for the event shown in Figure 5 are 0.11 and 0.17 in the case of the CBEnKF with or without the timing error correction scheme, respectively. This indicates 35% less adjustment to the ensemble mean state when the timing error correction scheme is applied prior to the assimilation. The CBEnKF without the transform model tends to over-adjust soil moisture to compensate for the timing error, which may be alleviated by using the transform model. The positive effect of the flow gradient-informed shifting of rainfall time series on correcting timing errors in the simulated flow may vary depending on the complexities of the problem. For example, the presence of both positive and negative streamflow timing errors in the same event may require a more sophisticated transform model than the one used in Section 3.2.2. Although the computational cost to implement the transform model described in Section 3.2.2. is low, the cost may be high for high-resolution models and multidimensional datasets. Quantifying the computation cost with different models and data is left for future research.

4. Discussions

A similarity estimate can varyingly influence analyses and predictions, depending on the assimilation problems tackled and data assimilation approaches used. For instance, the similarity estimate may help reduce the systematic bias in model states updated by a bias-aware filter [76], update states over a time window with smoothing techniques, or update parameters as well as states, subject to the definition of the control vector. Investigating and understanding the effects of the same transform model on different assimilation problems and approaches may require repeated applications involving a reference for performance evaluation, such as an open loop or the conventional assimilation without the transform model.

Quantifying similarity is unrestricted across dimensions and scales in space and time. The search domain and scale for quantifying similarity should consider various factors related to data paucity and quality as well as assimilation problems and approaches, in addition to data availabilities. The data quality is of particular importance for accurately quantifying similarity objects. Poor quality data not only hamper the credibility of quantified similarity objects but also reduce the beneficial effect of data assimilation for state updating [77,78]. With similarity objects (Table 1) quantified with reasonable accuracy, the formalism in Section 2 should apply to diverse geophysical problems without a loss of generality. It is worth noting here that the similarity objects summarized in Table 1 are directly linked to the physical property of a variable of interest. In the case of a hurricane, for example, the similarity object of “position” is the location of the hurricane center, not another dimensional representation of the data related to the position of the hurricane that may rely on dimensionality reduction techniques for feature extractions [20,79,80]. This attribute of similarity objects allows the straightforward interpretation of similarity objects, which may help increase the generality of the formalism.

As dimensions and scales increase, the characteristics of the similarity measure are likely complicated, due partly to the serial or spatial correlation and heteroscedasticity of the observations [81]. Errors in a similarity estimate may have a pronounced covariance with a conventional error metric if the same datasets are used in estimating both. This implies modeling error covariance may be useful for the proper accounting of error properties. Albeit in different domains, a position error is similar to a timing error in that both represent a degree of displacement. For a moving object such as a hurricane, its position error in space may create a timing error in rainfall time series at a fixed location [10]. In such a case, solving the position error problem also solves a timing error problem. As the spatial displacement of objects in two datasets, e.g., observation and simulation, becomes large, the position error becomes evident. In contrast, a large temporal displacement in observed and simulated hydrographs could be because of a different forcing, not because of a large timing error in simulations. Quantified similarity metrics may contain overlapping information with conventional metrics. Information theory and information decomposition methods may help expose the amount of the overlapping information and quantify the information contained in different similarity metrics, or information added by the transform model [82,83].

Besides data assimilation, utilizing similarity estimates may provide an opportunity to revisit hydrologic studies carried out with conventional metrics, including modeling [84,85,86,87], calibration [88,89,90,91,92], validation [93,94,95,96], classification [97], regionalization [98,99,100], and bias correction [101], among others. Especially, calibration has a commonality with data assimilation in the sense that both can refine model parameters using observations. Model calibration can apply the transform model prior to the calibration. Calibration with a multi-objective function [91,102] may adopt similarity metrics in its formulation. Using a long period of record, calibration may benefit notably from utilizing similarity metrics [30]. Similarity-based calibration can open room for analyses and discussions related to sensitivity, equifinality, identifiability, and uncertainty.

Similarity-based data assimilation may be stated as utilizing similarity information contained within observations in man-made models (of a similar but simpler structure than the part of the universe under consideration) [103] in order to obtain similarity-informed analyses and forecasts. It will be interesting to assess the extent to which similarity-based data assimilation can constrain hydrologic models in comparison to conventional techniques, especially for a highly under-determined system with a high-dimensional state space.

5. Concluding Remark

Similarity to reality is an essential property of geophysical models. Hence, incorporating similarity information into the assimilation procedure holds potential for advancing the area of hydrologic modeling and data assimilation. This paper presented a formalism that utilizes similarity within the existing theoretical framework of data assimilation. Two examples illustrated the usefulness of utilizing similarity in data assimilation. The theoretical example highlighted the importance of addressing the phase error to reduce the error in the amplitude estimate of the sine function. The real-world example demonstrated the benefit of using streamflow gradients to shift rainfall time series, improving streamflow timing without overly adjusting soil moisture states. With progresses seen in similarity measures and application studies across earth science disciplines, utilizing similarity in hydrologic data assimilation may serve as a promising avenue towards using multi-dimensional and cross-scale information to effectively account for predictive uncertainties. However, as each individual hydrology problem is unique and challenging, future applications are necessary to discover the extent of the benefit of similarity utilization in hydrology.

Author Contributions

Conceptualization, H.L. and H.S.; methodology, H.L. and H.S.; software, H.L. and H.S.; formal analysis, H.L., H.S. and Y.L.; investigation, H.L., H.S. and Y.L.; visualization, H.L. and H.S.; writing—original draft preparation, H.L., H.S. and Y.L.; writing—review and editing, Y.L.; supervision, H.L. and Y.L. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

The dataset used in this study is available upon request from the corresponding author.

Conflicts of Interest

Author Haksu Lee and Author Yuqiong Liu are employed by their companies. The authors declare no conflicts of interest.

Appendix A. Derivation of the Amplitude Estimate in the Sine Function Example in Section 3.1

For completeness, Equations (8)–(11) are rewritten as Equations (A1)–(A4) below. Equations (A1) and (A2) define the observation operator (

H_{s i n}

) and the truth (T), respectively. Equation (A3) describes

Z_{H_{s i n}}

as the noise-added truth, and Equation (A4) shows the objective function

J

H_{s i n} = X_{A} s i n (k + X_{P})

(A1)

T = X_{A, T} s i n (k + X_{P, T})

(A2)

where X_A and X_P are the amplitude and the phase of

H_{s i n}

; X_A,T and X_P,T are the truth corresponding to X_A and X_P, respectively; and k is the time index.

Z_{H_{s i n}} = X_{A, T} s i n (k + X_{P, T}) + ε_{k}

(A3)

where

ε_{k}

denotes the observational error at the k-th time step, which follows the normal distribution with a zero mean and a variance of

σ_{Z}^{2}

, or

ε_{k} ~ N (0, σ_{Z}^{2})

. The control vector

X

in the objective function in Equation (A4) includes

X_{A}

only in order to analyze the effect of the phase error (

X_{P}

) on the amplitude estimate

X_{A}^{+}

J (X_{A}) = {\frac{1}{2} R}_{X_{A}}^{- 1} {({Z_{X_{A}} - X}_{A})}^{2} + {\frac{1}{2} R}_{H_{s i n}}^{- 1} \sum_{k = K - L + 1}^{K} {(Z_{H_{s i n}} - H_{s i n})}^{2}

(A4)

In the absence of amplitude observations,

Z_{X_{A}}

, the best estimate of

Z_{X_{A}}

could be the a priori value of

X_{A}

, or

X_{A}^{-}

. Solving

\frac{d J}{d X_{A}} = 0

and substituting

Z_{X_{A}} = X_{A}^{-}

and

Z_{H_{s i n}} = X_{A, T} s i n (k + X_{P, T}) + ε_{k}

(Equation (A3)) yields the minimization solution

X_{A}^{+}

in Equation (A5).

X_{A}^{+} = \frac{σ_{Z}^{2} X_{A}^{-} + σ_{X_{A}}^{2} \sum_{k = K - L + 1}^{K} \sin (k + X_{P}) [X_{A, T} \sin (k + X_{P, T}) + ε_{k}]}{σ_{Z}^{2} + σ_{X_{A}}^{2} \sum_{k = K - L + 1}^{K} {s i n}^{2} (k + X_{P})}

(A5)

Without the phase error, i.e.,

X_{P} = X_{P, T}

X_{A}^{+}

converges to

X_{A, T}

as the observation is less uncertain than the a priori estimate, i.e.,

σ_{X_{A}}^{2} ≫ σ_{Z}^{2}

, and

ε_{k}

becomes negligible. In other words, the presence of the phase error

X_{P} \neq X_{P, T}

prevents

X_{A}^{+}

from converging to the ideal solution

X_{A, T}

even in the case of highly accurate observations. Without the observation error

ε_{k}

, the presence of a phase error

X_{P} \neq X_{P, T}

results in the error in

X_{A}^{+}

, or

X_{A}^{+} - X_{A, T}

X_{A}^{+} - X_{A, T} = X_{A, T} (\frac{\sum_{k = K - L + 1}^{K} \sin (k + X_{P}) \sin (k + X_{P, T})}{\sum_{k = K - L + 1}^{K} {s i n}^{2} (k + X_{P})} - 1)

(A6)

To find an analytic solution, Equation (A5) can be converted to an integral equation as follows. When

σ_{X_{A}}^{2} ≫ σ_{Z}^{2}

σ_{Z}^{2} \to 0

and

X_{P, T} = 0

, Equation (A5) can be rewritten as Equation (A7).

X_{A}^{+} = \frac{\sum_{k = K - L + 1}^{K} \sin (k + X_{P}) [X_{A, T} \sin (k)]}{\sum_{k = K - L + 1}^{K} {s i n}^{2} (k + X_{P})}

(A7)

In Equation (A7), multiply

\frac{∆ k}{∆ k}

at the right hand side, and find the limit as

∆ k \to 0

\frac{X_{A}^{+}}{X_{A, T}} = \frac{\underset{∆ k \to 0}{l i m} \sum_{k = K - L + 1}^{K} \sin (k + X_{P}) \sin (k) ∆ k}{\underset{∆ k \to 0}{l i m} \sum_{k = K - L + 1}^{K} {s i n}^{2} (k + X_{P}) ∆ k}

(A8)

Use

\lim_{∆ k \to 0} \sum_{k = a}^{k = b} f (k) ∆ k \equiv \int_{k = a}^{k = b} f (k) d k

[104] and change the summation from

K - L + 1

K

to an integral from

0

L

to be independent of

K

\frac{X_{A}^{+}}{X_{A, T}} = \frac{\int_{k = 0}^{k = L} \sin (k + X_{P}) \sin (k) d k}{\int_{k = 0}^{k = L} {s i n}^{2} (k + X_{P}) d k} = \frac{2 L \cos (X_{P}) - s i n (2 L + X_{P}) + s i n (X_{P})}{2 L - \sin 2 (L + X_{P}) + \sin ({2 X}_{P})}

(A9)

where

\int_{k = 0}^{k = L} \sin (k + X_{P}) \sin (k) d k = \frac{1}{2} \int_{k = 0}^{k = L} \{\cos (X_{P}) - \cos (2 k + X_{P})\} d k = \frac{1}{2} \{L \cos (X_{P}) - \frac{s i n (2 L + X_{P})}{2} + \frac{s i n (X_{P})}{2}\}

(A10)

\int_{k = 0}^{k = L} {s i n}^{2} (k + X_{P}) d k = \int_{k = 0}^{k = L} \{0.5 - 0.5 \cos (2 (k + X_{P}))\} d k = \frac{1}{2} L - \frac{1}{4} \{\sin 2 (L + X_{P}) - \sin ({2 X}_{P})\}

(A11)

Equation (A10) uses

\sin (x) \sin (y) = 0.5 \{c o s (x - y) - c o s (x + y)\}

, and Equation (A11) uses

{s i n}^{2} (x) = 0.5 - 0.5 \cos (2 x)

Appendix B. The Conditional Bias-Penalized Ensemble Kalman Filter (CBEnKF)

The CBEnKF [70] is an ensemble extension of the conditional bias-penalized Kalman filter (CBPKF) [71] that minimizes a weighted sum of the error variance and the expected value of the Type-II CB squared, as shown in Equation (A12).

Σ_{k} = \underset{e r r o r v a r i a n c e}{\underset{⏟}{E_{Y_{k}, Y_{k}^{*}} [(Y_{k} - Y_{k}^{*}) {(Y_{k} - Y_{k}^{*})}^{T}]}} + α \underset{T y p e - I I C B s q u a r e d}{\underset{⏟}{E_{Y_{k}} [(Y_{k} - E_{Y_{k}^{*}} [Y_{k}^{*} | Y_{k}]) {(Y_{k} - E_{Y_{k}^{*}} [Y_{k}^{*} | Y_{k}])}^{T}]}}

(A12)

where

Y_{k} = {[X_{k} G (X_{k})]}^{T}

is the (m

\times

1) augmented state vector at time k; X_k is the (

n_{c} \times 1

) state vector;

n_{c}

denotes the number of variables in the control vector; G(X_k) represents the generally nonlinear function which maps the state variables to the (n

\times

1) augmented state subvector;

Y_{k}^{*}

denotes the vector of the estimated states; and α denotes the weight given to the CB penalty. If α = 0, the CBEnKF is reduced to an EnKF [70]. With the state argumentation, the dynamical and observation equations solved in the CBEnKF can be written as Equations (A13) and (A14), respectively.

X_{k} = M (X_{k - 1}) + W_{k - 1}

(A13)

where M( ) denotes the dynamical model for the state variables, and W_k₋₁ denotes the dynamical model error at time step k − 1.

Z_{k} = H_{k} Y_{k} + V_{k}

(A14)

where Z_k denotes the (n

\times

1) observation vector, where n denotes the total number of observations; H_k is the (n

\times

m) observation matrix that consists of the (n

\times

n_c) zero matrix and (n

\times

n) identity matrix, i.e.,

H_{k} = [0 I]

; m = n_c + n; and V_k denotes the (n

\times

1) observation error vector at time step k with its covariance matrix R. The updated state vector of the ith ensemble member,

Y_{k | k}^{i}

, is given by Equation (A15).

Y_{k | k}^{i} = {Y_{k | k - 1}^{i} + K}_{k} [Z_{k} - H_{k} Y_{k | k - 1}^{i}]

(A15)

where

Y_{k | k - 1}^{i}

denotes the (m

\times

1) forecast state vector of the ith ensemble member, and K_k denotes the (m

\times

n) CB-penalized Kalman gain matrix. For brevity, presented below are the expressions for the Kalman gain K_k (Equation (A16)) and the error covariance

Σ_{k | k}

(Equation (A17)) matrices used in the CBEnKF, and for their derivations, the reader is referred to [71].

K_{k} = {[ϖ_{1, k} H_{k} + ϖ_{2, k}]}^{- 1} ϖ_{1, k}

(A16)

Σ_{k | k} = {[ϖ_{1, k} H_{k} + ϖ_{2, k}]}^{- 1} (ϖ_{1, k} R_{k} ϖ_{1, k}^{T} + ϖ_{2, k} Σ_{k | k - 1} ϖ_{2, k}^{T}) {[ϖ_{1, k} H_{k} + ϖ_{2, k}]}^{- 1}

(A17)

In Equations (A16) and (A17), the (m

\times

n) and (m

\times

m) weight matrices for the observations and model prediction,

ϖ_{1, k}

and

ϖ_{2, k}

, respectively, are given by

ϖ_{1, k} = {\hat{H}}_{k}^{T} Γ_{11, k} + Γ_{21, k}

(A18)

ϖ_{2, k} = {\hat{H}}_{k}^{T} Γ_{12, k} + Γ_{22, k}

(A19)

In Equations (A18) and (A19), the (m

\times

n) modified observation matrix,

{\hat{H}}_{k}^{T}

, and the (n

\times

n), (n

\times

m), and (m

\times

m) matrices,

Γ_{11, k}

Γ_{12, k} (= Γ_{21, k}^{T}),

and

Γ_{22, k}

, respectively, are given by

{\hat{H}}_{k}^{T} = H_{k}^{T} + α C_{k}^{T}

(A20)

Γ_{22, k}^{- 1} = Λ_{22, k} {- Λ}_{21, k} Λ_{11, k}^{- 1} Λ_{12, k}

(A21)

Γ_{11, k} = Λ_{11, k}^{- 1} + Λ_{11, k}^{- 1} Λ_{12, k} Γ_{22, k} Λ_{21, k} Λ_{11, k}^{- 1}

(A22)

Γ_{12, k} = - Λ_{11, k}^{- 1} Λ_{12, k} Γ_{22, k}

(A23)

In Equations (A20)–(A23), the (n

\times

m) gain matrix for the observation vector,

C_{k}

, and the (n

\times

n), (n

\times

m), and (m

\times

m) modified error covariance matrices,

Λ_{11, k}

Λ_{12, k} (= Λ_{21, k}^{T}),

and

Λ_{22, k}

, respectively, are given by

C_{k} = [H_{k} Σ_{k | k - 1} G_{2, k}^{- 1} + R_{k} H_{k}] {[{G_{2, k}^{- 1} Σ}_{k | k - 1} G_{2, k}^{- 1} + {2 (H}_{k}^{T} R_{k} H_{k} + Σ_{k | k - 1})]}^{- 1} G_{2, k}^{- 1}

(A24)

Λ_{11, k} = R_{k} + α (1 - α) C_{k} Σ_{k | k - 1} C_{k}^{T} + {Λ_{12, k} H}_{k}^{T} + H_{k} Λ_{21, k}

(A25)

Λ_{12, k} = - α C_{k} Σ_{k | k - 1}

(A26)

Λ_{22, k} = Σ_{k | k - 1}

(A27)

In Equations (A24)–(A27),

R_{k}

and

Σ_{k | k - 1}

denote the (n

\times

n) observation error covariance matrix,

C o v [V_{k}, V_{k}]

, and the (m

\times

m) forecast error covariance matrix, respectively, and

G_{2, k}^{- 1} = H_{k}^{T} H_{k} + I

. The forecast error covariance matrix is estimated by

Σ_{k | k - 1} = \frac{1}{n_{S} - 1} (Y_{k | k - 1} - \bar{Y_{k | k - 1}}) {(Y_{k | k - 1} - \bar{Y_{k | k - 1}})}^{T}

(A28)

where

n_{S}

denotes the ensemble size and

\bar{Y_{k | k - 1}}

denotes the ensemble mean of

Y_{k | k - 1}

. For further details, the reader is referred to [71].

The theoretically expected bounds for the sample

Σ_{k | k}

are used as a criterion to accept α [70]. If Equation (A29) is not satisfied, α is reduced to find a different

Σ_{k | k}

satisfying Equation (A29).

{[ϖ_{1, k} H_{k} + ϖ_{2, k}]}^{- 1} \leq Σ_{k | k} \leq B {[ϖ_{1, k} H_{k} + ϖ_{2, k}]}^{- 1}

(A29)

In Equation (A29),

B {[ϖ_{1, k} H_{k} + ϖ_{2, k}]}^{- 1}

represents the apparent CBPKF error covariance,

Σ_{k | k}^{a}

from the minimization solution for Equation (A12), where the (m

\times

m) matrix, B, is given by

B = α Σ_{k | k - 1} [ϖ_{1, k} {\hat{H}}_{1, k} + ϖ_{2, k}] + I

. The lower bound for

Σ_{k | k}

, i.e.,

{[ϖ_{1, k} H_{k} + ϖ_{2, k}]}^{- 1}

in Equation (A29), is obtained with

α = 0

or,

B = I

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Figure 1.

\frac{X_{A}^{+}}{X_{A, T}}

in Equation (12) as a function of the phase error

X_{P}

and the length of the assimilation window

L

. With

X_{P, T} = 0

in Equations (8)–(10), negative

X_{P}

values render the modeled sine curve leading the observations, and positive

X_{P}

values make the modeled sine curve trailing the observations.

Figure 1.

\frac{X_{A}^{+}}{X_{A, T}}

in Equation (12) as a function of the phase error

X_{P}

and the length of the assimilation window

L

. With

X_{P, T} = 0

in Equations (8)–(10), negative

X_{P}

values render the modeled sine curve leading the observations, and positive

X_{P}

values make the modeled sine curve trailing the observations.

Figure 2. Percent reduction in the error in

X_{A}^{+}

as quantified by

(1 - \frac{|X_{A}^{+} - X_{A, T}|}{|X_{A, X_{P} = 0.25 π}^{+} - X_{A, T}|}) \times 100

when the phase error

X_{P}

is reduced from

0.25 π

prior to the assimilation in the case of

L = π

Figure 2. Percent reduction in the error in

X_{A}^{+}

as quantified by

(1 - \frac{|X_{A}^{+} - X_{A, T}|}{|X_{A, X_{P} = 0.25 π}^{+} - X_{A, T}|}) \times 100

when the phase error

X_{P}

is reduced from

0.25 π

prior to the assimilation in the case of

L = π

Figure 3. The selected event from 1998 in the Madisonville basin in Texas for the assimilation experiment, with similarity information at an hourly time step.

Figure 4. A schematic of implementing data assimilation with or without similarity information, where

Δ_{k}

X_{k}^{-}

, and

X_{k}^{+}

denote a transform model, a state forecast, and a state analysis, respectively.

Figure 4. A schematic of implementing data assimilation with or without similarity information, where

Δ_{k}

X_{k}^{-}

, and

X_{k}^{+}

denote a transform model, a state forecast, and a state analysis, respectively.

Figure 5. Comparison of streamflow and SAC soil moisture states with or without utilizing similarity information in assimilating streamflow observations.

Table 1. Examples of similarity objects.

Similarity Object	Description	Example
Gradient	A change in the magnitude from one point to another	Streamflow fluctuation
Shape	Characristic surface configuration	Rainfall area
Pattern	Recurring feature	Vegetation
Timing	A particular (period of) time of an event	Peak flow timing
Distribution	Distribution of values of a physical variable over a space and/or time domain	Soil moisture
Connectivity	State of being (inter)connected	River network
Position	Place located	Hurricane center
Direction	Course of movement	Wind direction

Table 2. Description of selected dissimilarity measures.

Name	Description	Similarity Object	Reference
Discharge change rate	Fluctuation in discharge at a given time interval	Gradient, e.g., streamflow fluctuations	[44,52]
Eigenshape analysis	A morphometric procedure for describing changes in shape	Shape, e.g., soil moisture or rainfall fields	[53]
Shannon’s diversity index	A measure of variety of different kinds within a domain	Pattern, e.g., vegetation spatial structure	[54]
Cross wavelet transform (XWT)-based timing error estimation	Technique to estimate a timing difference between two time series based on the phase information from the XWT of the two time series into a two-dimensional time-scale space	Timing, e.g., streamflow timing	[29]
Earth Mover’s distance (EMD)	The minimum cost to turn a probability distribution (assumed as a pile of dirt) to another where the cost is estimated by the amount of dirt moved multiplied by its moving distance	Distribution, e.g., soil moisture or rainfall fields	[55]
Connectivity function	Lag-dependent probability that a pixel is spatially connected to another pixel by the continuous path of neighboring pixels exceeding a given threshold	Connectivity, e.g., soil moisture fields	[12]
Phase correlation	A method based on the Fourier transform to align two images with a displacement error	Position, e.g., hurricane	[56]
Direction-based similarity measure	A direction-based similarity measure for trajectory clustering	Direction, e.g., wind	[57]

Table 3. Statistics of ensemble mean streamflow time series in Figure 5, where RMSE, MAE, E_P, and E_T denote root mean squared error, mean absolute error, peak flow magnitude error, and peak flow timing error, respectively; negative E_P means simulation underestimating observations; negative E_T means simulated peak flow issued earlier than observed peak flow.

Metrics	Open Loop	CBEnKF	CBEnKF with a Flow Timing Error Correction Scheme
RMSE (m³/s)	32	10	6
MAE (m³/s)	25	5	3
E_P (m³/s)	−11	−23	−11
E_T (h)	−7	5	4

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Lee, H., Shen, H., & Liu, Y. (2024). Toward Utilizing Similarity in Hydrologic Data Assimilation. Hydrology, 11(11), 177. https://doi.org/10.3390/hydrology11110177

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Toward Utilizing Similarity in Hydrologic Data Assimilation

Abstract

1. Introduction

2. Formalism for Utilizing Similarity Within the Existing Theoretical Data Assimilation Framework

3. Examples

3.1. Example 1: A Sine Function

3.2. Example 2: A Flood Event

3.2.1. Hydrologic Models

3.2.2. The Transform Model $∆_{k}$ That Addresses Streamflow Timing

3.2.3. Results

4. Discussions

5. Concluding Remark

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

Appendix A. Derivation of the Amplitude Estimate in the Sine Function Example in Section 3.1

Appendix B. The Conditional Bias-Penalized Ensemble Kalman Filter (CBEnKF)

References

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Toward Utilizing Similarity in Hydrologic Data Assimilation

Abstract

1. Introduction

2. Formalism for Utilizing Similarity Within the Existing Theoretical Data Assimilation Framework

3. Examples

3.1. Example 1: A Sine Function

3.2. Example 2: A Flood Event

3.2.1. Hydrologic Models

3.2.2. The Transform Model ∆ k That Addresses Streamflow Timing

3.2.3. Results

4. Discussions

5. Concluding Remark

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

Appendix A. Derivation of the Amplitude Estimate in the Sine Function Example in Section 3.1

Appendix B. The Conditional Bias-Penalized Ensemble Kalman Filter (CBEnKF)

References

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3.2.2. The Transform Model $∆_{k}$ That Addresses Streamflow Timing