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Optimized Determination of Viscous Mud Properties Using a Nonlinear Wave–Mud Interaction Model NAVID TAHVILDARI AND JAMES M. KAIHATU Zachry Department of Civil Engineering, Texas A&M University, College Station, Texas (Manuscript received 6 February 2011, in final form 11 May 2011) ABSTRACT The complex process of surface wave propagation over areas of cohesive sediments has generally been treated by assuming a particular rheological behavior for the mud layer, thereby fixing the description of the mud characteristics into the specification of parameters relevant to the selected rheology. The capability of inverting data to recover these parameters is investigated here. Representing the mud layer as a thin viscous fluid, a nonlinear wave–mud interaction model, coupled with a nonlinear optimization technique (Levenberg– Marquardt), is used to deduce mud characteristics from estimates of wave energy. A set of numerical tests with a deterministic phase-coherent cnoidal wave are conducted to individually estimate viscosity and mud layer depth (keeping one fixed while estimating the other), and to determine the limits of convergence of the inversion algorithm. It is shown that instances of convergence or nonconvergence can be traced to the shape of the dissipation rate curve as a function of the parameter under consideration as well as the location of the initial guesses of the target parameter along that curve. It is found that the estimation of viscosity is less problematic than the estimation of mud layer depth. Tests with random waves are also performed, using both root-mean-square wave height (representation of wave energy) and wave skewness (representation of nonlinear wave properties) as input for the inversion. The use of random waves appears to ameliorate many of the convergence difficulties encountered with the cnoidal wave tests, while the use of wave skewness, while promising, is somewhat less successful. Finally, the inversion algorithm is tested against laboratory data and the deduction of both mud layer depth and viscosity proceed well. Implications for general mud property deduction are discussed.
1. Introduction The interaction of surface waves with bottom sediments has been widely studied, with a significant priority on sandy sediments. However, it has been estimated (Holland and Elmore 2008) that only about 40% of the world’s coastline can be considered sandy. Vast areas of mud, silt, and fine sediments line the adjacent shorelines of many areas of the world (e.g., the coastlines of Louisiana, the Korean Peninsula, the Amazon River delta, and the Persian Gulf). Generally, the immediate effect of these bottom muds on the surface waves has been that of very strong damping, as reported by Wells and Coleman (1981) off the coast of Surinam. To describe the nature of the damping, the characteristics of the mud must be specified a priori. Apart from
Corresponding author address: Navid Tahvildari, Zachry Department of Civil Engineering, Texas A&M University, 3136 TAMU, College Station, TX 77843-3136. E-mail:
[email protected] DOI: 10.1175/JTECH-D-11-00025.1 Ó 2011 American Meteorological Society
the use of sedimentological approaches (e.g., Chou et al. 1993) to obtain the dissipation properties, mud properties are usually specified via selection of a rheological model to represent mud behavior. Most of the analytical and numerical attempts in modeling wave–mud interaction assume a density-stratified two-fluid system in which clear water overlies the mud layer. In addition to the assumption of a viscous Newtonian fluid (Gade 1958; Dalrymple and Liu 1978; Ng 2000), the mud state has variously been assumed to be a Bingham plastic (Mei and Liu 1987), a viscoelastic material (Macpherson 1980; Hsiao and Shemdin 1980; Jiang and Mehta 1995, 1996; Mei et al. 2010), or a non-Newtonian fluid (Foda et al. 1993; de Wit 1995). Jain and Mehta (2009) discuss the range of applicability of the commonly used rheological models. The dissipation characteristics of each rheological model often have a notable dependence on wave frequency; Kaihatu et al. (2007) showed for their application of the dissipation mechanism of Ng (2000) that the dissipation rate had a broad distribution over a range of relative water depths kh (where k is the wavenumber and h
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the water depth), and that the peak rate occurred near kh ; 1. Representation of the wave attenuation by mud in numerical wave models has recently been done for phase-averaged spectral models (Winterwerp et al. 2007; Kranenburg 2008) and phase-resolving wave models (Kaihatu et al. 2007; Huang and Chen 2008). This has allowed investigations of the effects of mud on various nearshore wave processes. For instance, the model by Kaihatu et al. (2007) accounts for the effect of a thin mud layer on near-resonant nonlinear interactions over the entire wave spectrum. They showed that damping of highfrequency waves seen in the data of Sheremet and Stone (2003) could be explained by nonlinear subharmonic interactions. Damping of low-frequency energy by bottom mud drains high-frequency energy through these interactions, thus attenuating the energy in wave spectral frequencies, which are not kinematically coupled to the bottom. This was observed by Elgar and Raubenheimer (2008) in field measurements on the Atchafalaya (Louisiana) shelf, though they observed the highest dissipation rates in the 0.2 # kh # 0.3 range. Alternatively, shortwave modulations over fluid mud in intermediate water depth can also result in attenuation of the associated bounded low-frequency waves (Mei et al. 2010; TorresFreyermuth and Hsu 2010). In addition to the damping of waves by mud through direct coupling with the sea bottom or indirect damping resulting from nonlinear interactions, surface waves can lose energy in resonant interaction with interfacial waves over the lutocline. These interactions in a two-layer fluid system have been investigated to the second order by Wen (1995), Hill and Foda (1998), and Jamali et al. (2003); third-order effects in the resonance are discussed in Tahvildari and Jamali (2009). The present numerical models each predict the wave behavior and coastal processes in a forward mode, with all mud characteristics specified in advance. This implies that the parameters specifying the assumed rheological behavior for the mud are known with sufficient certainty for all areas in which the processes are active. However, because mud behavior is a function of its state of consolidation and dynamic forcing from the surface wave, it is often not clear which rheological model works well for a particular situation. The sensitivity of the wave response to different parameter ranges in the mud description has not been studied to any great degree. Concomitantly, the use of remotely sensed data [e.g., video (Lippmann and Holman 1990) and X-band radar (Bell 1999)] of free surface kinematics has brought in a wealth of data, which can be used for deduction of relevant environmental parameters; this deduction would employ the model in an ‘‘inverse’’
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mode as an engine for data assimilation. It is thus of use to determine whether free surface information can be used to glean the parameters describing the mud, and what the range of applicability of these inverse methods might be. Data assimilation and inverse methods, while well established in meteorology and oceanography (e.g., Le Dimet and Talagrand 1986; Bertino et al. 2003), are relatively new to the study of nearshore processes. Existing studies have concentrated on using data to obtain estimates of various environmental variables, including water depth (Grilli 1998; Kennedy et al. 2000; Narayanan et al. 2004), bottom friction roughness lengths (Keen et al. 2007), and wave-induced currents and setup (Feddersen et al. 2004). Recently, Rogers and Holland (2009) used a phase-averaged forward model and an inverse technique to deduce mud properties at Cassino Beach, Brazil. In this study we use a phase-resolving nonlinear wave model and a mud dissipation mechanism, in conjunction with the Levenberg–Marquardt numerical optimization method (e.g., Press et al. 1986) to deduce the mud characteristics from free surface data. One of the goals of this study, aside from creating and testing the inversion algorithm, is to see whether instances of nonconvergence (where the inversion algorithm does not converge upon the expected mud parameter values) have a physical meaning. We find that, for relatively simple pseudomonochromatic wave propagation cases, nonconvergence can be linked to the characteristics of a curve detailing dissipation as a function of the parameter under consideration. In addition, we also look at cases of random wave transformation over areas of mud, using both energy-based (root-mean-square wave height) and wave shape–based (skewness) statistics as a convergence metric. The focus here is on investigating how the deduction of mud characteristics from data may be limited by the physical nature of mud-induced damping. These probable difficulties may be exacerbated by measurement uncertainty and other sources of error; while the algorithm can be altered to incorporate these aspects, here our focus is on the physics and the concomitant effect on the deduction of mud parameters. It is noted that, of the previously mentioned studies, those of Keen et al. (2007) and Rogers and Holland (2009) are closest in scope to our interest here. However, the forward model in both studies was Simulating Waves Nearshore (SWAN; Booij et al. 1999), which is a spectral phase-averaged model. The SWAN model does not include any representation of the nonlinear triad energy transfer inherent in the nearshore zone, aside from a parameterized energy transfer (Eldeberky and Battjes 1995).
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2. Forward and inverse modeling a. Model for surface wave propagation in shallow water The model used here is based on the work of Kaihatu et al. (2007). The surface wave model is a nonlinear, phase-resolving, frequency domain parabolic mild-slope model (reduced here to one dimension), which describes wave transformation over varying bathymetry and nearshore nonlinear triad interactions between surface waves (Kaihatu and Kirby 1995), with full dispersion in the kinematics. In this model, the free surface h is expressed as N
h(x, t) 5
å A2n ei(
Ð
kn dx2vn t)
1 c:c. ,
(1)
n51
where x is the horizontal coordinate and An is the complex amplitude of the free surface elevation, and c.c. represents complex conjugate terms. The frequency vn is associated with the wavenumber kn through the linear dispersion relation v2n 5 gkn tanhkn h,
(2)
where g is the gravity acceleration and h is the water depth (in the case of a muddy bottom, h is the distance from the free surface to the top of the mud line). The boundary value problem for water waves is expanded to second order in (5ka), where a is a representative wave amplitude. Using (1), (2), and the boundary value problem, Kaihatu and Kirby (1995) derived the following equation for one-dimensional nonlinear wave transformation: Anx 1
(CCg )nx (2CCg )n
An 1 Dn An
2 n21 Ð i 4 52 RAl An21 ei( (kl 1kn2l 2kn )dx 8(kCCg )n l51
å
N2n
12
å SA*l An1l e
l51
Ð
3
i( kn1l 2kl 2kn )dx 5
,
(3)
where C is the phase velocity, Cg is the group velocity, R and S are nonlinear interaction coefficients, and the subscript x refers to spatial differentiation. The interested reader is referred to Kaihatu and Kirby (1995) for more details on the forward model. The term Dn in (3) represents energy dissipation in a generic sense. For the dissipation of wave energy resulting from bottom mud, Kaihatu et al. (2007) used the dissipation model of Ng
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(2000) to specify the form of Dn. The following section will describe salient features of Ng (2000). The model (3) is capable of simulating a variety of periodic waveforms. For this study we use two different incident wave conditions: a cnoidal-like wave solution of permanent form (Kaihatu 2001) and a series of random waves. The permanent form solution represents a simple input condition, and allows us to connect the convergence characteristics of the algorithm to the dissipation while still incorporating the effect of the wave–mud interaction on the nonlinear energy transfer of the surface wave, as seen by Kaihatu et al. (2007). The random wave incident conditions used here are more representative of actual field conditions, and determine the robustness of the algorithm in instances where the connection between convergence and dissipation is not as clear.
b. Model for mud-induced energy dissipation The target goal of much previous work on wave damping by mud is the development of a dispersion relation, in which the wavenumber k is complex (e.g., Dalrymple and Liu 1978; Macpherson 1980). For an expansion such as (1), the real part of the resulting wavenumber would reflect the wavelength over the muddy bottom, and the imaginary part would dictate the dissipation rate. As previously mentioned, the wave–mud interaction model used in this study is that of Ng (2000). Because of the explicit expression of the dissipation rate, this model for mud-induced dissipation can be easily included in predictive wave models (Kaihatu et al. 2007; Rogers and Holland 2009). The model is a thin-layer reduction of the viscous fluid dissipation model of Dalrymple and Liu (1978), in which the Stokes boundary layer thickness of the mud is of the same order as the thickness of the mud layer. Despite this apparent limitation, Kaihatu et al. (2007) showed favorable model comparisons with the laboratory experiment of deWit (1995) for the experimental parameters beyond the thin-layer assumption. The analytical results of Ng (2000) suggest that the maximum mud-induced damping occurs when the mud layer thickness is about 1.5 times its boundary layer thickness, confirming the laboratory result of Gade (1958) and the numerical result of Dalrymple and Liu (1978). pffiffiffiffiffiffiffiffiffiffi The Stokes boundary layer is computed as d 5 2n/v, where n is the kinematic viscosity. In accordance with the equivalence between the Stokes boundary layer thickness and the mud thickness, it is established that ka 5 kdm 5 kdm 1,
(4)
where a is the surface wave amplitude, dm is the depth of the mud layer, and subscript m refers to mud. The
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FIG. 2. Dissipation effect of viscous mud with rm 5 1111.11 kg m23, nm 5 1.3 3 1024 m2 s21, and dm 5 0.2 m extending from x 5 300 m to x 5 800 m on the free surface permanent form cnoidal waves with period T 5 10 s. FIG. 1. Dissipation rate, D (1/m), contour against mud layer thickness and viscosity, g 5 0.9.
ordering employed by Ng (2000) allowed for the total wavenumber k to be expressed as k 5 k1 1 k2 ,
(5)
where k1 is the solution to (2) and the leading order term. The second-order term k2 is complex, and its imaginary part is the dissipation rate of the wave resulting from its interaction with the muddy bottom. This dissipation rate can be calculated as follows (Ng 2000): D 5 Im(k2 ) 5
2Im(B)k1 , sinh(k1 h) cosh(k1 h) 1 k1 h
(6)
where B is a function of the depth of the mud layer dm, the ratio ofpthe Stokes boundary layer thicknesses ffiffiffiffiffiffiffiffiffiffiffiffi ffi z 5 dm /dw 5 nm /nw , and the density ratio g 5 rw/rm. The change in the density, viscosity, and depth of the mud would affect the mud damping behavior accordingly. Figure 1 illustrates a contour of variations of D against mud layer thickness and viscosity for rm 5 1111.11 kg m23. The dissipation effect of the mud on the free surface elevation h is illustrated in a model output sample in Fig. 2. In this example, cnoidal waves of permanent form with periods of T 5 10 s propagate over the mud patch with a kinematic viscosity of nm 5 1.3 3 1024 m2 s21, a thickness of dm 5 0.2 m, and a density of rm 5 1111.11 kg m23 extended from x 5 300 m to x 5 800 m. Despite the utility of the Ng (2000) formulation for mud damping, the model is restricted to
the viscous behavior of mud in a layer that is thin relative to the Stokes boundary layer.
c. Data The data used for inversion can theoretically be any measurable characteristic of the surface wave field, measured from any platform (either in situ or remotely sensed). We will first use wave energy (integrated over the spectrum) as the output of the wave–mud interaction model and regard it as our synthetic data. In a later section we will investigate the effect of bottom mud on the wave shape statistics, which goes toward addressing the nonlinearity of the surface wave field and their change in the face of mudinduced dissipation.
d. Inversion scheme We use the Levenberg–Marquardt method (a damped Gauss–Newtonian scheme) for the inversion. The advantage of the Levenberg–Marquardt method is its ability to control the iterations toward convergence under the constraint that the errors not increase. This offers increased accuracy at the expense of further iterations. The scheme has been used by Narayanan et al. (2004) to deduce bathymetric profile parameters from a wave propagation model and has provided satisfactory results. The algorithm starts with an initial guess and the steps toward the answer will be governed by the error vector e at each step. The error to be minimized is e 5 [xd (x) 2 x(x)][xd (x) 2 x(x)]T ,
(7)
where xd(x) is the data row vector (vector of measurements), and x(x) is the iterative row vector of outputs
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from the model generated by the use of trial mud parameters. Here, xd(x) is the output of the forward model with a set of given mud parameters. The values of these parameters are the values to be recovered using the inverse model. The mud parameter value corresponding to the minimum error is the primary output of the inverse nonlinear least squares scheme and is compared to the data. Each step in the optimization is governed by the variation of the output with respect to a change in parameter. To control the magnitudes of the iterative steps (and improve the robustness of the algorithm), a relaxation parameter l is used. The following equation is solved successively: hi11 5 hi 1 (JJT 1 l)21 JT (xd 2 xi ),
(8)
where h is the parameter sought, indices i are the step counters, and J is an n 3 m sensitivity matrix defined as J5
›xj
›hk
,
j 5 1, 2, 3, . . . , n, k 5 1, 2, 3, . . . , m, (9)
where j is the number of measurement points for x and k is the number of parameters to be determined. The iteration continues until the error e is smaller than a userdefined value. It is noted that two values of hk and corresponding xj are required to build J at each step. The initial l (positive) is typically chosen as being proportional to the maximum element in the matrix J0 JT0 , where J0 is the initial sensitivity matrix. The algorithm is not sensitive to this proportionality; however, successive decreases in l as the algorithm progresses are a good indication of convergence. We slightly modify the Levenberg–Marquardt algorithm to keep the parameters in physically realistic space. It is possible, for example, to obtain parameter values that are negative and, hence, unacceptable as the input of the forward model for the next iteration. To ameliorate this, the generic Levenberg–Marquardt scheme is altered to allow a trial iterative step to be rejected and then recalculated with a larger l. Because the scheme is gradient based, success of the convergence can depend on the presence of local maxima and minima in the response of the model to variations in the parameter under consideration. To simplify the analysis, it is presently assumed that the mud viscosity and density are physically independent; we explore possible interdependencies and their implications in a later section.
e. Tests using permanent form cnoidal waves In this section we perform inverse deduction of mud depth and viscosity using measurements of wave field
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energy as input. For this initial test, the waves are permanent cnoidal surface waves and the interaction with viscous muddy bottom is incorporated into the wave model. The input cnoidal wave was generated using a permanent form solution of model (3). The goal of this set of tests is to investigate the robustness of the algorithm for a deterministic case, allowing focus on the mechanics of the algorithm. The damping model of Ng (2000) uses mud layer thickness dm, mud density rm, and viscosity nm as the primary descriptors of dissipation rate. In the tests done throughout this study we assume that the density of the mud is known and deduce the viscosity and the mud layer depth. One reason for this is that Kaihatu et al. (2007), using a test case similar to that used here, showed that the wave responses to mud density specification in the range 0.5 # rw/rm # 0.99 are quite similar. This similarity indicates a reduced sensitivity of wave damping to changes in mud density, which would make deduction via the algorithm difficult. For each sought parameter, we bracket the target value with two initial guesses and apply the inversion algorithm. We then describe the inversion for the mud parameter using physical properties in terms of the theory of thin viscous mud layer.
1) INVERSION FOR MUD DEPTH In practical terms, the thickness of the mud layer underneath a wave can be difficult to quantify. Possible responses to wave forcing can range from fluidization near the top of the mud layer to complete suspension of the mud layer (Jaramillo et al. 2009). The estimation of this thickness, therefore, can be considered an estimate of the ‘‘effective’’ depth, or the depth of mud required to provide the response observed in the overlying surface waves. These substantial uncertainties in the mud layer estimates are neglected in our inversion exercises here, though they can be incorporated in future extensions of the inversion scheme. We now examine the inversion for mud depth using wave energy for three different mud viscosities, as expressed by their nondimensional representation z (z 5 10, 50, 100). Mud density rm is assumed to be 1111.11 kg m23, a value that Ng (2000) and Kaihatu et al. (2007) demonstrated to produce significant damping. The wave period is 10 s, wave height is 0.1 m, and water depth is 1 m. Various initial values were assumed and several convergent and nonconvergent cases of inversion were observed. If the data and the initial guesses on mud depth are smaller than about 6% of dmax, where dmax denotes the mud depth corresponding to the maximum damping rate, then the scheme does not converge to a result. Figure 3 illustrates a few examples of inversion model results. Figure 3a shows a successful convergence for
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FIG. 3. Inverse model results for mud depth deduction, with four different mud depths as data and two random initial guesses for each data value. In (a) z 5 10, in (b) and (c) z 5 50, and in (d) z 5 100. In all cases g 5 0.9.
a target depth of 0.005 m. The number of iterations required toward convergence depends on the initial estimate for the parameter value and the initial assumption for relaxation factor l. Figure 3b shows a convergence for z 5 50 and ddata 5 0.07 m, where ddata is the target mud depth corresponding to the synthetic data. In this case the scheme that started with a smaller initial guess converged quickly. However, using the larger initial guess (d1 ’ dmax) requires control from the relaxation parameter to converge. Without this control, the low sensitivity of the overlying waves to the mud depth in the larger initial value causes the first trial depth to be negative. The use of relaxation control is manifested in the linear trend of the trial mud depths toward the target (seen for the upper initial estimate in Fig. 3b. Target mud depth is further increased to 0.4 m, and Fig. 3c shows convergence to the target value from an initial guess larger than the target, but not for one smaller than the target. Other inversion model outputs exhibit similar behavior to the cases illustrated in Fig. 3. Figure 4 summarizes the results of inversion tests for z 5 10, 50, 100. The convergence regions are hatched. The inverse model recovers the target mud depth only if the initial guess and target value are located in the same
convergence region. These regions can be explained by examining the dissipation rate as a function of mud layer thickness, according to the mud dissipation theory of Ng (2000) used in the forward model. Figure 5 shows the relationship between the damping rate of the wave and thickness of the mud layer for three different mud viscosities. In all three cases, damping increases with mud depth until it reaches a maximum value at dm ’ 1.55dm. Further increases in mud depth will cause a decrease in wave damping effect until a point is reached where a further increase in mud depth will not have a significant effect on damping rate. However, there are a local minimum and maximum in this region that give rise to two convergence regions (2) and (3). In Fig. 5 the regions with the same gradient signs for the case of z 5 100 are indicated. It is also observed that the larger the mud viscosity, the larger the mud thickness at which the maximum damping will occur. If the target mud depth is in a steep region of the dissipation rate curve, then convergence can be a problem if one of the initial guesses falls near the peak dissipation rate where the slope of the curve is reduced. This is shown in Fig. 3b. The target depth of this particular case is well below the peak, but the larger initial
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FIG. 4. Invertibility regions for mud depth with different viscosity values: For z 5 10, dmax 5 0.0316 m, for z 5 50, dmax 5 0.158 m, and for z 5 100, dmax 5 0.316 m.
guess is located near the peak, where the slope is small (indicative of small sensitivity). This is the reason that the initial trial encompassed negative values of mud depth, necessitating relaxation control. As previously mentioned, nonconvergence to the target mud depth was evident in some of these simulations (e.g., Fig. 3c). This can occur if the initial guesses and the target mud depth were on opposite sides of a local maximum or minimum. From Fig. 5, it is apparent that the curves detailing the dependence of dissipation rate on mud layer depth have an iconic shape. Neglecting the very thin mud layer (less than 6% of the mud depth at maximum dissipation rate, or dmax), the following four regions depicted in Fig. 4 a–c can be recognized by close examination of the curve:
locations of the initial guesses or the target depth will not lead to convergence to the target depth. Figure 3c is an example of such a nonconvergence condition. In this example, the target depth is located in region 3, while the first and second initial guesses are located at regions 2 and 4, respectively. As observed in the figure, the scheme starting with the smaller initial guess converges to a depth in region 2 with the same damping rate as the target depth. However, the larger initial guess converges to the target. This convergence indicates that the target value is located near the local minima and in region 4. It can be gathered from Fig. 5 that if the mud viscosity is
(i) a region with positive slope 0.06dmax # dm # dmax, (ii) a region with negative slope dmax # dm # 2dmax, (iii) a region with a slight positive slope 2dmax # dm # (3.13 2 4.00)dmax, and (iv) a region with a negative slight slope region at dm $ (3.13 2 4.00)dmax. The location of the local minima and maxima changes slightly with changes in viscosity or density of mud. The identified regions and the general shape of this curve help explain the instances of convergence or nonconvergence to the target values of the mud layer depth. In general, convergence appears to be ensured when both the target depths and the initial guesses fall in the same region 1, 2, 3, or 4, with relaxation control ameliorating the effects of smaller slopes near the peak of the dissipation rate curve. Other permutations of the
FIG. 5. Variations of dissipation rate Dm/w against mud depth dm for different mud viscosities: z 5 10(nm 5 1.3 3 1024 m2 s21), z 5 50(nm 5 3.25 3 1023 m2 s21), and z 5 100(nm 5 1.3 3 1022 m2 s21). In regions 1–4, the gradient does not change sign.
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FIG. 6. Inverse model outputs for mud viscosity, with four different viscosity values and two random initial guesses for each data value; dm 5 0.2 m, rm 5 1111.11 kg m23.
increased, then the convergence range of mud depth increases, as the extent of region 1 and 2 across dm increases with viscosity. It is also noteworthy that in a highly viscous case of z 5 100 (e.g., Fig. 3d), the gradient of the curve in region 4 increases compared to weaker viscosities. Although in idealized inversion the smallest gradients drive the scheme to convergence to the target value, this gradient increase (increase in sensitivity) makes the inversion on mud depth more robust for higher viscosities when the data are noisy. We note here that the aforementioned convergence regions are present in the Ng (2000) formulation of thin viscous mud layer; other damping mechanisms may have other dissipative trends with increases in mud layer thickness and, as a result, different convergence regions. For instance, if the thin-layer assumption is removed, then the damping depth diagram would have two regions with distinct positive and negative gradients (see Dalrymple and Liu 1978; their Fig. 2). Similarly, the semianalytical model proposed by Jain and Mehta (2009) also gives two regions of different gradient signs in dissipation rate–mud depth diagram. The inversion algorithm applied to these two forward models should have two convergence regions.
2) INVERSION FOR MUD VISCOSITY Figure 6 shows some cases of convergence for various target values of viscosity. We assume a mud density of rm 5 1111.11 kg m23 and mud layer depth of dm 5 0.2 m, corresponding to a high degree of damping (Kaihatu et al. 2007). Based on target values of n 5 0.0005 m2 s21 (Fig. 6a), n 5 0.007 m2 s21 (Fig. 6b), n 5 0.02 m2 s21 (Fig. 6c), and n 5 0.1 m2 s21 (Fig. 6d), we see that convergence appears to take place from both initial values. The implemented relaxation control is apparent for the larger bracketing initial guess in Fig. 6a. Figure 7 and Table 1 provide the convergence and nonconvergence instances of inverse model output for mud viscosity. The viscosity corresponding to the maximum damping rate is denoted by nmax in Table 1. Figure 7 is a generic plot of the convergence regions where two regions with different gradient signs are distinguished in the damping diagram. In this diagram, p denotes the parameter, mud depth, or viscosity, and pmax denotes the value of the parameter corresponding to maximum damping, which is the boundary of the regions with distinct gradient signs. Similar to the inversion for mud
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FIG. 7. Invertibility regions of mud viscosity using cnoidal waves and mud depth using random waves; rm 5 1111.11 kg m23.
depth, the convergent and nonconvergent cases can be explained by examining the behavior of the damping rate. Figure 8 shows the relationship between the damping rate and the viscosity of the mud layer for four different mud depths. As expected, in each diagram in Fig. 8, the following two regions with different slopes are distinguishable: (i) a region with a positive slope to the left of the maximum damping rate, where the increase in mud viscosity will lead to the increase in mud damping rate; and (ii) a region to the right of the maximum, where a further increase in mud viscosity will cause a decrease in the wave damping effect. In cases corresponding to Fig. 6, the target viscosity and both initial guesses are located in regions 1 (Figs. 6a,b) and 2 (Figs. 6c,d). Similar to the inversion for mud depth, if the data and initial guesses are located on the same region, then convergence is expected. Nonconvergence happens otherwise. It is noteworthy that unlike mud depth, the dependence of damping rate of a monochromatic wave on mud viscosity has two regions with slopes of comparable absolute magnitude. This property gives a wider invertibility range for mud viscosity relative to that for mud depth. It is also apparent that, as the mud depth increases, the shape of the viscosity curve becomes more amenable to inversion, because the gradient to the right
of the peak increases and the region to the left of the peak dissipation rate increases its range across nm. As mentioned in section 2d, the optimization algorithm requires two values to build the sensitivity matrix. Therefore, an initial guess should include two values to build the sensitivity matrix J0 at the first step. If the first initial value corresponds to pmax, then the second one will dictate the direction of marching toward the parameter value that gives the same damping rate as the target parameter. In this condition, if the second value falls in the same region with the target parameter value, then the algorithm converges. The inversion is nonconvergent otherwise. Therefore, as it will be seen in section 3b, the inversion algorithm can potentially converge to two distinct values depending on the initial guess. Some judgment on the value of the parameter in the environment is required to choose the proper result.
f. Random waves The primary result of the prior section was an understanding of the sensitivity of the damping process to variations in the depth and viscosity of the mud layer TABLE 1. Values of parameter pmax (mud depth or viscosity) in Fig. 7. nmax(m2 s21)
dmax(m)
p dm 5 0.1 m dm 5 0.2 m dm 5 0.3 m dm 5 0.4 m z 5 100 0.0026 0.0101 0.0224 0.0398 0.1850 pmax
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FIG. 8. Variation of dissipation rate Dm/w against mud viscosity nm for different mud layer depths.
under deterministic, phase-coherent conditions. Random waves in the ocean, however, comprise many different frequencies with varying degrees of phase coherence. Furthermore, for a given set of mud characteristics, mud dissipation rate is dependent on wave frequency. In a broadbanded wave spectrum, the effect of mud on the general dissipative characteristics of the wave field (outside of total integrated energy loss) is not obvious; thus, it is not immediately clear whether the cross-spectral dissipation has similar behavior to the monochromatic case and whether it is the dominant factor on inversion properties in random waves. In this section we investigate the inversion process for mud properties under random waves. We at first use root-mean-square wave height Hrms as our synthetic data for the inversion. However, in recognition of the presence of strong cross-spectral interactions in shallowwater waves and the effect of dissipation on these interactions (Sheremet et al. 2005; Kaihatu et al. 2007), we also investigate the inversion using wave skewness as an input. Though it is unlikely that wave skewness would be available as a measurement in the absence of first-order quantities (like Hrms), this would help quantify the effect of nonlinearity on the tenability of the inversion process. For the random wave spectrum, we use the Texel– Marsden–Arsloe (TMA) spectrum (Bouws et al. 1985), in which we specify a peak period Tp 5 10 s and an Hrms 5 0.14 m, in concert with the monochromatic wave condition of the previous tests. The water depth is (as before) specified as 1 m. The phase-averaged TMA spectrum is decimated into a time series and then input into a fast Fourier transform, which is then put into the forward model. We used 500 frequencies in our model runs, leading to
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a maximum frequency of nearly 1 Hz, or 10 times the peak frequency. Because of the nonlinear summation term, the model is quite computationally intensive; for N frequencies, the number of computations is O(N2). Generally, multiple realizations of the initial time series are run through the model, and then averaged to increase the number of degrees of freedom of the result. However, this would entail a significant increase in computer time for no real gain, because Hrms generally does not vary greatly between realizations. Thus, we use a single realization of the time series for modeling. It can be argued that skewness estimates will be adversely affected by the lack of multiple realizations for averaging. However, this is essentially a proof of concept; our primary reason for using skewness as input data for the inversion is to see whether the nonlinear behavior of the waves could be exploited to determine mud characteristics. As such, a single realization should be sufficient to determine this. To reduce unnecessary computations, we have also shortened the modeled domain relative to the monochromatic case; the overall domain now is 550 m long, with mud starting at x 5 50 m.
1) INVERSION FOR MUD DEPTH We use the random wave model to invert Hrms for the depth of the mud layer for z 5 100. We use the same target and initial guess depths as of the monochromatic case. Looking at Figs. 9a,b, we see that all of the cases demonstrate convergence to the target. The random wave analog of the monochromatic wave tests appears to converge more readily to the target mud depths. The inversion of wave skewness to deduce mud depth is somewhat problematic. Figures 9c,d show some results for random wave inversion using skewness as input. While most cases appear to converge readily (e.g., Fig. 9d), one case did not converge at all (Fig. 9c). As a monochromatic case, the convergence behavior in a random wave condition can be explained by examining the damping rate variation with mud viscosity. Figure 7 and Table 1 summarize the general convergence regions for mud depth. One possible reason for more robust inversion when using Hrms of the wave spectrum instead of H of the monochromatic wave is that the combination of the frequency-dependent dissipation rates have a curve that is more amenable to inversion. This appeared to be a general trend throughout these tests. Figure 10 shows the variations of the root-mean-square of damping Drms against dm compared with the monochromatic D with T 5 Tp. It appears that, unlike with the monochromatic case, the dissipation curve has only two regions with different slope signs leading to two convergence regions (Fig. 7). It is worth noting that in random wave conditions,
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FIG. 9. Inversion for mud depth using (a),(b) Hrms and (c),(d) skewness of random waves, with three different mud depths as data and two random initial guesses for each data value; z 5 100, rm 5 1111.11 kg m23.
the maximum dissipation rate is larger and occurs at a thinner mud layer than the monochromatic wave condition. To investigate the inversion using wave skewness, we inspected the evolution of skewness with distance (expressed here as output ‘‘gauge’’ number; the distance between gauges is 20 m) to determine any tendencies that may help to drive convergence. Figure 11 shows the time series, spectral density functions, and skewness variations for a nonconvergent (corresponding to Fig. 9c) and a convergent (corresponding to Fig. 9d) case. As seen in Fig. 11d, the evolution shows a significant recovery of skewness values in the lee of the mud patch for dm 5 0.80 m. This recovery aids convergence in that it is highly variable. In this case, the skewness, and hence the nonlinear effects, appears to be significant in the entire studied domain. Therefore, skewness can be a suitable measure in inversion. In contrast, a much smoother evolution with no recovery of skewness is observed for the nonconvergent case of dm 5 0.15 m. It appears that, in the lee of the mud patch, nonlinear effects on the shape of the wave are significantly reduced. In this case
the skewness is not a desirable measure of inversion, and while there is some nonmonotonicity in the skewness evolution profile, it does not appear to be sufficient to influence the convergence characteristics. Overall, however, it is interesting that a direct measure of nonlinear characteristics of the wave field can be used to deduce mud properties. Kaihatu et al. (2007) showed that the nonlinear characteristics of wave evolution are affected by the presence of mud, and this exercise directly incorporates this observation into the inversion process. Consequently, although using the wave skewness exhibits promising results in inversion, Hrms appears to be a more suitable measure for inversion. It is evident in Fig. 11b that the wave spectrum corresponding to the convergent case (the dash–dot curve) has a harmonic structure, while the nonconvergent case lacks such a characteristic. In addition to the examination of skewness variation, this result suggests that the wave spectrum with more harmonic shape ameliorates the convergence of the inversion algorithm. An alternate approach would be to apply the inversion algorithm to the individual spectral densities rather than
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2) INVERSION FOR MUD VISCOSITY
FIG. 10. Variations of root-mean-square dissipation rate Drms against dm for different mud viscosities, compared with monochromatic D; Tp 5 10 s, z 5 100, and z 5 0.9.
integrated quantity. However, several complications would be added to the problem. First, there will be additional degrees of freedom, one for each frequency. The energy at each frequency would have to match the data to have successful convergence. Investigating the damping for an explanation of convergence cases seems to be generally successful. However, in optimizing the shape of the spectrum, change in the shape resulting from the nonlinear energy transfer over mud would be an additional complication. It would be difficult to identify whether the change in spectral shape is due to direct coupling with the muddy bottom, energy transfer to other harmonics, or some combination thereof. In addition, it is unclear whether the entire frequency range of the spectrum would need to be matched to ensure sufficient convergence. Such complications are beyond the scope of the present paper. It is instructive to evaluate the sensitivity of the deduced mud parameters to the measurement errors in data (e.g., Hrms). Here, we apply a 10% error to the mud layer thickness and measure the Hrms output of the forward model. One value of mud depth is chosen in each side of the peak of the Drms diagram (dm 5 0.09 m for dm , dmax and dm 5 0.9 m for dm . dmax; see Fig. 10). The resulting Hrms errors are shown in Table 2. As expected, while the mud layer thickness is thinner than dmax, the error in dm results in large errors in Hrms. Conversely, the errors in Hrms resulting from the errors in dm are relatively small when dm . dmax. Such variations in the error are expected because the gradient of the D versus the dm diagram is much larger when dm , dmax than in thicker mud layers where dm . dmax. The inverse model retrieves the original 10% error in the mud depths when the erroneous Hrms, given in the second column of Table 2, is used as input.
We first use Hrms to perform the inversion; results are seen in Figs. 12a,b. We also used skewness as a basis for deduction of viscosity. Figures 12c,d show these results, all of which converge. The variation of damping against mud depth is similar to Fig. 8; thus there are two regions of convergence, as illustrated in Fig. 7. Viscosities are normalized using the nmax of the corresponding monochromatic case. Table 3 shows the errors in Hrms resulting from a 10% error in mud viscosity. Similar to Table 2, one value of mud viscosity is chosen on each slope of the D versus nm diagram (Fig. 8). It is found that that the errors in the Hrms do not change considerably. This can be explained by noting that the magnitudes of the gradients at each side of the peak of the diagram do not vary significantly. The results of the inversion for cnoidal and random waves indicate that invertibility of a mud parameter depends upon on the dissipation-parameter curve. In the previous sections, we fixed one of the parameters and investigated the inversion for the other. As illustrated in Fig. 1, the dissipation rate is not a unique function of mud depth or viscosity and various combinations of these parameters can result in the same wave damping rate. Methods, such as the imposition of smoothness constraints, are available to alleviate this problem and determine the two-parameter inversion. However, such approaches are beyond the scope of the present study.
3. Applying the scheme to data In this section, the inversion scheme is applied to the laboratory data of de Wit (1995). Kaihatu et al. (2007) showed that the forward model compared well against the data. We use the inversion scheme to obtain the mud depth and/or mud viscosity, which provides the best comparison with the measured data. The flume of the de Wit (1995) experiment is 40 m long, 0.8 m wide, and 0.8 m deep. The wave period and initial wave height are 1.5 s and 0.045 m, respectively. Mud layer thickness is 0.115 m and water depth is equal to 0.325 m. Further mud and water properties are nm 5 2.6 3 1023 m2 s21, rm 5 1300 kg m23, nw 5 1.3 3 1026 m2 s21, and rw 5 1000 kg m23. The measured wave heights are shown in Fig. 13. Of note is the increase in wave height near the downwave end of the flume (rightmost two gauges). Because of the damping effect of mud, the continuous decay of wave height is expected in the flume, and it is unclear what is causing the increase in wave height at these two locations; no reason is given by de Wit (1995). The forward model of Kaihatu et al. (2007) does not replicate this sudden increase in wave
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FIG. 11. (a),(c) Time series, (b) spectral density function, and (d) skewness variation along the wave flume with variation of mud depth for successful inversion (dm 5 0.015 m) and unsuccessful inversion (dm 5 0.80 m); z 5 100 and rm 5 1111.11 kg m23.
height (their Fig. 3), so inversion using this data is not successful. While the technique described here can be adapted to account for measurement uncertainty, we have not done so here. We exclude these points with the understanding that this can be overcome with some errorbased weighting. It is additionally noted that other experiments in de Wit (1995) exhibited larger degrees of wave height amplification near the downwave end of the mud patch than the test used here, possibly resulting from liquefaction effects (de Wit 1995), which the model does not reproduce.
convergence are both in region 2, but the larger initial guess is located in region 3. Thus, convergence is not expected.
b. Mud viscosity deduction In this section, the inversion scheme is applied to the experimental measurements to deduce mud viscosity. Similar to the previous subsection, the two most downwave measurements are discarded. Figure 15a shows a successful convergence to the experimental data of mud viscosity.
a. Mud depth deduction Figure 14 shows the result for the initial guess (dm 5 0.06 m), which is smaller than the target value. While the inversion converges, it is clear that control is required; the small wave heights in the lee of the high damping region have likely affected convergence. When an initial guess (dm 5 0.18) larger than the target mud depth was used, however, the scheme failed to converge. Referring to the convergence regions shown in Fig. 4, we see that the target value and the initial guess leading to
TABLE 2. The dm and resulting Hrms; imposed and resulting errors are noted in parentheses. dm (m)
Hrms (m)
0.081 (0.09 2 10%) 0.09 0.099 (0.09 1 10%) 0.81 (0.9 2 10%) 0.9 0.99 (0.9 2 10%)
0.0477 (0.0391 1 22%) 0.0391 0.0321 (0.0391 2 17%) 0.0021 (0.002 1 5%) 0.002 0.0019 (0.002 2 5%)
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FIG. 12. Inversion for mud depth using (a),(b) Hrms and (c),(d) skewness of random waves, with two different target viscosity values and two random initial guesses for each target viscosity values and two random initial guesses for each target values; dm 5 0.2 m and rm 5 1111.11 kg m23.
Interestingly, if a large enough initial guess is used, then the scheme will converge to a value of viscosity that is larger than the target value. An example of this behavior is shown in Fig. 15b where the scheme converges to nm 5 0.121 m2 s21 rather than nm 5 0.0026 m2 s21. Similar to the cnoidal wave case in Fig. 8, a curve showing dissipation rate as a function of viscosity can be drawn (Fig. 16) using the other parameters of the experiment. It is apparent that two values of viscosity can be deduced for a single value of dissipation rate. Because the initial guess of viscosity was to the right of the dissipation rate peak and the algorithm will drive the iteration toward lower error, it is sensible that the inversion mechanism focuses on the higher value of viscosity.
4. Summary and conclusions In this study, we detail the development of an optimization technique for the inverse deduction of mud properties from free surface information. The forward model, which represents the physics of the wave–mud interaction, is that of Kaihatu et al. (2007), which couples the nonlinear
triad interaction model of Kaihatu and Kirby (1995) with the thin-layer viscous mud dissipation mechanism of Ng (2000). The Levenberg–Marquardt method, a Gauss–Newtonian method with adjustable relaxation, is used for the optimization. Our synthetic data for input to the system are either bulk spectral energy measures (wave energy Hrms) or wave shape statistics (wave skewness). The latter addresses the fact that mud has a considerable effect on the nonlinear energy transfer (Kaihatu et al. 2007), which in turn would also affect the wave shape and associated statistics. Our intent is to apply the technique
TABLE 3. The nm and resulting Hrms; imposed and resulting errors are noted in parentheses. nm(m2 s21)
Hrms(m)
0.0045 (0.005 2 10%) 0.005 0.0055 (0.005 2 10%) 0.0117 (0.013 2 10%) 0.013 0.0143 (0.013 1 10%)
0.0094 (0.091 1 0.3%) 0.0091 0.0089 (0.0091 2 2.2%) 0.0094 (0.0096 2 2.1%) 0.0096 0.0099 (0.0096 1 3.1%)
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FIG. 13. Wave height measurement from an experiment of de Wit (1995).
FIG. 14. Convergence in inversion for mud depth data in the experiment; dm 5 0.115 m.
to several cases of wave–mud interaction, noting not only the successful inversion cases, but also determining the causes for nonconvergence in terms of the physics of wave–mud interaction. Of the three parameters used to specify mud characteristics, we chose to focus on the deduction of the mud layer depth dm and the viscosity nm. We first apply the methodology to the case of cnoidal wave propagation. Although one would not expect to see phase-coherent waves propagate for long distances in the ocean, these cases do help make the connection between (non-) convergence and dissipation characteristics very clear. Tests for deduction of the mud layer depth dm revealed a strong correlation between instances of convergence and the location of the initial estimates and target values of dm along the curve-relating dissipation rate to dm (Fig. 5). Convergence to the target value always occurred when initial estimates and target values were in the same region of convergence (Fig. 4), though some reliance on the relaxation control was required for conditions near the peak dissipation rate. In contrast, mud viscosity nm appeared to more readily converge to the target value than dm. The shape of the curve-relating dissipation rate to nm (Fig. 8) has more structure than the corresponding curve for dm; there are only two regions in the dissipation curve, and this structure aids in convergence to target viscosity values. Random waves were then introduced into the inversion procedure as a means of incorporating more realistic scenarios. Both Hrms and wave skewness were used as input to the inversion; the wave conditions were chosen to be random wave analogs to the cnoidal wave conditions of the prior section. Unlike D (monochromatic wave dissipation rate), Drms has two regions with
distinct slope signs. Therefore, convergence for dm using Hrms of random waves appeared to be more robust than the cnoidal waves. Wave skewness input led to some instances of nonconvergence for the target dm; this seemed to be dependent on the variation of skewness with distance (Fig. 11), with highly structured, positive skewness patterns (Fig. 11a) leading more reliably to convergence. In addition, it appears that the convergent case has a more harmonic structure than that of the nonconvergent case. Use of skewness appeared to work well for deducing nm. While it is unlikely that measured wave skewness would be known in the absence of energy measures such as Hrms, this exercise does directly incorporate the effect of the mud on nonlinear energy transfer, and how this varies with changes in the mud properties. We then tested the algorithm against laboratory data (de Wit 1995). The algorithm was able to converge upon the correct dm and nm, though it was determined that a large initial guess for nm might converge to a value larger than the target value. This is due to the nonuniqueness of the dissipation rate, as seen in Fig. 16. Given the close correlation between the characteristics of the dissipation rate curves and the convergence tendencies of the algorithm, it can reasonably be asked why the inversion technique is required. For instance, it is possible to estimate the degree of energy dissipation from the surface wave measurements and use that information to determine the unknown parameters of the mud via the given curves developed from the theory of Ng (2000). However, the primary intent behind the comparisons of that section, as stated previously, is to verify that the inversion scheme returns reasonable results that fit with the mud dissipation theory used. Furthermore, cases of convergence and nonconvergence can be explained by use of the dissipation rate curves, the location of the data, and
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FIG. 15. (a) Convergence (b) nonconvergence in inversion for mud viscosity in the experiment; nm 5 2.6 3 1023 m2 s21.
iterative guesses on the curves. This gives us confidence that the methodology is consistent with the embedded dissipation theory and will additionally be helpful for other dissipation theories, particularly if they do not have
as straightforward an implementation as that of Ng (2000). Moreover, it was shown that a strong link exists between the dissipation rate curve and inversion characteristics for random waves as well; these encompass
FIG. 16. Nonunique solution in inverting for mud viscosity using experiment (de Wit 1995) data.
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a multitude of frequencies, all of which have a specific response to the presence of mud. It is evident from Fig. 1 that the wave damping rate is a nonunique function of mud parameters. Therefore, in investigating the invertibility of mud parameters, we had to fix one of the parameters and invert along the axis of the other. There are methods to approach a twoparameter inversion problem (such as smoothness constraints). However, investigating such approaches is beyond the scope of the present paper. In a general field situation more uncertainty is present, both in the physics (wind wave generation, bottom friction dissipation, and wave breaking may play a role in addition to mud-induced dissipation) and in the measurements used as input to drive the inversion. Contributions to the error vector e in (7) could be weighted in accordance with the level of certainty to account for this. Additionally, different mud dissipation mechanisms could also be incorporated as well as address of field data. These will serve as a basis for future work. Acknowledgments. This work was supported by the Texas Engineering Experiment Station (TEES). The authors thank the anonymous reviewers for their comments, which increased the clarity of the paper. REFERENCES Bell, P. S., 1999: Shallow water bathymetry derived from an analysis of X-band marine radar images of waves. Coastal Eng., 37, 513–527. Bertino, L., G. Evensen, and H. Wackernagel, 2003: Sequential data assimilation techniques in oceanography. Int. Stat. Rev., 71, 223–241. Booij, N., R. C. Ris, and L. H. Holthuijsen, 1999: A third-generation wave model for coastal regions 1. Model description and validation. J. Geophys. Res., 104 (C4), 7649–7666. Bouws, E., H. Gu¨nther, W. Rosenthal, and C. L. Vincent, 1985: Similarity of the wind wave spectrum in finite depth water. 1. Spectral form. J. Geophys. Res., 90, 975–986. Chou, H.-T., M. A. Foda, and J. R. Hunt, 1993: Rheological response of cohesive sediments to oscillatory forcing. Nearshore and Estuarine Cohesive Sediment Transport, A. J. Mehta, Ed., Coastal Estuarine Studies, Vol. 42, Amer. Geophys. Union, 126–148. Dalrymple, R. A., and P.L.-F. Liu, 1978: Waves over soft muds: A two layer fluid model. J. Phys. Oceanogr., 8, 1121–1131. de Wit, P. J., 1995: Liquefaction of cohesive sediment by waves. Ph.D. dissertation, Delft University of Technology, Department of Civil Engineering, 211 pp. Eldeberky, Y., and J. A. Battjes, 1995: Parameterization of triad interactions in wave energy models. Proc. Coastal Dynamics Conf. ’95, Gdansk, Poland, ASCE, 140–148. Elgar, S., and B. Raubenheimer, 2008: Wave dissipation by muddy seafloors. Geophys. Res. Lett., 35, L07611, doi:10.1029/ 2008GL033245. Feddersen, F., R. T. Guza, and S. Elgar, 2004: Inverse modeling of the one-dimensional setup and alongshore current in the nearshore. J. Phys. Oceanogr., 34, 920–933.
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