1. Introduction
Measurement of fluxes of water mass, heat, chemicals, and biological organisms is fundamental to the understanding of ocean circulation, marine ecology, and global climate (Dickson et al. 2007; Münchow et al. 2006; Cuny et al. 2005; Ganachaud and Wunsch 2003; Tanhua and Olsson 2006; Holmes et al. 2000; Spurrier and Kjerfve 1988; Bryden et al. 2005). For example, measurement of mass and heat transports is key to the studies of the California Current System (CCS) and coastal upwelling (Lentz 1987; Rudnick and Davis 1988). On the basin scale, high freshwater inputs to the northern Atlantic are believed to be the primary triggers for some climate changes in history (Broecker 2003). These examples underscore the importance of being able to both measure transport and quantify the error associated with that measurement. However, temporal and spatial variabilities of the ocean introduce a challenge, because fixed platforms sample sparsely in space, while mobile platforms might not be fast enough to synoptically capture ocean fields.
Traditional methods of flux studies employ moorings or research vessels that make measurements along sections (Hansen and Østerhus 2000; Cokelet et al. 2008). Measurements by an array of moorings are synoptic (i.e., made simultaneously) and of long durations (on the order of years). Their spatial resolution, however, is limited by the number of deployed moorings under the curb of equipment and logistic costs. As a consequence, spatial aliasing will lead to an error in flux estimation (Petrie and Buckley 1996). A ship is a moving platform, so it can make denser measurements on a transect, thus improving the spatial resolution. Still, the density of occupied stations is limited by the ship’s high operational cost (Rhein et al. 2002).
Autonomous underwater vehicles (AUVs; encompassing propeller-driven vehicles and buoyancy-driven gliders) are cost-effective mobile platforms. Their relatively low cost (compared with a ship) means that they can be deployed in numbers. AUVs provide high-resolution measurements in both time and space, making them useful for flux estimation. They have been used to measure ocean fluxes in a number of field programs. To investigate freshwater flux from the Arctic to the Atlantic, two Seagliders conducted hydrographic surveys in Davis Strait (Lee et al. 2006) to derive the geostrophic current velocity shear. In some other experiments, current velocity measurements from AUVs were directly used for deriving vertical heat flux (Morison and McPhee 1998; Yoerger et al. 2001). In the Adaptive Sampling and Prediction (ASAP) Experiment (Princeton University 2008; Godin et al. 2006), which was a major component of the Monterey Bay 2006 Experiment, researchers dispatched over a dozen gliders and one propeller-driven AUV to survey a control volume around an upwelling center to study the heat transfer. The vehicles were equipped with acoustic Doppler current profilers (ADCPs) for current velocity measurement. Note that during a survey by a moving platform, the ocean field varies. Thus, nonsynopticity in the measurements will cause an error in flux estimation (as will be elaborated upon in section 2b): the faster the field varies, the larger the error.
This paper addresses the need for quantitative design of sampling strategy for flux estimation. Flux equals the spatial integration of some flux variable [see Eq. (2)]. Because integration is essentially low-pass filtering, some high-frequency variations in the flux variable are filtered out. Hence, the sampling density requirement for flux estimation is less demanding than that for mapping the flux variable itself (i.e., using measurements to estimate the flux variable at unmeasured locations). Consequently, the same number of moorings can do a better job for estimating the flux than for mapping the flux variable, as verified by simulations in Y. Zhang and J. G. Bellingham (2006, unpublished manuscript). In the literature, some considerations for flux experiment designs are based on correlation scales of the flux variable, but we have not seen a rigorous error analysis on the flux. In Cuny et al. (2005), the authors used an array of moorings to estimate the volume, freshwater, and heat fluxes across Davis Strait. They calculated the correlations of temperature, salinity, and current velocity between adjacent moorings. Based on the low correlation values, they determined that the moorings were too sparse to capture all the spatial variability of the flux variable. However, for estimating the flux (i.e., the spatial integration of the flux variable), how densely should we deploy an array of moorings? A further question is, given some temporal and spatial scales of the flux variable, what type of platforms (fixed versus mobile) should we choose for a better flux estimation performance? To answer these questions, we need to find a way to quantitatively assess the flux estimation error.
Empirical methods in the literature take the flux estimate’s temporal variability within some time window as the flux estimation error. In Cuny et al. (2005), temperature, salinity, and current velocity were measured in hourly intervals by moorings. The raw measurements were averaged in each month to provide monthly flux estimates, and the flux’s standard deviation within each month was considered the estimate’s uncertainty. In Münchow et al. (2006), freshwater flux was estimated by ship-based measurements. The flux’s standard deviation in a 36-h time window was taken as the estimate’s uncertainty. In Hayes and Morison (2002), an AUV was used to measure vertical heat flux. Instantaneous flux was averaged over a 100-m-distance window, and the standard deviation within that window was regarded as the estimate’s uncertainty.
If the true flux is a constant within the time window, its variance is zero (i.e., Vartrue = 0). Thus, we have Vartotal = Varest_err. In this case, we can take the observed Vartotal as Varest_err. However, if the true flux varies significantly within the time window (i.e., Vartrue is large), we have Vartotal = Vartrue + Varest_err > Varest_err. Hence, it is no longer appropriate to take Vartotal as Varest_err; in other words, Vartotal would be an overestimate of Varest_err. This is often the case in coastal waters, where temporal and spatial variabilities are high.
In this paper, we present a method of analyzing the flux estimation error and designing the sampling strategy for fixed or mobile platforms (Zhang et al. 2006). In section 2, we derive the mean-square error of the flux estimate, based on the flux variable’s temporal–spatial autocovariance function. We start from flux estimation on a line and then extend the analysis to a section. Utilizing the error analysis, one can evaluate the flux estimation performance by fixed or mobile platforms and choose the best sampling strategy, as discussed in section 3.
In section 4, we verify the error analysis using synthesized ocean fields. We estimate the flux by moorings and AUVs at different numbers and speeds. The estimation errors agree very well with the analytical predictions. In section 5, as an application, we run simulated AUV/mooring surveys in Monterey Bay, California, using the Regional Ocean Modeling System (ROMS) reanalysis output (Chao et al. 2009). Flux estimation performances by moorings and AUVs at different numbers and speeds agree with the analytical predictions. In multi-AUV deployment, the vehicles may be “out of phase” between each other. In section 6, we consider the impact of such phase differences on flux estimation. We summarize findings in section 7.
2. Error analysis and sampling strategy design for lateral flux estimation
a. Estimating lateral flux using fixed platforms
Note that measurements made by an array of moorings are synoptic (i.e., made simultaneously). Hence, in Eqs. (5) and (6), we leave out the time variable t for conciseness.
- (iii) The third term in Eq. (11) vanishes because
Using the above settings and assuming, for simplicity, that μ(x) is a linear function of x (so that flux estimation by moorings is unbiased), ηmoorings as a function of the number of moorings is shown in Fig. 2. The error is large when Xs is large relative to λ0 (i.e., the spatial aliasing is severe). The error drops with the decrease of Xs (i.e., increase of the number of moorings). The relative error for two moorings is greater than 1, when Xs = 60 km ≫ 12.8 km. Using three moorings (Xs = 60 km/(3 − 1) = 30 km), the relative error drops to about 0.4. The error levels off beyond four moorings.
b. Estimating lateral flux using mobile platforms
1) Estimating flux using one AUV
If we shorten the survey line by a factor of N and reduce the AUV speed from υ to υ/N, the survey duration remains the same: (L/N)/(υ/N) = L/υ. Will the relative estimation error η1_AUV remain unchanged? No, it actually decreases with N. An explanation is as follows. The flux estimation error stems from the temporal and spatial variations of the flux variable. For different values of N, the survey duration remains the same, so the temporal variation is the same. The spatial variation, however, is smaller for a larger N because the survey line is shortened to L/N. Therefore, the spatial variation’s contribution to the flux estimation error is smaller for a larger N. Consequently, η1_AUV decreases with N. Consider the case where L = 60 km; C(τ, r) = σ2e−(|τ|/τ0)e−(r 2/λ02), where τ0 = 16 h and λ0 = 12.8 km, and υ = 1 m s−1. The decrement of η1_AUV with N is shown in Fig. 4.
The above observation indicates that if we divide a survey line into a number of segments and have each segment surveyed by a slower AUV, we may obtain a more accurate flux estimate. This will be shown in Fig. 6.
2) Estimating flux using multiple AUVs
Suppose we divide the survey line into two equal segments, each of length L/2. On each segment, we run one AUV at speed υ to estimate the flux. We take the sum of the two segments’ fluxes as an estimate for the whole line. As illustrated in Fig. 5, the two AUVs run in synchrony: at time t − (L/4υ), they start from x = −L/2 and x = 0, respectively; at time t, they reach x = −L/4 and x = L/4; and at time t + (L/4υ), they finish at x = 0 and x = L/2. Thus, the total survey duration is L/2υ for both vehicles.
For L = 60 km, C(τ, r) = σ2e−(|τ|/τ0)e−(r 2/λ02) (τ0 = 16 h and λ0 = 12.8 km), we compute ηN_AUVs for different combinations of vehicle number and single-AUV speed, as shown in Fig. 6, where we have defined cumulative speed υcum ≜ single-vehicle speed υ × number of vehicles N. For moorings, we show two sets of results. The first set (shown by the inner-circle sizes, which correspond to the numbers to the left) is for μ(x) being a linear function of x, in which case the estimates by moorings are unbiased (the same as in Fig. 2). The second set (shown by the outer-circle sizes, which correspond to the numbers to the right) is for μ(x) in the ROMS dataset (to be presented in section 5), in which case the estimates by moorings are biased. On the flux estimation performance by AUVs, we make the following observations:
(i) For a given number of AUVs, the flux estimation error drops with the increase of the single-vehicle speed; for a given single-vehicle speed, the error decreases when running more AUVs (note that with more AUVs, each segment is shorter). Having faster AUVs and/or shorter segments means that each AUV can complete surveying its segment in a shorter time, thus reducing the flux estimation error caused by temporal smearing.
(ii) In Fig. 6, we draw four contours, each corresponding to some cumulative speed υcum; υcum determines how fast one or multiple AUVs can complete a line survey, τone_survey = L/υcum, where L is the total length of the line. For example, there are two points on the 0.5 m s−1 cumulative-speed contour: one for one AUV at speed 0.5 m s−1 and the other for two AUVs each at speed 0.25 m s−1. For a given cumulative speed, the estimation error is smaller when running more AUVs; that is, it is beneficial to run a larger number of slower vehicles as opposed to a smaller number of faster vehicles. As discussed at the end of section 2b(1) and illustrated by Fig. 4, when we divide a survey line into N segments and have each segment covered by an AUV at speed υ/N, the flux estimate for each segment is more accurate than that for the entire line estimated by one AUV at speed υ. Consequently, the sum of the flux estimates on all segments gives a more accurate estimate for the entire-line flux. On the other hand, with the increase of the cumulative speed, the benefit margin of running a larger number of slower AUVs shrinks.
c. Extending error analysis from a line to a section
i. Estimating the section flux using moorings:
Suppose N moorings, each equipped with a vertical string of instruments with a vertical spacing of Δd, provide equispaced (spacing = Xs) measurements on each of the K lines, as shown by the dots in the top panel in Fig. 7. By Eq. (6), the flux estimate on the ith line isThe total flux on the section is given by the sum of the fluxes on the K lines,By Eqs. (37) and (40), the true and estimated section fluxes are both formulated as line fluxes, so that we can directly utilize the analysis developed in section 2a. The bias and the mean-square estimation error can be evaluated by Eqs. (9) and (15), where μ
P (x) takes the place of μ(x) and CP (r) takes the place of Cspatial(r).ii. Estimating the section flux using AUVs.
When we run an AUV on a sawtooth (i.e., “yo-yo”) track on the section, the vehicle actually samples the K horizontal lines in turn, as shown by the dots in the lower panel in Fig. 7. On each line, the maximum spacing between samples is denoted by Δxmax. As long as the horizontal sampling on each line is dense enough (i.e., Δxmax ≪ λ0, where λ0 is the flux variable’s spatial correlation scale in the horizontal direction), the sampling can be regarded as if each line were surveyed by one dedicated vehicle (i.e., as if K AUVs ran in parallel on the K lines).
Using Eq. (20), the flux estimate on the ith line by one AUV at a horizontal speed υ iswhere the integral is approximated by the summation of the interpolated AUV samples on the ith line. As discussed above, as long as the horizontal sampling is dense enough, this approximation error is small. Note that υ is the horizontal component of the AUV’s cruise speed υcruise: υ = υcruise cos(θpitch), where θpitch is the vehicle’s pitch angle.The total flux on the section is given by the sum of the fluxes on the K lines:Using Eq. (36), Eq. (42) is rewritten asBy Eqs. (37) and (43), the true and estimated section fluxes are both formulated as line fluxes, so that we can directly utilize the analysis developed in section 2b, and Q̂section_by_1_AUV(t) is an unbiased estimate of E[Qsection(t)]. The mean-square estimation error can be evaluated by Eq. (23), where C
P (τ, r) takes the place of C(τ, r).We can horizontally divide the section into N subsections and run one yo-yoing AUV at horizontal speed υ to estimate the lateral flux across each subsection. Then, we sum the N subsections’ fluxes as an estimate for the whole section. The flux estimate is unbiased. The mean-square estimation error can be evaluated by Eq. (33).
If we still assume L = 60 km, C
3. Performance comparison and platform choice
The mechanism of flux estimation by moorings is using measurements at discrete points to approximate the corresponding segments (each segment’s length equals mooring spacing Xs) and then taking the summation, as expressed by Eq. (6). Hence the flux estimation error is caused by spatial aliasing, the severity of which is reflected by the ratio (Xs/λ0). The mechanism of flux estimation by AUVs is using measurements over some duration to approximate a snapshot and then taking the spatial integration, as expressed by Eqs. (20) and (32). Hence, the flux estimation error is caused by temporal smearing, the severity of which is reflected by the ratio [L/(Nυ)]/τ0. The two different mechanisms are illustrated in Fig. 8.
Flux estimates by AUVs are statistically unbiased regardless of μ
Still using the settings L = 60 km, C
When the number of platforms N ≤ 3, AUVs perform much better than moorings. For moorings, when N = 2 (or 3), Xs = 60 (or 30) km is much larger than λ0 = 12.8 km. Such severe spatial aliasing leads to a big MSE in flux estimation. In contrast, AUVs sweep the survey section with negligible spatial gaps (presuming Δxmax ≪ λ0, as discussed in section 2c). Although the temporal smearing leads to a flux estimation error, the error is smaller than that caused by severe spatial aliasing. When the number of platforms ≥4, moorings start to perform better than AUVs. With denser mooring placement, spatial aliasing is quickly overcome. However, for AUVs, the impact of temporal smearing fades slowly.
When we desire to achieve some specified performance by the fewest platforms, which type of platforms should we choose, moorings or AUVs? This sampling strategy question can be answered by the flux estimation error analysis results. When the spatial scale is small and the temporal scale is large, AUVs are advantageous because moorings suffer from spatial aliasing. Conversely, when the spatial scale is large and the temporal scale is small, moorings are advantageous because AUVs’ performance is curbed by temporal smearing.
4. Verification of error analysis using a synthesized random process
For comparison with the analytical results in section 2, we still use settings L = 60 km, C(τ, r) = σ2e−(|τ|/τ0)e−(r 2/λ02) (τ0 = 16 h and λ0 = 12.8 km) for generating Psynthesized(t, x). To closely approximate the desired temporal–spatial PSD, we set Mη and Nv to a large number of 4000. We estimate the flux using moorings and AUVs at different numbers and speeds. For each deployment scheme, we generate 1000 realizations of the random process so as to run 1000 surveys for evaluating the mean-square errors of flux estimation. Because the flux variable Psynthesized(t, x) has a zero mean, the flux estimates by moorings are unbiased. The relative errors
5. Application to lateral heat flux estimation in Monterey Bay
The science goal of the Monterey Bay ASAP Experiment (Princeton University 2008; Godin et al. 2006) in 2006 was to study heat transfer in a control volume around an upwelling center off Point Año Nuevo. The control volume was a box about 40 km long and 20 km wide (shown in Fig. 11), with a variable depth down to several hundred meters. Two classes of AUVs were deployed to measure temperature, conductivity, and current velocity on the sides and in the interior of the box: a propeller-driven Dorado AUV (Kirkwood 2007) with a speed of about 1.5 m s−1 and buoyancy-driven Spray and Slocum gliders (Rudnick et al. 2004) with a horizontal speed of about 0.25 m s−1.
For the lateral heat flux estimation, we conduct simulated AUV/mooring surveys using the ROMS output (Chao et al. 2009) for Monterey Bay in August 2006. The model has assimilated in situ and remote sensing observations with enhanced quality control, thus producing the “reanalysis fields.” The original instantaneous output from ROMS has a time interval of 1 h, a spatial resolution of 0.02° in latitude and longitude, and a nonuniform depth grid from the surface to 2000 m. In the upper 100 m (which is used for heat flux estimation in this paper), the depth resolution is 5 m in the 0–20-m depth range, 10 m in the 20–60-m depth range, 15 m in the 60–75-m depth range, and 25 m in the 75–100-m depth range.
In Chao et al. (2009), it was shown that the reanalysis outputs of temperature and salinity in an earlier ROMS dataset (for Monterey Bay in August 2003) agreed well with the gliders’ measurements: the root-mean-square (RMS) difference of temperature was lower than 0.5°C at most depths and the RMS difference of salinity was lower than 0.1 psu. For current velocity, the temporal evolution of the model output agreed with the mooring measurements although the absolute values had discrepancies. The ROMS model data quality in the Monterey Bay 2006 dataset is similar to that in the 2003 dataset. Note that our objective in this paper is to use the model output for demonstrating the flux estimation error analysis, instead of validating the model.
The hourly ROMS fields are obtained from daily forecasts initialized at 0300 UTC each day. The first 24 h from each daily forecast are concatenated to form a contiguous time series for the month. The initial conditions for each daily forecast are the output of a four-stage (each 6 h long) data assimilation procedure run over the 24 h prior to the initialization time of that forecast. Because the data assimilation procedure is successively but separately conducted on each daily forecast, there are some discontinuities (i.e., “jumps”) in temperature and current velocity values at the junctions between 24-h blocks. Although these discontinuities are undesirable, it should be noted that, because data assimilation is used, the resulting ocean fields are more realistic than those from a month-long “free run” that is initialized only once at the beginning of the month. However, such temporal discontinuities would adversely affect heat flux estimation in a simulated AUV survey when the survey duration crosses the time boundary between successive 24-h blocks. To mitigate the effect, we low-pass filter the raw model output by 5-point moving–averaging. For instance, for hour M, we use the average at hours M − 2, M − 1, M, M + 1, and M + 2 to replace the raw model output at hour M. The width of the moving–averaging window is thus 5 − 1 = 4 h. If the window is too wide, the semidiurnal tidal component will be filtered out too much; if the window is too narrow, the discontinuities cannot be sufficiently mitigated. The 5-point moving–averaging turns out to be a good balance. In the resultant temporal autocovariance function (as shown in the top panel in Fig. 13), the semidiurnal tidal component is well preserved.
To estimate the lateral heat flux, we run AUVs on a yo-yo track from the sea surface to the 100-m depth on the southwest side (40 km) of the control volume, with an extension of 20 km, as marked by the solid line in Fig. 11. The length of the extension equals that of the southeast side of the control volume. Thus, the total length of the survey section is L = 40 km + 20 km = 60 km, which equals half of the perimeter of the control volume. The yo-yo track with a pitch angle of 20° is depicted in the top panel of Fig. 12. Each yo-yo cycle spans a horizontal distance of about 549 m [100 m × cot(20°) × 2]. We set the vertical distance between the AUV sampling points on the yo-yo track to 10 m [so that, correspondingly, the horizontal distance between the sampling points is 10 m × cot(20°) = 27 m]. We do linear interpolation in time and space to generate AUV measurements at the sampling points on the yo-yo track.
Corresponding to Fig. 7, we have Δd = 10 m and Δxmax = 549 m. Because Δd ≪ λvertical and Δxmax ≪ λ0, the conditions for applying the line-to-section extension of the error analysis are met well. Using the temporal and spatial scales of
At a cumulative horizontal speed vcum, the time it takes one or multiple AUVs to complete one survey is τone_survey = L/υcum, where L = 60 km is the horizontal distance. For example, it takes about 33 h for one AUV of horizontal speed 0.5 m s−1 to complete one survey. Over this duration, the ocean field has varied. We call the instant when the AUV reaches the midpoint of the survey, the “midpoint time.” The AUV-measured heat flux over the survey duration is taken as the estimate of the true synoptic heat flux at the midpoint time. In Fig. 14, we compare the nonsynoptic surface current velocity measured by one 0.5 m s−1 AUV in one survey and the synoptic surface current velocity at the midpoint time of that survey. The discrepancy is due to the temporal variation of the field. We call the difference Vsynoptic − Vnonsynoptic the nonsynopticity error. The higher the AUV speed, the smaller the discrepancy. Figure 15 shows the nonsynopticity error in AUV-measured normal current velocity and temperature on the whole section (normal velocity is defined as pointing southwest). The nonsynopticity error will cause an error in flux estimation.
To compute the lateral heat flux across the section in a yo-yo survey, at each depth we linearly interpolate T and V⊥ samples on the yo-yo track to a uniform grid at that depth. The summation of [TV⊥] at all depths gives the heat flux estimate for the midpoint time of that survey. Assuming nonstop AUV runs, the total number of surveys in August 2006 is ⌊one month/τone_survey⌋, where ⌊·⌋ takes the integer part.
In the top panel of Fig. 16, we plot heat flux densities (flux density = flux/area) estimated by one 0.5 m s−1 AUV and two 0.25 m s−1 AUVs. The quoted speed is the horizontal component of the vehicle’s cruise speed [as noted below Eq. (41)]. For θpitch = 20°, the horizontal speed υ is slightly lower than the cruise speed υcruise: υ = υcruise cos(θpitch) = 0.94υcruise. For conciseness, speed quoted henceforth will be the horizontal component. The two deployment schemes have the same cumulative speed 0.5 m s−1. For each survey, the flux estimate is plotted at the midpoint time of that survey, as marked by the circles. By linear interpolation, we fill in the estimated values between the midpoint times. The estimates are compared with the true heat flux density.
(i) Bias:
As analyzed in section 2a, flux estimates by moorings using the trapezoidal approximation are generally biased, and the upper bound of the bias decreases quadratically with the number of moorings. Plotted in Fig. 18 is biassection for two, three, or four moorings. The decrement of the bias is approximately bounded by the 1/(N − 1)2 curve [Eq. (10)].
As analyzed in section 2b, flux estimates by AUVs are statistically unbiased. However, at a low cumulative speed 0.5 m s−1, we see a nonzero bias in the bottom panel of Fig. 16. An explanation is as follows: First, as noted at the beginning of this section, the ROMS output values have some discontinuities between 24-h blocks because of data assimilation that is separately conducted on each 24-h block. After the 5-point moving–averaging, the discontinuities are smoothed out to some extent. Still, when the AUV survey duration crosses the time boundary between 24-h blocks, an error is introduced in flux estimation, because the vehicle experiences an artifact jump in data values. This error, in turn, increases the bias. For example, the relatively large error in flux estimation by AUVs on 2 August (shown in Figs. 16 and 17) is caused by such a jump. Note that the flux estimates by moorings are not impacted by the discontinuities between 24-h blocks, because measurements at the series of moorings are made at the same time instant. Second, the unbiasedness derived in Eq. (22) is in a statistical sense (i.e., for a large number of samples). If the sample size is small, the sample mean will not be an accurate representation of the ensemble mean; consequently, the apparent bias will not be zero. This explains why the apparent bias drops with the increase of the cumulative speed υcum. When υcum is low, τone_survey is large. Then, the total number of surveys in one month ⌊one month/τone_survey⌋ is small. Thus, from a statistical point of view, if we regard the flux estimate in one survey as one sample, the sample size is small for a low υcum. Because of the small sample size, the apparent bias (i.e., the time average)
(ii) MSE:
As expressed by Eq. (15), the MSE of a flux estimate by moorings is composed of two parts: the square of the bias and the variance of the error. For different number/speed settings of moorings and AUVs, the relative error
At the low cumulative speed 0.5 m s−1, the MSE by one or two AUVs is higher than that by three moorings, but at a higher cumulative speed 2 m s−1, the MSE by one or two AUVs becomes lower than that by three moorings. We also see that for the same cumulative speed, a larger number of slower vehicles achieve a lower MSE than a smaller number of faster vehicles, but the performance margin shrinks with the increase of the cumulative speed.
The relative errors in Fig. 20 agree reasonably well with the analytical predictions shown in Fig. 6. The discrepancies are mainly due to the difference between [TV⊥]’s actual temporal–spatial autocovariance function in the ROMS dataset and its simplified formulation in Eq. (45) used for the analytical predictions, as displayed in Fig. 13. The discrepancies are relatively large for two and three moorings. In particular, when using only two moorings placed at the two ends of the 60-km section to estimate the lateral flux across the section, whereas the spatial e-folding scale of [TV⊥] is only 12.8 km, the severe spatial aliasing leads to a big estimation error (relative error >1 in both results), and the discrepancy between the two results is also large.
For flux estimates by AUVs, the MSE comes mostly from the variance (i.e., noisiness) because the long-term bias is essentially zero. For flux estimates by moorings, however, the presence of a bias can make a significant contribution to the MSE, especially when the total number of moorings is small. The contrast is shown in Table 1. For a given cross section, if we are constrained by a very small number of platforms, fast AUVs will have an advantage over moorings for estimating unbiased flux. For instance, the flux estimates by one 2 m s−1 AUV and by three moorings have similar MSEs, but the bias of the former is smaller than that of the latter by a factor of 5. Note that, if there were no discontinuity between 24-h blocks, the flux estimation bias by AUVs would be even smaller.
6. Impact of phase differences between AUVs on flux estimation
In a multi-AUV deployment scheme, we divide a survey line into segments and have each segment covered by one vehicle. Discussions so far are based on the presumption that the AUVs are in phase as shown in Fig. 5: they start and finish their individual segments at the same time.
For L = 60 km, C(τ, r) = σ2e−(|τ|/τ0)e−(r 2/λ02) (τ0 = 16 h and λ0 = 12.8 km), we calculate flux estimation errors for two cumulative speeds, 0.5 and 2 m s−1. We set x0 to (L/2) × [0%, 10%, 20%, 30%]. The analytical results are shown in the left panel of Fig. 22. The variation range of η2_AUVs_out_of_phase is shown by the vertical bars. The estimation errors for one AUV (in the left panel) are taken from Fig. 6. It is seen that the phase difference induces an increase in the flux estimation error, but as long as x0 < (L/2 × 30%), two AUVs still provide a more accurate flux estimate than one AUV under the same cumulative speed.
To verify the error analysis, we run simulated surveys on a line using a synthesized random process, as in section 4. On a synthesized survey line, we run one AUV or two AUVs at the same cumulative speed. In the two-AUV scheme, we set the second vehicle’s start location x0 to L/2 × [0%, 10%, 20%, 30%]. Two cumulative speeds are tested: 0.5 and 2 m s−1. For each cumulative speed and x0 setting, 1000 surveys are run. To be consistent with Eq. (49) for the two-AUV scheme, in each run, the instant when AUV 1 reaches the midpoint of its segment is taken as the “nominal midpoint time” (when AUV 2 is at a distance of x0 away from the midpoint of its segment). The true flux is calculated at the nominal midpoint time. The relative errors 
To minimize phase differences between AUVs so as to minimize the induced increment of flux estimation error, synchronous coordination of the vehicles is required. The Glider Coordinated Control System (Paley et al. 2008; Leonard et al. 2007) provides an effective coordination mechanism that has been field tested in Buzzards Bay, Massachusetts, and in Monterey Bay.
7. Conclusions
For using fixed or mobile platforms to estimate lateral flux in the ocean, we have developed a method of analyzing the estimation error and designing the sampling strategy. When using moorings, the flux estimation error is caused by spatial aliasing; when using AUVs, the error is caused by temporal smearing. When the spatial scale is small and the temporal scale is large, AUVs perform better because moorings suffer from spatial aliasing. Conversely, when the spatial scale is large and the temporal scale is small, moorings are advantageous because AUVs’ performance is curbed by temporal smearing. We point out that the flux estimate by moorings using the trapezoidal rule generally carries a bias that drops quadratically with the number of moorings. We also find that a larger number of slower AUVs may achieve a more accurate flux estimate than a smaller number of faster AUVs under the same cumulative speed, but the performance margin shrinks with the increase of the cumulative speed. We run simulated surveys using a synthesized random process, and the flux estimation errors agree very well with the analytical predictions. Using a ROMS ocean model dataset, we estimate the lateral heat flux across a section in Monterey Bay by AUVs and moorings. The flux estimation performance agrees with the analytical prediction. Using the presented method, one can choose platform type and design the sampling strategy to optimize the flux estimation performance under resource constraints.
In the paper, we assume a constant horizontal overground speed for an AUV. This assumption is reasonable for a fast (relative to the current velocity) propeller-driven AUV because of its capability of dynamically adjusting thrust and attitude. However, it is difficult for a slow buoyancy-driven glider to maintain a constant overground speed in the current. It is thus desirable to extend the flux estimation analysis for a variable AUV speed.
To sustain the advantage of multiple AUVs for flux estimation, coordinated control is important. If the vehicles are out of phase, the flux estimation error will increase, which will diminish the advantage of running more AUVs. For the two-AUV scheme, we have derived the formula for evaluating the flux estimation error when the vehicles have different phases while maintaining the same speed. It is shown that as long as the phase difference is controlled under some level, two vehicles still provide a more accurate flux estimate than one vehicle under the same cumulative speed. Multi-AUV coordination is more difficult in strong and spatially varying current, which induces speed discrepancies between vehicles, in addition to phase differences. The impact of speed discrepancies on flux estimation needs further investigation.
Acknowledgments
This work was supported by the Office of Naval Research (ONR) under Grant N00014-02-1-0856 and by the David and Lucile Packard Foundation. The ROMS ocean modeling research described in this publication was carried out by the Jet Propulsion Laboratory (JPL), California Institute of Technology, under a contract with the National Aeronautics and Space Administration (NASA). The authors are thankful to Russ Davis for his insightful comments on flux calculations and sampling considerations. The authors are thankful to John Farrara and Zhijin Li for their assistance in providing the ROMS reanalysis output and particularly to John Farrara for the remarks on the dataset. The authors appreciate the helpful comments from Thomas Curtin, Steven Ramp, Naomi Leonard, Michael Godin, Sharan Majumdar, David Fratantoni, and other ASAP team members. The authors are thankful to the anonymous reviewers for their very valuable comments and suggestions for improving the paper.
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APPENDIX
Generation of Temporal and Spatial Frequency Components for Synthesizing a Random Process
(i) Generate Mη random numbers ui (i = 1, … , Mη) that are uniformly distributed on [0, 1].
- (ii) Find the corresponding temporal frequency ηi that satisfies
In the following two steps, νj is generated:
(i) Generate Nν random numbers qj (j = 1, … , Nν) that are uniformly distributed on [0, 1].
- (ii) Find the corresponding spatial frequency νj that satisfies
Estimating flux by measurements from an array of moorings.
Citation: Journal of Atmospheric and Oceanic Technology 27, 3; 10.1175/2009JTECHO700.1
Relative error ηmoorings of flux estimation by moorings.
Citation: Journal of Atmospheric and Oceanic Technology 27, 3; 10.1175/2009JTECHO700.1
Estimating flux on a line by one AUV.
Citation: Journal of Atmospheric and Oceanic Technology 27, 3; 10.1175/2009JTECHO700.1
If we shorten the survey line by a factor of N and reduce the AUV speed also by a factor of N, the flux estimation error decreases with N.
Citation: Journal of Atmospheric and Oceanic Technology 27, 3; 10.1175/2009JTECHO700.1
Estimating flux on a line by two AUVs.
Citation: Journal of Atmospheric and Oceanic Technology 27, 3; 10.1175/2009JTECHO700.1
Relative error (by analysis) of flux estimation at different number/speed settings of moorings and AUVs. For moorings, the numbers to the left (corresponding to the inner-circle sizes) are for μ(x) being a linear function of x (so that the estimates are unbiased) and the numbers to the right (corresponding to the outer-circle sizes) are for μ(x) in the ROMS dataset.
Citation: Journal of Atmospheric and Oceanic Technology 27, 3; 10.1175/2009JTECHO700.1
Flux estimation on a section using moorings or AUVs.
Citation: Journal of Atmospheric and Oceanic Technology 27, 3; 10.1175/2009JTECHO700.1
Mechanisms of flux estimation by moorings vs AUVs.
Citation: Journal of Atmospheric and Oceanic Technology 27, 3; 10.1175/2009JTECHO700.1
Comparison of flux estimation performance by moorings and AUVs.
Citation: Journal of Atmospheric and Oceanic Technology 27, 3; 10.1175/2009JTECHO700.1
Relative error of flux estimation at different number/speed settings of moorings and AUVs using a synthesized random process.
Citation: Journal of Atmospheric and Oceanic Technology 27, 3; 10.1175/2009JTECHO700.1
In the paper, the lateral heat flux out of the southwest side (solid line, with an extension) of the heat budget control volume in the Monterey Bay ASAP Experiment is estimated by moorings or yo-yoing AUVs. The other three sides are marked by the dashed lines.
Citation: Journal of Atmospheric and Oceanic Technology 27, 3; 10.1175/2009JTECHO700.1
(top) AUV yo-yo tracks from surface to 100-m depth on a 60-km section and (bottom) μ
Citation: Journal of Atmospheric and Oceanic Technology 27, 3; 10.1175/2009JTECHO700.1
Temporal and horizontal spatial autocovariance functions of the depth-averaged [TV⊥].
Citation: Journal of Atmospheric and Oceanic Technology 27, 3; 10.1175/2009JTECHO700.1
Comparison of the nonsynoptic surface current velocity measured by one 0.5 m s−1 AUV in one survey and the synoptic surface current velocity at the midpoint time (i.e., the instant when the AUV reaches the midpoint of the survey).
Citation: Journal of Atmospheric and Oceanic Technology 27, 3; 10.1175/2009JTECHO700.1
Nonsynopticity error in normal current velocity and temperature measured by one 0.5 m s−1 AUV in one survey.
Citation: Journal of Atmospheric and Oceanic Technology 27, 3; 10.1175/2009JTECHO700.1
The true heat flux density compared with the estimates by one or two AUVs with cumulative speed 0.5 m s−1 and by three moorings.
Citation: Journal of Atmospheric and Oceanic Technology 27, 3; 10.1175/2009JTECHO700.1
The true heat flux density compared with the estimates by one or two AUVs with cumulative speed 2 m s−1 and by three moorings.
Citation: Journal of Atmospheric and Oceanic Technology 27, 3; 10.1175/2009JTECHO700.1
The flux estimation bias decreases quadratically with the number of moorings.
Citation: Journal of Atmospheric and Oceanic Technology 27, 3; 10.1175/2009JTECHO700.1
Bias of flux estimation at different number/speed settings of moorings and AUVs using the ROMS dataset.
Citation: Journal of Atmospheric and Oceanic Technology 27, 3; 10.1175/2009JTECHO700.1
Relative error of flux estimation at different number/speed settings of moorings and AUVs using the ROMS dataset.
Citation: Journal of Atmospheric and Oceanic Technology 27, 3; 10.1175/2009JTECHO700.1
Two AUVs out of phase.
Citation: Journal of Atmospheric and Oceanic Technology 27, 3; 10.1175/2009JTECHO700.1
Relative error of flux estimation by two out-of-phase AUVs as compared with that by one AUV at the same cumulative speed. In the two-AUV scheme, the variation range of the error level is caused by a phase difference of up to 30% of each AUV’s segment length: (left) by analysis and (right) using a synthesized random process.
Citation: Journal of Atmospheric and Oceanic Technology 27, 3; 10.1175/2009JTECHO700.1
Bias of the flux density estimate and its contribution to the MSE.